Noise and Random Telegraph
Signals in Nanoelectronic
Devices
Zeynep Çelik-Butler
Electrical Engineering Department
University of Texas at Arlington
Arlington, Texas, 76019
[email protected]
Outline

Motivation: Problems Encountered as the Devices
Shrink, Frequencies Increase, and Voltages Reduce
Improved Model for 1/f Noise in MOSFETs
 Random Telegraph Signals in MOSFETs

 Complex
RTS
 Extraction of trapping parameters using RTS
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
2
UTA - Noise Characterization
Facilities

6' x 6' x 8' Shielded Room

3 Spectrum and Signal Analyzers,
f=1 mHz - 20 GHz.

3 Cryostats, T= 2 K to 350 K.

Various Lock-ins, Preamps, System
Controllers, Battery Operated Sources
etc.

Optical Equipment

Computer Software for Modeling
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
3
Problems Encountered as the Devices Shrink,
Frequencies Increase, and Voltages Reduce

Signal-to-noise ratio decreases.
Noise models based on large number of electrons
break down.


Quantum effects become dominant.
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
4
Signal to Noise Ratio Decreases
For
a MOSFET
Start from W=100mm, L=10mm, tox=800Å, NSS=4x1010 eV1cm-2.
Assume scaling factor is K.
Assume trap and surface state densities remain the same.
W  W K , L  L K , tox  tox
K
in noise level due to the K1/2 law chosen for tox.
Unpredictability of noise level for K>20.
NSS is actually a two dimensional Poisson variable.
Increase
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
5
Large Area Noise Models Break Down
Single electron, single trap effects.
NSS=4x1010 eV-1 cm-2, W=1mm, L=0.1mm.
EC
kT=26 meV
EF
EV
Si
1 trap per channel
SiO2
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
6
Large Area Noise Models Break Down
Break-down of large-area models for sub-micron channel length.

kTq 2 I d m eff 
NO  N *
1
2
2
S Id ( f ) 
A
ln

B
(
N

N
)

C
N

N

O
L
O
L
2
*
2
fL Cox 
NL  N
•
•
•
•
A=Nt (cm-3 eV-1)
B=ameffNt (cm-1 eV-1)
C=a2meff2Nt (cm eV-1)
A=B2/(4C)




