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On 1ƒ noise and random telegraph noise in very small electronic devices
Physica B 164 (1990) 331-334 North-Holland
ON l / f NOISE AND RANDOM TELEGRAPH NOISE IN VERY SMALL ELECTRONIC DEVICES T.G.M. KLEINPENNING Eindhoven University of Technology, Eindhoven, The Netherlands Received 16 April 1990
We demonstrate that random telegraph signal (RTS) noise can only be observed in small electronic devices where the number of free charge carriers is smaller than 1/{ a In(f= r,)). Here a is the Hooge 1If noise parameter, f= the bandwidth of the measuring system and r~ the sum of the mean capture and emission time of the charge carriers. The current idea that 1/f noise in larger devices is the result of a superposition of RTS noise due to individual trapping is doubtful.
1. Introduction In very small area silicon metal-oxidesemiconductor field-effect transistors (MOSFETs) the effect of a single electron trapped in the gate oxide on the device resistance has been observed by many authors [1-5]. This effect leads to discrete switching of the resistance, referred to as a random telegraph signal (RTS). No such switching was observed in larger devices, which showed only the ubiquitous 1/f resistance fluctuations. RTS fluctuations have a Lorentzian spectrum. Many authors [1-5] claim that the 1 if noise observed in larger devices is the result of a simple superposition of RTS noise from a large number of traps, with a broad distribution of capture and emission times of the trapped and detrapped electrons. On looking at the fluctuating signals observed in small devices by RaUs et al. [1], by Uren et al. [2, 3, 5] and by Restle [4], what one observes is just a normal fluctuating signal with bursts. Strasilla and Strutt [6] have described a method which allows the power spectra of the normal noise superimposed on the burst noise to be measured separately. They found the normal noise (1 If and white noise) to be lower than the Lorentzian burst noise. In the present paper we develop a simple model which shows that RTS noise appears only in dex4ces with a small wJmber of free charge carders, in general less than about 104. The
experimental results from the literature will be checked against our model.
2. Two-level random telegraph signals Let us consider an n-channel MOST of channel length L and operating in the Ohmic region. If we have only 1/f noise and Nyquist noise, the open-circuit voltage noise of the source-drain voltage is [7]
(1)
S v ( f ) = crV2/(fN) + 4 k T R .
Here a is the Hooge 1/f noise parameter, V the applied source-drain bias, N the total number of free electrons in the channel, f the frequency and R the channel resistance. For MOSTs, the value of a is in the range of 10 -6 to 10 -3 (refs. [7, 8]). The fluctuations in V versus time t are sketched in fig. 1. The rms value of the fluctuating voltage is [9] r f
]1;2
from
= [otVZ l n ( f m T ) / N
+ 4 k T R ( f m - l / T ) ] I'2
(2) Here fmax =fm is the bandwidth of the measuring system and fmi. = 1 / T is the reciprocal observation time T.
0921-4526/90/$03.50 ~ 1990- Elsevier Science Publishers B.V. (North-Holland)
T.G.M. Kleinpenning / I l l noise and RTS noise
332
~v(t)
N ~
(a)
Ze
Zc
(b) m
1;$
.
T
Fig. 1. Fluctuations in voltage AV(t) vs. time t of a signal with only 1If and Nyquist noise (a), of a signal with only RTS noise (b) and of a signal with all three types of noise (c).
Let us consider an n-MOST with only RTS noise as a result of single-electron switching between the channel and an interface state. The capture time of the electron is r c and the emission time is r e . The fluctuating voltage is sketched in fig. 1. The voltage step is given by a~, ~= I ( d V / d N ) A N I
=
I(V/N)ANI=
V/N
(3)
.
Here we assume that the electron mobility does not depend on N. If we have 1/f, Nyquist and RTS noise at the same time, we have the situation as sketched in fig. 1(c). Here we take the view that 1If noise and RTS noise have different physical sources. This point of view is based on two experimental findings. Firstly, the results described by Strasilla and Strutt [6] (sec also section 1). Secondly, the empirical result that the 1If noise is in all probability caused by mobility fluctuations [10]. From fig. 1 we can conclude that, according to eqs. (2) and (3) a voltage step with repetition time r~ can be observed on the condition that AV >i Orm~, thus
V { a V 2 ln(fml"s) + 4kTR(fm _ ~.~-1)} N>~ N
(5)
a In(fmrs) "
If the Nyquist noise dominates the 1/f noise, we find the stronger condition
ql~ V 2 1 N ~ 4kTL2f m < a I n ( f mr~) .
