rts
TRANSCRIPT
Chapter 9
1/f Noise and Random TelegraphSignals
In practically all electronic and optical devices, the excess noise obeying the inverse fre-quency power law exists in addition to intrinsic thermal and quantum noise. An enormousamount of experimental data has been accumulated on 1/f noise in various materials andsystems. However, a physical mechanism for 1/f noise has not been identified clearly yet.We have several mathematical models that lead to the 1/f power law but they do notnecessarily shed light on the physical mechanism for the 1/f noise.
In very small electronic devices the alternate capture and emission of carriers at anindividual defect site generates discrete switching in the device resistance—referred toas a random telegraph signal (RTS). The study of RTS has provided a powerful meansof investigating the capture and emission kinetics of single defects, has demonstrated thepossible microscopic origins of low-frequency (1/f) noise in these devices, and has providednew insight into the nature of defects at an interface.
As a consequence of recent advances in processing technology, it has now become pos-sible to produce devices in which the active volume is so small that it contains only a smallnumber of charge carriers. The examples are small-area silicon metal-oxide-semiconductorfield-effect transistors (MOSFETs) and metal-insulator-metal (MIM) tunnel junctions.Figure 9.1 shows an example of the random telegraph signal (RTS) measured in the draincurrent of a MOSFET as a function of time; the times in the high- and low-current statescorrespond to carrier capture and emission times, respectively.
The bias-voltage dependence of the capture and emission times allows one to determinethe location of the defects. In MOSFETs they are found to reside in the oxide up to a fewnanometers from the interface and hence within tunnelling distance of the inversion layer.For the MIM tunnel junctions, the traps are also located in the insulator. Through thestudy of the temperature and bias-voltage dependence of these times for a single defect,one can extract parameters such as capture cross-section, activation energy for captureand emission, and the temperature dependence of the trap energy level (trap entropy).
A principal theme of this chapter is the relationship between these defects and the1/f noise found in large devices. During the past two decades the origin of 1/f noise hasbeen the subject of extensive investigation[1]-[6]. Despite this intensive effort, the subjectof 1/f noise has been notorious for several reasons: first, there has been a lack of data
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open to unambiguous interpretation; secondly, there has been a long-running and rathersterile debate over ‘mobility-fluctuation’ versus ‘number-fluctuation’ models. The basicreason why no consensus has emerged is that little detailed information comes from theconventional ensemble-averaged power spectrum. We shall discuss the recent results onthe noise properties of microstructures in which the averaging process is incomplete andindividual fluctuators can be resolved. In the case of MOSFETs and MIM diodes, it willbe shown conclusively that the 1/f noise in large devices is caused by the summation ofmany RTSs due to the defects in the insulator. In addition, the distribution of physicalcharacteristics measured for the defects accounts easily for the wide range of time constantsnecessary to generate 1/f noise.
Figure 9.1: Random telegraph signal. Change in current against time. Activearea of MOSFET is 0.4 µm2. VD = 10 mV, VG = 0.94 V, ID = 6.4 nA, T =293 K.
The electrical activity of defects at the Si/SiO2 interface is normally studied usingcapacitance-voltage or conductance-voltage techniques. Recent experiments that haveused the conductance technique show that there are two classes of interface defect: thefirst includes those defects normally seen, and which presumably reside at the interface,and are characterized by a single time constant; the second class incorporates defectsresiding in the oxide, which have a wide range of time constants and are responsible forthe 1/f noise.
9.1 Characteristics of 1/f Noise
9.1.1 Scale invariance
A 1/f noise form, x(t), is characterized by a power spectral density function:
Sx(ω) = C/ω , (9.1)
where C is a constant. The integrated power in the spectrum between ω1 and ω2 is givenby
Px(ω1, ω2) =12π
∫ ω2
ω2
Sx(ω)dω
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=C
2πln
(ω2
ω1
), (9.2)
This result shows that for a fixed frequency ratio ω2/ω1, the integrated noise power isconstant. Thus the total noise powers in between any decade of frequency, say 0.1 Hz to1 Hz or 1 Hz to 10 Hz or 10 Hz to 100 Hz, are identical. This property of 1/f noise isknown as scale invariance.
9.1.2 Stationarity
Consider a 1/f noise, x(t), has the band-pass filtered power spectral density,
Sx(ω) =
{C/ω for ω1 ≤ ω ≤ ω2
0 otherwise. (9.3)
The autocorrelation function of x(t) is obtained by using the Wiener-Khintchine theorem,
φx(τ) =C
2π
∫ ω2
ω1
cosωτ
ωdω
=C
2π[Ci(ω2τ)− Ci(ω1τ)] , (9.4)
whereCi(z) =
∫ z
−∞cos y
ydy , (9.5)
is the cosine integral. The series expansion of Ci(z) is
Ci(z) = γ + ln(z) +∞∑
k=1
(−1)kz2k
(2k)!2k, (9.6)
where γ = 0.5772 · · · is Euler’s constant. Thus, in the limit of z → 0, the cosine integralreduces to Ci(z) ' ln z. The mean-square of x(t) is thus given by
φx(τ = 0) =C
2πln
(ω2
ω1
). (9.7)
It is evident from the above argument that the band-pass filtered 1/f noise is statisticallystationary because it has the second-order quantities depend only on the delay time τ andnot on the absolute time at which the ensemble average is performed.