Independent parameters:
a and Nt
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
7
Large Area Noise Models Break Down
-10
-11
10
Vgs-VT= -1 V
Vds= -50 mV
S
Vd
2
(1Hz) (V /Hz)
10
-12
10
-13
10
-14
10
0.1
1
Channel length (mm)
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
10
8
Large Area Noise Models Break Down
20
10
2 channel region model
uniform channel model
19
-1
N (cm eV )
10
18
-3
10
17
t
10
16
10
15
10
0.2
0.4
0.6
0.8
L (mm)
1
1.2
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
1.4
9
Large Area Noise Models Break Down
Modified 1/f noise model that takes into account threshold variation
along the channel.
• For simplicity assume two regions:
– DV, DL, VT2,, A2, B2, C2
– Vds-DV, L-DL, VT1, A1, B1, C1
–
–
–
–
–
DL<<L, VTVT1
A1 = A2, since Nt1 = Nt2
B12/C1 = B22/C2 = 4A
I1 = I2 = Id
meff1 = meff2,
Independent parameters:
Nt, a1, a2, VT2, and DV
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
10
Large Area Noise Models Break Down
Modified 1/f noise model that takes into account threshold variation along the
channel.
-17
10
-18
-19
10
Id
2
S (A /Hz)
10
L=0.32mm
-20
10
L=0.45mm
-21
10
L=1.0mm
-22
10
0
0.5
1
1.5
2
|V -V | (V)
gs
T
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
11
RTS in MOSFETs
Random Telegraph Signals: single electron switching.
t1
RTS (Arbitrary Units)
0.0002
0.00015
0.0001
5 10
-5 10
DId
-5
0
-5
-0.0001
5.2
5.3
5.4
5.5
Time (sec)
5.6
t0
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
5.7
12
RTS in MOSFETs
Time Scale  seconds
PSD
Random Telegraph Signals (RTS) with a Lorentzian on 1/f spectum.
Frequency
Time Scale
 milliseconds
S( f ) 
(f)
4DI 2
t0  t1  1 t0  1 t1 2  2f 2 
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
13
NMOS,W/L(mm)=5/0.23, VDS=175mV, VGS=0.60V
10-9
10-10
Sv (V2/Hz)
10-11
10-12
Sv = 6.11e-12 / ( 1 + f / 1260 ) 2
10-13
1 RTS process
10-14
10-15
100
101
102
103
104
105
Frequency (Hz)
DV (10-4 V)
2
0
-2
-4
-6
2 RTS levels
-8
-10
0
1
2
3
4
5
6
7
8
Time (ms)
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
14
PMOS T1 W/L=5/0.25 VDS=150mV VGS=0.9V
10-9
2
Sv = 1.4e-10 / ( 1 + f / 1.8)
10-10
Sv (V2/Hz)
10-11
2
Sv = 2.3e-14 / ( 1 + f / 23700)
10-12
10-13
2 RTS processes
10-14
10-15
10-1
100
DV (10-4 V)
2
101
102
103
104
105
Frequency (Hz)
1
0
3 RTS levels
-1
0
1
2
3
Time (ms)
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
15
NMOS ,W/L(mm)=5/0.23 ,VDS=150mV,VGS=0.775V
10-8
Sv = 1.11e-10 / ( 1 + f / 3.12 ) 2
10-9
Sv = 1.18e-11 / ( 1 + f / 48)
2
Sv = 1.08e-12 / ( 1 + f / 1345)
10-11
2
2
Sv (V /Hz)
10-10
4 RTS processes
10-12
Sv = 2.93e-14 / ( 1 + f / 36 780) 2
10-13
10-14
10-15
10-1
100
101
102
103
104
105
3
Frequency (Hz)
DV (10-4 V)
2
1
0
-1
5 RTS levels
-2
-3
0
1
2
3
4
5
6
7
8
Time (ms)
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
16
COMPLEX RTS
9.E-03
(a)
Voltage (V)
7.E-03
level 4
5.E-03
level 3
3.E-03
level 2
1.E-03
level 1
-1.E-03
2
2.02
2.04
2.06
2.08
2.1
Time (s)
Complex random telegraph signals due to multiple traps
SI ( f )
I
2
N traps
 
k 1
DI I 2k
t0  t1 k 1 t0  1 t1 2k  2f 2 
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
17
RTS in MOSFETs
RTS can be used to characterize trapping sites.
RTS modeling.

EC
S( f ) 
ECox-ET
EFp
xT
4DI 2
t0  t1  1 t0  1 t1 2  2f 2 
qVc
EFn
qs
EFg
gate
qVgs
S( f ) 
oxide
AF  I d2
2f 
2
 KF 2

silicon
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
18
RTS in MOSFETs
RTS can be used to characterize trapping sites.
•
•
•
•
Position of the trap along the channel, yT
Position of the trap in the oxide, xT
Trap energy, ECox - ET
Screened scattering coefficient, a
Vc  y Vds L
DI d DN Dm
1
1





am

Id
N
m
Weff Leff  N




t
xT
1 


ln c  
E

E

E

E

qV



q


q
V

V


 Cox
T
C
Fp
c
0
s
gs
FB
s 
te
kT 
Tox

Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]