(6)
Here we have used fm ">r~t and R = L2/qt~N with /~ the electron mobility. Taking a = 10 -4, fm = 106 Hz and r, = 1 s, we find from eqs. (5) and (6) that N < 10 3. The voltage noise spectrum of the RTS fluctuations is Lorentzian and given by [5]
,I t
0
1
1/2
(4)
Here we have used T = % + ,,-~= ~-s. If the 1/f noise dominates the Nyquist noise, we find
S v ( f ) = (4VelN2)(rplrs) 1 + (2'rrfrp) 2 with % 1 = r e- 1 -I- T c- .! Around f = 1/(2xr%) ratio of RTS noise and 1 I f noise is
rp/rs SRTs/SI,I
= ~aN
"
(7) the
(8)
Taking % = % and a - 10 -4 the ratio is approximately equal to 1 0 a / N . Thus, for N < 10 3 the RTS noise exceeds the 1 I f noise at f--- 1/(2~r%).
3. Comparison with literature Rails et al. [1] have observed discrete resistance switching in submicron Si-MOSFETs. In small-area devices they found A R / R = &V/V= 1 / N "~ 2.5 x 10 -3, thus N ~ 400. In large-area devices they did not observe RTS fluctuations. The ratio A V R T S / U r m s , with Urms defined in eq. (2), was found to be aLout 4, which implies in(freTs) ~ Iv . Uren et al. [2] have investigated 1/f and RTS noise in n-MOSFETs at T = 3 0 0 K . For the largest devices, where N - L /2q l ~ R --- 10,6 the,V found 1/f noise with ~ ~ 4 x 10 -s. However, for the smaller devices with N ~ 103 they found noise spectra with both Lorentzian components due to RTS fluctuations and I / f noise with a 3 × 10 -~. The change in current AIDs(t ) versus
T.G.M. Kleinpenning / I / f noise and RTS noise
time t at various currents IDs and at constant voltage VDs has been presented for the smallest devices. The current steps, A/us = VDsA(1/R) = ql~ VDS/L 2 ,~ 20 p A , are found to be almost independent of the current /us- The current noise superimposed on the RTS was found to increase with the current, which is in accordance with the l l f noise formula as given by
102
333
1
1
|
1 I
aR/rrms
I0
SIDS= al ,s/ fN = aql~iDsVDslf Lz
/US"
(9) -
At los = 99.3 nA Uren et al. [2] observed Al---" 4irms, with irms the rms value of the 1 I f current fluctuations. According to our model we have Al = {Nal
1 (?,n %," ' ' ' 2 "
(10)
Using N = l o s / A I D s ~ 5 X 10 3 and ln(fml"~) ~ 10 we obtain a ~ 10 -6. In ref. [3] Uren et al. reported on RTS noise in small-area Si-MOSFETs. According to fig. 1 in their paper A l l l ~ 0.012 and A l / i r m s "~ 8. The number of electrons follows from both N = I/ A I ~ 8 0 and from N = L Z / q l x R ~ 7 0 . Here the mobility is assumed to be 500 cmZ/Vs, which can be considered typical for n-MOSTs at room temperature. Using eq. (10) we obtain a = 2 x 10 -5. In very small area field-effect transistors Restie [4] found RTS fluctuations with A R I R ~ 10 -3 and ARIrm~'~ 3. The time % was found to be 0.1 s. He did not present spectra. With the help of eq. (!0) we find a ~ 10 -5. Kirton and Uren [5] have presented a great deal of data on RTS noise in solid state microstructures. A typical random telegraph signal of a MOSFET with an active area of 0.42 ixm 2 was presented in their fig. i. For this device we have A I / I = 0.06, ~'~= 5 s and Al/irm ~~ 25, which resuits in a ln(fmr~) w 10 -4. We have summarised a number of experimental data obtained from MOSFETs in fig. 2. Here we have plotted AR/rm~ versus A R / R , with AR the resistance step due to RTS and r~m~ the rms value of the 1 / f resistance fluctuations. The solid line represents the relation
1
~
__
163
~ aRIR
-
i
l
10 a
10"
Fig. 2. The quantity ARlr,m ~ vs. the relative RTS step in the resistance ARIR. Here r,m~ is the rms value of the l l f resistance fluctuations. The data are obtained from MOSFETs. Solid line: a ln(.f,,~,)= 10 -4, a from ref. [1], b from ref. [2] (fig. 2, I = 99 nA), c from fig. 1 in ref. [31, d from ref. [4] (fig. 1, A = 0.42 I~m:), e from fig. 1 in ref. [5], f from ref. [5] (fig. 40, ! = 97 pA) and g from fig. 48 in ref. [5].