However, there is no experimental evidence for the existence of the low frequency limitω1 for 1/f noise. The reason is that an observation time T is always finite in practice and soa lower frequency region of the spectrum, ω ≤ 2π
T , cannot be observed. The autocorrelationfunction and the mean square value that can be measured in actual experiments are thusgiven by replacing the low frequency limit ω1 with 2π/T in Eqs. (9.4) and (9.7).
The Wiener-Levy process, discussed in Chapter 2, is a cumulative process of randomwalk. The power spectrum obeys 1/ω2 law. By the very nature of the process, the Wiener-Levy process is statistically nonstationary and there is no possibility for the low-frequencylimit ω1 to exist. The corner (roll-off) frequency in the calculated spectrum (Chapter 2)is an artifact associated with the finite gate time T . In the case of 1/f noise, however, alow-frequency limit ω1 may or may not exist. The stationarity of the process is still opento question.
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9.2 Mathematical Model of 1/f Noise
9.2.1 A random pulse train model of 1/f noise
The power spectral density of a random pulse train, x(t), is given by Carson’s theorem,
Sx(ω) = 2νa2|F (iω)|2 , (9.8)
where F (iω) is the Fourier transform of the pulse shape function, ν is the mean rate of thepulses and a2 is the mean-square value of the pulse height. Thus the frequency dependenceof Sx(ω) is entirely determined by f(t). Consider the fictitious pulse shape function,
f(t) = u(t)t−(1−α2 )e−ωxt , (9.9)
where α and ωx are positive and independent of time, and u(t) is the unit step function.The Fourier transform of f(t) is
F (iω) =∫ ∞
0t−(1−α
2 )e−(ωx+iω)tdt
=Γ
(α2
)
(ωx + iω)α2
, (9.10)
where Γ(x) is the gamma function. Using Eq. (9.10) in Eq. (9.8), we obtain
Sx(ω) =2νa2Γ2
(α2
)
(ω2x + ω2)
α2
. (9.11)
It shows the approximate 1/f noise characteristic at ω À ωx,
Sx(ω) ' C/ωα , (9.12)
when α ' 1. Here C = 2νa2Γ2(
α2
). However small ωx may be, provided it is non-zero,
Sx(ω) has the flat spectrum at ω ≤ ωx, which indicates such a random pulse train isstatistically stationary.
The autocorrelation function of this random pulse train is given by
φx(τ) =12π
∫ ∞
0Sx(ω) cos(ωτ)dω
=C
2πK0(ωxτ) , (9.13)
where α = 1 and K0(z) is the modified Bessel function of the second kind of zero order.The series expansion of K0(z) is
K0(z) = −γ + ln 2− ln(z) + · · · . (9.14)
For small ωxτ, φx(τ) varies as ln(ωxτ) and takes a finite value except at the origin τ = 0.This logarithmic infinity is associated with the infinite extension of the 1/f noise spectrumto high frequencies, which is of course unrealistic because any finite response time in asystem introduces a cut-out characteristic beyond the certain frequency and the spectrumusually rolls off with a 1/ω2 dependence. Figures 9.2(a) and 9.2(b) show schematicallySx(ω) and φx(τ) of this random pulse train.
It is clear from the above argument that a random pulse train in which the pulse shapevaries as t−
12 shows the 1/f noise behavior. However, the physical origins of such a pulse
shape are not clear at all.
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Figure 9.2: (a) Sx(ω), normalized by Sx(0), vs. ω/ωx. (b) φx(τ), normalizedby C/2π, vs. ωxτ .
9.2.2 Superposition of relaxation processes
If the noisy form z(t) is a random relaxation process with a time constant τz, the powerspectral density has the general form:
Sz(ω) =g(τz)
1 + ω2τ2z
, (9.15)
where g(τz) depends on the physical mechanism of the noise. Suppose x(t) is constructedfrom such linear superposition of relaxation processes, whose decay constants are dis-tributed between upper and lower limits τ1 and τ2 with a probability density p(τz). Theoverall power spectral density is then
Sx(ω) =∫ τ2
τ1Sz(ω)p(τz)dτz =
∫ τ2
τ1
p(τz)g(τz)(1 + ω2τ2
z )dτz . (9.16)
If the numerator p(τz)g(τz) is independent of τz and equal to a constant P , the aboveintegral reduces to
Sx(ω) = P[tan−1(ωτ2)− tan−1(ωτ1)
]/ω . (9.17)
When the two time constants τ2 and τ1 satisfy ωτ2 À 1 and 0 ≤ ωτ1 ¿ 1, respectively, thetwo terms in the numerator are approximately equal to π/2 and 0. Thus, a superpositionof relaxation processes can give rise to a spectrum
Sx(ω) =(
πP
2
)/ω . (9.18)
If the product p(τz)g(τz) is proportional to τα−1z , the power spectral density has a more
general form of ω−α.