19
tc / t e
Trapping Parameters Through RTS in
MOSFETs
10
1
10
0
yT/L=0.6
ECox-ET=3.04 eV
(b)
0.01
0.1
Drain Voltage (V)
xT=2.7 nm
10
2
10
1
10
0
1
t /t
c
e
10
-1
Forward
Reverse
(a)
10
-1
0.25
0.3
0.35
0.4
Gate Voltage (V)
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
0.45
0.5
20
Trapping Parameters Through RTS in MOSFETs
10
-1
10
-2
10
-3
10
-4
ds
DV / V
ds
D N/N
10
-1
0.04
Forward
Reverse
0.06 0.08 0.1
V -V
10
-2
10
-3
T
(V)
ds
DV / V
ds
gs
0.3
0.01
0.1
Drain Voltage (V)
1
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
21
Scattering Coefficient (V-s)
Trapping Parameters Through RTS in
MOSFETs
6 10
-14
5 10
-14
4 10
-14
3 10
-14
2 10
-14
1 10
-14
Forward
Reverse
0.04
0.06 0.08 0.1
V -V (V)
gs
0.2
T
a  K1  K 2 ln N
Noise and Reliability Laboratories, Zeynep Celik-Butler,
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22
Effects of Quantization
•Increase in effective energy
band-gap: change in te and tc
• Shift in carrier distribution:
change in Cox
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
23
3-D Treatment of RTS
1
1
tc 

cn  n3D  n (3D)  Vth  n3D
exp ( E F  ET ) / k BT 
te 
 n (3D)  Vth  n3D
cn  n 3D  Vth
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
24
2-D Treatment of RTS - tc and te
tc 
1
cn  n2 D  0z
p( z )
dz
z
1

 n (2 D)  Vth  n2 D  0z
p( z )
dz
z
1
exp ( EF  ET ) / k BT 
te 

en  (2 D) V  n  z p( z ) dz
n
th 2 D 0
z
cn  n 2D Vth
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
25
2-D Treatment of RTS
• From Stern - Howard wave-function:
b3 2
p( z ) 
z exp  bz 
2
1/ 3
 12qml 
11

b 2
  QB   Qinv 
32

   Si 0 
z  3/ b
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
26
2-D Treatment of RTS
• Calculate the inversion carrier concentration
assuming they are located primarily at E0:
1
1

N  n2 D  p( z )dz
 2k BTmt

z

exp  ECS  DE0  EF  / k BT  0 p( z )dz 
2
 

1/ 3
 2 

DE0  
 2m 
 l
 9q 


8


 Si 0 
2/3
1
2 Si 0qN B (VSB  2F )1/ 3
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
27
2-D Treatment of RTS - tc and te
te 
tc 
exp ( ECS  ET  DE0 ) / k BT 
n (2D) Vth  (2k BTmt b / 5 2)
exp ( ECS  EF  DE0 ) / k BT 
n (2D) Vth  (2k BTmt b / 5 2)
 tc 

1 
zT




ln    
E

E

E

E



q


q
V

V


T
CB
F
0
s
gs
FB
s 
 Cox
t 
k
T
T
B 
ox

 e


• To first order, the ratio is not affected by quantization.
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
28
RTS Measurements
•
•
•
•
•
•
MDD n-MOSFETs
Weff  Leff = 1.37  0.17 mm2
Tox = 4 nm
VT = 0.375 V for VSB = 0 V
strong inversion, linear region VDS = 100 mV
VSB = 0 - 0.4 V, VGS = 0.5 - 0.75 V
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
29
ECox-ET and zT from tc and te
 tc 

1 
zT


ln

Vgs  VFB   s 
ECox  ET   ECB  EF   0  q s  q
t 
k BT 
Tox

 e


3.5
VSB=0 V
3
t
ln( t / )
c e
ln(t
c/te)
2.5
2
1.5
1
0.5
0
0.45
0.5
0.55
0.6
0.65
V (V)
0.7
0.75
0.8
GS
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
30
ECox-ET and zT from tc and te
 tc 