AR
1
rrms
[ N a In(freTs)] i / 2 -
1
[(R/AR)a ln(f
Z,)]
with a l n ( f m~:~)= 10 -~.
4. Conclusions We have shown that RTS noise is only observed in MOSFETs if the number of free charge carriers in the channel is lower than roughly 1/(10~) with ~ the Hooge 1If noise parameter. We are very'doubtful about the claims made in the literature that 1 I f noise in MOSFETs is the result of superposition of random telegraph signals from individual ~arrier trapping events [ 2 , 4 , 5 , 11]. What we wish to entphasize is that, in any discu~sion of 1 If ,; oise sc~rces, one has to make a dis°~r,;tion betw, en mo~iity fluctuations and numbe~ ~ , c t u ~ ; i ' ~
cknowledgeL~=t ~: This work was sv ?po-ted in part by the European Economic (~ mrnunity in the ESPRIT Basic Research Aciign.
'
334
T.G.M. Kleinpenning / I I.f noise and RTS noise
References [1| K,S, Rails, W.J, Skocpol, L.D. Jackel, R.E. Howard, L,A. Fetter, R.W. Epworth and D.M. Tennant, Phys. Rev. Left. 52 (1984) 228. [2] M.J. Uren, D.J. Day and M.J. Kirton, Appl. Phys. Lett. 47 (1985) 1195. [3] M.J. Uren, M.J. Kirton and S. Collins, Phys. Rev. B 37 (1988) 8346. [4] P. Restle, Appl. Phys. Lvtt. 53 (1988) 1862. [5l M.J. Kirton and M.J. Ut'en, Adv. Phys. 38 (1989) 367.
[61 U.J. Strasilla and M.J.O. Strutt, Proc. IEEE 62 (1974) 1711.
171 L.K.J. Vandamme and R.G.M. Penning de Vries, Solid State Electron. 28 (1985) 1049.
[81 T.G.M. Kleinpenning, in: Proc. Tenth Int. Conf. on
Noise in Physical Systems, A. Ambr6zy, ed. (Akad+miai Kiad6, Budapest, 1990) p. 443. |91 T.G.M. Kleinpenning and A.H. de Kuijper, J. Appl. Phys. 63 (1988) 43. llO] F.N. Hooge, T.G.M. Kleinpenning and L.K.J. Vandamme, Rep. Prog. Phys. 44 (1981) 479. [111 W.J. Skocpol, Physics Today 38 (1985) S19.
ON l / f NOISE AND RANDOM TELEGRAPH NOISE IN VERY SMALL ELECTRONIC DEVICES T.G.M. KLEINPENNING Eindhoven University of Technology, Eindhoven, The Netherlands Received 16 April 1990
We demonstrate that random telegraph signal (RTS) noise can only be observed in small electronic devices where the number of free charge carriers is smaller than 1/{ a In(f= r,)). Here a is the Hooge 1If noise parameter, f= the bandwidth of the measuring system and r~ the sum of the mean capture and emission time of the charge carriers. The current idea that 1/f noise in larger devices is the result of a superposition of RTS noise due to individual trapping is doubtful.
1. Introduction In very small area silicon metal-oxidesemiconductor field-effect transistors (MOSFETs) the effect of a single electron trapped in the gate oxide on the device resistance has been observed by many authors [1-5]. This effect leads to discrete switching of the resistance, referred to as a random telegraph signal (RTS). No such switching was observed in larger devices, which showed only the ubiquitous 1/f resistance fluctuations. RTS fluctuations have a Lorentzian spectrum. Many authors [1-5] claim that the 1 if noise observed in larger devices is the result of a simple superposition of RTS noise from a large number of traps, with a broad distribution of capture and emission times of the trapped and detrapped electrons. On looking at the fluctuating signals observed in small devices by RaUs et al. [1], by Uren et al. [2, 3, 5] and by Restle [4], what one observes is just a normal fluctuating signal with bursts. Strasilla and Strutt [6] have described a method which allows the power spectra of the normal noise superimposed on the burst noise to be measured separately. They found the normal noise (1 If and white noise) to be lower than the Lorentzian burst noise. In the present paper we develop a simple model which shows that RTS noise appears only in dex4ces with a small wJmber of free charge carders, in general less than about 104. The
experimental results from the literature will be checked against our model.
2. Two-level random telegraph signals Let us consider an n-channel MOST of channel length L and operating in the Ohmic region. If we have only 1/f noise and Nyquist noise, the open-circuit voltage noise of the source-drain voltage is [7]
(1)
S v ( f ) = crV2/(fN) + 4 k T R .
