Transcript 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 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:
Snm (k|| , k ) 
'
||
2
'
||
M (k|| , k )
Transition rate from a state k|| in a
subband n into a state k||’ into a
subband n’

E  E '  

nn '
2
Matrix element for scattering between
state k|| in a subband n into a state k||’
into a subband n’
i  k||  k||' r
1
2
*
M  k|| , k    d re
dz m ( z ) H q ( R) n ( z )

A
'
||
Acoustic Phonon Scattering
• The matrix element for acoustic phonon
scattering in the bulk phonon approximation is:
Restricts to longitudinal
modes only
H ( R)   ac 
q ,
2MN q
q  eq  aq eiqR  aq e  iqR 
where q  (q|| , qz ), R  (r , z )
After integrating over the phonon coordinates, the matrix
element for scattering between states k|| and k||’, in
subbands n and m becomes:
M  k|| , k||'    ac
1

 Nq 
2 V q 
2
1/ 2
1

2
  k||  k||'  q   dz m* ( z )e  iqz z n ( z )
Inm(qz)
S nm  k|| , k  
'
||
2
2
q
 2ac
2 V q
1

 Nq 
2


1
2
 I nm (qz )
2
  k||  k||'  q   En  Ek||  Em  Ek '  q
||

• In the elastic and equipartition approximation, the total scattering
rate out of state k is of the form:
2D DOS function
2

k
T

1
1
B
ac

g 2 D  En  Em  Ek || 
2
 n (k|| ) m
 vs Wnm

1
  dz n2 ( z ) m2 ( z )
Wnm 
• Effective extent of the interaction in the z-direction
2   nm
1

• For infinite well:
, L is the well width
Wnm
L
Interface-Roughness Scattering
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.
z
G
W
S
x
D
y
oxide
n+
L
p-type SC
n+
dFy
dFx

dx
dy
Square-Law Theory
• The charge on the gate is completely balanced by QN(x), i.e:
Q N ( x )  Ctot VG  VT  V ( x ) 
VS= 0
EFS
x=0
V(x)
x
VD
EC
EFD= EFS - VD
• Total current density in the channel:
dn
dV
J n  qn n F ( x )  qDn
  qn n
dx
dx

negligible
• Integrating the current density, we obtain drain current ID:
Effective Mobility
W
yc ( x )
0
0
I D    dz
dV
W
dx

dV 

dy   qn( x, y ) n ( x, y )
dx 

yc ( x )

qn( x, y ) n ( x, y )dy
0
 QN ( x ) eff
dV
dV
 QN ( x) eff W
 CoxW eff VG  VT  V ( x ) 
dx
dx
Effective electron mobility, in
which interface-roughness is
taken into account.
Mobility Characterization due to
Interface Roughness
2.71 Å
3.84 Å
High-resolution transmission electron micrograph of the interface
between Si and SiO2 (Goodnick et al., Phys. Rev. B 32, p. 8171, 1985)
Interface Roughness
Phonon
17
N A  7  10 cm
500
0
3
2
Mobility [cm2/V-s]
1000
Mobility [cm /V-s]
Coulomb
1500
10
10
16
10
17
10
-3
Doping [cm ]
18
s
)
depl
N
300
200
100
15
-1
(aN + bN
400
experimental data
uniform
step-like (low-high)
retrograde (Gaussian)
12
10
13
10
-2
Inversion charge density N [cm ]
s
Interface-roughness
Bulk samples
-1/3
s
Si inversion layers
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 specular and 50% to be diffusive
• In k-space treatments of interface roughness, the perturbing potential is evaluated from:
V ( z ')  z '  z    r 
V ( z  )  V ( z ) 
V
 r  
z

H ( R)  eE ( z )   r 
 V ( z )  eE ( z )  r  
• The matrix element for scattering between states k|| and
k||’ is:
Fnm
i  k||  k||' r
1 2
M  k|| , k   e  ( z ) E ( z ) m ( z )dz   d r (r )e
A
2
i  k||  k||'  r1  r2 
'
2 2 1
2
2
M  k|| , k||   e Fnm 2  d r1  d r2 e
 (r1 ) (r2 )
nm
A
'
|| nm
*
n
Random variable that is characterized
by its autocovariance function which is
obtained by averaging over many
Samples – R(r)
M  k|| , k
'
||

2
nm
e F
2
2
nm
i  k||  k||'  r1  r2 
1
2
2
d r1  d r2e
(r1 )(r2 )
2 
A
When the random process is stationary, the autocorrelation
function depends only upon the difference of the variables
r1 and r2.
If R(r) is the autocorrelation function, then its power spectral
density is S(q||) and the transition rate is:
S (k|| , k ) 
'
||
2
2
e Fnm
2
S (q|| )
A
 ( E  E ')
For exponential autocorrelation function we have:
R(r )  2e r / L  S  q||  
2 L2
1  12 L2 q||2
Δ = Roughness correlation length
L = Ruth Mean Square (r.m.s.) of the roughness
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