Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.
-
Upload
abel-lewis -
Category
Documents
-
view
225 -
download
0
Transcript of Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.
![Page 1: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/1.jpg)
Scattering Rates for Confined Carriers
Dragica Vasileska
Professor
Arizona State University
![Page 2: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/2.jpg)
Outline
• General comments on matrix element calculation
• Examples of scattering rates calculation– Acoustic phonon scattering– Interface roughness scattering – dominant
scattering mechanism in nanoscale MOSFETs
![Page 3: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/3.jpg)
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
![Page 4: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/4.jpg)
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
![Page 5: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/5.jpg)
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)
![Page 6: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/6.jpg)
• 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
![Page 7: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/7.jpg)
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.
![Page 8: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/8.jpg)
)()( 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:
![Page 9: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/9.jpg)
• 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.
![Page 10: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/10.jpg)
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
![Page 11: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/11.jpg)
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
![Page 12: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/12.jpg)
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
![Page 13: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/13.jpg)
• 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.
![Page 14: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/14.jpg)
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
![Page 15: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/15.jpg)
![Page 16: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/16.jpg)
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
![Page 17: Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University.](https://reader035.fdocuments.in/reader035/viewer/2022062301/5697bf9d1a28abf838c93f9b/html5/thumbnails/17.jpg)
No exponential is forever…. But we can delay forever….
Gordon E. Moore