Here a is the Hooge 1/f noise parameter, V the applied source-drain bias, N the total number of free electrons in the channel, f the frequency and R the channel resistance. For MOSTs, the value of a is in the range of 10 -6 to 10 -3 (refs. [7, 8]). The fluctuations in V versus time t are sketched in fig. 1. The rms value of the fluctuating voltage is [9] r f
]1;2
from
= [otVZ l n ( f m T ) / N
+ 4 k T R ( f m - l / T ) ] I'2
(2) Here fmax =fm is the bandwidth of the measuring system and fmi. = 1 / T is the reciprocal observation time T.
0921-4526/90/$03.50 ~ 1990- Elsevier Science Publishers B.V. (North-Holland)
T.G.M. Kleinpenning / I l l noise and RTS noise
332
~v(t)
N ~
(a)
Ze
Zc
(b) m
1;$
.
T
Fig. 1. Fluctuations in voltage AV(t) vs. time t of a signal with only 1If and Nyquist noise (a), of a signal with only RTS noise (b) and of a signal with all three types of noise (c).
Let us consider an n-MOST with only RTS noise as a result of single-electron switching between the channel and an interface state. The capture time of the electron is r c and the emission time is r e . The fluctuating voltage is sketched in fig. 1. The voltage step is given by a~, ~= I ( d V / d N ) A N I
=
I(V/N)ANI=
V/N
(3)
.
Here we assume that the electron mobility does not depend on N. If we have 1/f, Nyquist and RTS noise at the same time, we have the situation as sketched in fig. 1(c). Here we take the view that 1If noise and RTS noise have different physical sources. This point of view is based on two experimental findings. Firstly, the results described by Strasilla and Strutt [6] (sec also section 1). Secondly, the empirical result that the 1If noise is in all probability caused by mobility fluctuations [10]. From fig. 1 we can conclude that, according to eqs. (2) and (3) a voltage step with repetition time r~ can be observed on the condition that AV >i Orm~, thus
V { a V 2 ln(fml"s) + 4kTR(fm _ ~.~-1)} N>~ N
(5)
a In(fmrs) "
If the Nyquist noise dominates the 1/f noise, we find the stronger condition
ql~ V 2 1 N ~ 4kTL2f m < a I n ( f mr~) .
(6)
Here we have used fm ">r~t and R = L2/qt~N with /~ the electron mobility. Taking a = 10 -4, fm = 106 Hz and r, = 1 s, we find from eqs. (5) and (6) that N < 10 3. The voltage noise spectrum of the RTS fluctuations is Lorentzian and given by [5]
,I t
0
1
1/2
(4)
Here we have used T = % + ,,-~= ~-s. If the 1/f noise dominates the Nyquist noise, we find
S v ( f ) = (4VelN2)(rplrs) 1 + (2'rrfrp) 2 with % 1 = r e- 1 -I- T c- .! Around f = 1/(2xr%) ratio of RTS noise and 1 I f noise is
rp/rs SRTs/SI,I
= ~aN
"
(7) the
(8)
Taking % = % and a - 10 -4 the ratio is approximately equal to 1 0 a / N . Thus, for N < 10 3 the RTS noise exceeds the 1 I f noise at f--- 1/(2~r%).
3. Comparison with literature Rails et al. [1] have observed discrete resistance switching in submicron Si-MOSFETs. In small-area devices they found A R / R = &V/V= 1 / N "~ 2.5 x 10 -3, thus N ~ 400. In large-area devices they did not observe RTS fluctuations. The ratio A V R T S / U r m s , with Urms defined in eq. (2), was found to be aLout 4, which implies in(freTs) ~ Iv . Uren et al. [2] have investigated 1/f and RTS noise in n-MOSFETs at T = 3 0 0 K . For the largest devices, where N - L /2q l ~ R --- 10,6 the,V found 1/f noise with ~ ~ 4 x 10 -s. However, for the smaller devices with N ~ 103 they found noise spectra with both Lorentzian components due to RTS fluctuations and I / f noise with a 3 × 10 -~. The change in current AIDs(t ) versus
T.G.M. Kleinpenning / I / f noise and RTS noise
time t at various currents IDs and at constant voltage VDs has been presented for the smallest devices. The current steps, A/us = VDsA(1/R) = ql~ VDS/L 2 ,~ 20 p A , are found to be almost independent of the current /us- The current noise superimposed on the RTS was found to increase with the current, which is in accordance with the l l f noise formula as given by