1 
zT


ln

Vgs  VFB   s 
ECox  ET   ECB  EF   0  q s  q
t 
k BT 
Tox

 e

3.5

VSB=0.4 V
3
t
t/ )
c )e
ln(tln(c/t
e
2.5
2
1.5
1
0.5
0
0.45
0.5
0.55
0.65
0.6
0.7
0.75
0.8
V (V)
GS
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
31
ECox-ET and zT from tc and te
Tox =4 nm
VSB (V)
VT (V)
zT (Å)
ECox-ET (eV)
0
0.375
11.22
3.09
0.1
0.382
11.53
3.08
0.2
0.393
11.37
3.08
0.3
0.401
11.64
3.07
0.4
0.408
11.08
3.08
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
32
Dependence of te on VSB
te 
te(s)
te (s)
2 10
exp ( ECS  ET  DE0 ) / k BT 
n (2D) Vth  (2k BTmt b / 5 2)
-3
10
-3
8 10
-4
6 10
-4
VGS=0.75 V
VGS=0.55 V
4 10
-4
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
V (V)
SB
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
33
Dependence of tc on VSB
tc 
10
exp ( ECS  EF  DE0 ) / k BT 
n (2D) Vth  (2k BTmt b / 5 2)
-2
VGS=0.55 V
(s)
tc t(s)
c
VGS=0.65 V
VGS=0.75 V
10
-3
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
V (V)
SB
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
34
cn Extracted from tc and te

Vth  8k BT / mn
n  0 exp  DEB k BT 
-11
cn  n 2D Vth
n
3
capture coefficient c (cm /s)
10

* 1/ 2
10
-12
10
-13
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
V (V)
GS
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
35
2-D Treatment of RTS - Amplitude
 1 DN 1 m 
DI D
1
1

 

DNt  
  am
ID
Weff  Leff  N

 DN DNt m DNt 
m
1
1
1
 m n  mt
1
 m n  aNt
• Question: How does quantization affect number
and mobility fluctuations?
– Number fluctuation through N
– Mobility fluctuations through oxide charge scattering,
mt.
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
36
Extraction of Scattering Coefficient
• Mobility Fluctuations:
– Using Surya’s 2D surface mobility fluctuations model,
2
exp(

4
kz
sin

)
sin

/ 2
1
mt 
dNt E , z 
 dz  dE 0
2
c 2
8av E p
(sin   )
2k
mn*q 3
k  0.82 aSi 
   2 N  
2q 2 d v mn* 




c

1  exp  
2
* 
4 si 
  k BTd v mn  


Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
37
Calculation of Scattering Coefficient
• Considering a single trap:
Nt(E,z) = Nt(E-ET)  (z-zT)
2
sin

/ 2
1
mt 
exp( 4kzT sin )dNt
0
2
c 2
8av E p
(sin   )
2k
mn*q 3
a
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
38
RTS Amplitude
-4
DS
D V /V
DS
3 10
-4
2 10
V =0V
SB
V =0.1V
SB
V =0.2V
SB
V =0.3V
SB
V =0.4V
SB
-4
1 10
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
V (V)
GS
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
39
Extraction of Scattering Coefficient
2.4 10
-14
Tox =4 nm
2 10
-14
experimental data @V
=0V
SB
fitting with z =0.11nm
t
a (V-s)
fitting with z =0.12nm
t
1.6 10
-14
1.2 10
-14
8 10
-15
6 10
a = 2.91x10-13 - 9.93x10-15 ln(N)
-11
10
12
-2
N(cm )
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
10
40
13
Extraction of Scattering Coefficient
-14
Scattering Coefficient (V-s)
1 10
Tox =8.6 nm
-15
8 10
W L = 1.2  0.35 mm2
-15
6 10
-15
4 10
experimental results of Hung et al.
8
fitting of Pacelli et al.11
-15
2 10
theoretical calculation from 2-D
mobility fluctuation model
11
10
zT =0.25 nm
12
10
-2
N(cm )
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
13
10
41
Possible Reasons for Discrepancy
• Threshold non-uniformity along the channel is not
taken into account.
• Location of the trap along the channel
• Variation of the channel voltage from source to drain
is neglected.
• DN/DNt  1 is not valid, even in strong inversion, for
very thin oxides.
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
42
ACKNOWLEDGEMENTS
• This work has been supported by NSF, THECB-ATP,
SRC, TI, Legerity, Motorola and ST-Microelectronics.
Noise and Reliability Laboratories, Zeynep Celik-Butler,
[email protected]
43