102
333
1
1
|
1 I
aR/rrms
I0
SIDS= al ,s/ fN = aql~iDsVDslf Lz
/US"
(9) -
At los = 99.3 nA Uren et al. [2] observed Al---" 4irms, with irms the rms value of the 1 I f current fluctuations. According to our model we have Al = {Nal
1 (?,n %," ' ' ' 2 "
(10)
Using N = l o s / A I D s ~ 5 X 10 3 and ln(fml"~) ~ 10 we obtain a ~ 10 -6. In ref. [3] Uren et al. reported on RTS noise in small-area Si-MOSFETs. According to fig. 1 in their paper A l l l ~ 0.012 and A l / i r m s "~ 8. The number of electrons follows from both N = I/ A I ~ 8 0 and from N = L Z / q l x R ~ 7 0 . Here the mobility is assumed to be 500 cmZ/Vs, which can be considered typical for n-MOSTs at room temperature. Using eq. (10) we obtain a = 2 x 10 -5. In very small area field-effect transistors Restie [4] found RTS fluctuations with A R I R ~ 10 -3 and ARIrm~'~ 3. The time % was found to be 0.1 s. He did not present spectra. With the help of eq. (!0) we find a ~ 10 -5. Kirton and Uren [5] have presented a great deal of data on RTS noise in solid state microstructures. A typical random telegraph signal of a MOSFET with an active area of 0.42 ixm 2 was presented in their fig. i. For this device we have A I / I = 0.06, ~'~= 5 s and Al/irm ~~ 25, which resuits in a ln(fmr~) w 10 -4. We have summarised a number of experimental data obtained from MOSFETs in fig. 2. Here we have plotted AR/rm~ versus A R / R , with AR the resistance step due to RTS and r~m~ the rms value of the 1 / f resistance fluctuations. The solid line represents the relation
1
~
__
163
~ aRIR
-
i
l
10 a
10"
Fig. 2. The quantity ARlr,m ~ vs. the relative RTS step in the resistance ARIR. Here r,m~ is the rms value of the l l f resistance fluctuations. The data are obtained from MOSFETs. Solid line: a ln(.f,,~,)= 10 -4, a from ref. [1], b from ref. [2] (fig. 2, I = 99 nA), c from fig. 1 in ref. [31, d from ref. [4] (fig. 1, A = 0.42 I~m:), e from fig. 1 in ref. [5], f from ref. [5] (fig. 40, ! = 97 pA) and g from fig. 48 in ref. [5].
AR
1
rrms
[ N a In(freTs)] i / 2 -
1
[(R/AR)a ln(f
Z,)]
with a l n ( f m~:~)= 10 -~.
4. Conclusions We have shown that RTS noise is only observed in MOSFETs if the number of free charge carriers in the channel is lower than roughly 1/(10~) with ~ the Hooge 1If noise parameter. We are very'doubtful about the claims made in the literature that 1 I f noise in MOSFETs is the result of superposition of random telegraph signals from individual ~arrier trapping events [ 2 , 4 , 5 , 11]. What we wish to entphasize is that, in any discu~sion of 1 If ,; oise sc~rces, one has to make a dis°~r,;tion betw, en mo~iity fluctuations and numbe~ ~ , c t u ~ ; i ' ~
cknowledgeL~=t ~: This work was sv ?po-ted in part by the European Economic (~ mrnunity in the ESPRIT Basic Research Aciign.
'
334
T.G.M. Kleinpenning / I I.f noise and RTS noise
References [1| K,S, Rails, W.J, Skocpol, L.D. Jackel, R.E. Howard, L,A. Fetter, R.W. Epworth and D.M. Tennant, Phys. Rev. Left. 52 (1984) 228. [2] M.J. Uren, D.J. Day and M.J. Kirton, Appl. Phys. Lett. 47 (1985) 1195. [3] M.J. Uren, M.J. Kirton and S. Collins, Phys. Rev. B 37 (1988) 8346. [4] P. Restle, Appl. Phys. Lvtt. 53 (1988) 1862. [5l M.J. Kirton and M.J. Ut'en, Adv. Phys. 38 (1989) 367.
[61 U.J. Strasilla and M.J.O. Strutt, Proc. IEEE 62 (1974) 1711.
171 L.K.J. Vandamme and R.G.M. Penning de Vries, Solid State Electron. 28 (1985) 1049.
[81 T.G.M. Kleinpenning, in: Proc. Tenth Int. Conf. on
Noise in Physical Systems, A. Ambr6zy, ed. (Akad+miai Kiad6, Budapest, 1990) p. 443. |91 T.G.M. Kleinpenning and A.H. de Kuijper, J. Appl. Phys. 63 (1988) 43. llO] F.N. Hooge, T.G.M. Kleinpenning and L.K.J. Vandamme, Rep. Prog. Phys. 44 (1981) 479. [111 W.J. Skocpol, Physics Today 38 (1985) S19.