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9.2.3 Distributed trapping model
The most popular model of 1/f noise is the trapping model with a wide spread of timeconstants. If a free carrier is immobilized by falling into a trap, it is no longer available forcurrent transport. The modulation of carrier numbers has the form of random telegraphsignal with a Poisson point process as shown in Fig. 9.3. The probability of observing m
Figure 9.3: A random telegraph signal produced by a carrier trap.
telegraphic signals in the time interval T is given by
p(m,T ) =(νT )m
m!e−νT , (9.19)
where ν is the mean rate of transitions per second. If τ+ and τ− are the average timesspent in the upper and lower states, respectively, the probability distributions of the uppert+ and lower state times t− are
p(t±) = τ−1± exp
(− t±
τ±
). (9.20)
The product x(t)x(t + τ) is equal to +a2 if an even number of transitions occur in theinterval (t, t + τ) and to −a2 if an odd number of transitions occur in the same interval.Therefore, the autocorrelation function is
φτ (x) = a2 [p(0, τ) + p(2, τ) + · · ·]−a2 [p(1, τ) + p(3, τ) + · · ·]
= a2e−ντ
[1− ντ +
(ντ)2
2!− (ντ)3
3!+ · · ·
]
= a2e−2ντ . (9.21)
The power spectrum is thus calculated by the Wiener-Khintchine theorem,
Sx(ω) = 4∫ ∞
0φx(τ) cos(ωτ)dτ
=2a2/ν
(1 + ω2/4ν2)
= a2 4τz
(1 + ω2τ2z )
. (9.22)
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Here τz = 1/2ν is the time constant of the trap. If τz is distributed according to thefunction p(τz), the power spectral density of the total carrier number fluctuation is
Sn(ω) = 4φn(τ = 0)∫ ∞
0
τzp(τz)(1 + ω2τ2
z )dτz . (9.23)
Here it is assumed∫∞0 p(τz)dz = 1.
Suppose the carrier trap occurs by the tunneling of carriers from semiconductors tothe traps inside the oxide layer at depth w, the time constant obeys
τz = τ0 exp(γw) , (9.24)
where τ0 and γ are constants. If the traps are homogeneously distributed between thedepth w1 and w2, corresponding to the time constants τ1 and τ2, we obtain
p(τz)dτz =
{dτz/τz
ln(τ2/τ1) (τ1 ≤ τz ≤ τ2)0 (otherwise)
. (9.25)
Using Eq. (9.25) in Eq. (9.23), the power spectral density of the total number fluctuationis given by
Sn(ω) =4φn(0)
ln(τ2/τ1)
∫ τ2
τ1
dτz
(1 + ω2τ2z )
=4φn(0)
ln(τ2/τ1)× tan−1(ωτ2)− tan−1(ωτ1)
ω. (9.26)
As we discussed before, Eq. (9.26) shows 1/f power law in the frequency range of ωτ2 À 1and 0 ≤ ωτ1 ¿ 1.
The above argument applies also for the intrinsic bulk transport property of the hop-ping conduction. The essential requirement to obtain the 1/f power law is the Poissoniantelegraphic event with a distributed time constant which obeys 1/τz distribution function.
9.3 Theoretical Background for RTSs
The main purpose of the following subsections is to provide a framewark for the detailedanalysis of random telegraph signals (RTSs) and the capture and emission kinetics ofindividual defects.
9.3.1 Probability distribution of RTS
Referring back to Fig. 9.1, we shall take the high-current state of the RTS to be state 1 andthe low-current state to be state 0. We shall assume that the probability (per unit time)of a transition from state 1 to state 0 (i.e. from up to down) is given by 1/τ1, with 1/τ0
being the corresponding probability from 0 to 1 (i.e. from down to up). The transitionsare instantaneous. We now intend to show that these assumptions imply that the timesin states 0 and 1 are exponentially distributed, that is, the switching is a Poisson process.
Let p1(t)dt be the probability that state 1 will not make a transition for time t, thenwill make one between times t and t + dt. Thus
p1(t) = A(t)/τ1 , (9.27)
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where A(t) is the probability that after time t state 1 will not have made a transition and1/τ1 is the probability (per unit time) of making a transition to state 0 at time t. However,
A(t + dt) = A(t)(1− dt/τ1); (9.28)
that is, the probability of not making a transition at time t + dt is equal to the productof the probability of not having made a transition at time t and the probability of notmaking a transition during the interval from t to t + dt. We can rearrange Eq. (9.28) togive
dA(t)dt
= −A(t)τ1
. (9.29)
Integrating both sides of Eq. (9.29), we find
A(t) = exp(−t/τ1) , (9.30)
such that A(0) = 1. Thus
p1(t) =1τ1
exp(− t
τ1
). (9.31)
p1(t) is correctly normalized such that∫ ∞
0p1(t)dt = 1 .
The corresponding expression for p0(t) is
p0(t) =1τ0
exp(− t
τ0
). (9.32)
Hence, on the assumption that the up and down times are characterized by single attemptrates, we expect the times to be exponentially distributed. The mean time spent in state1 is given by ∫ ∞
0tp1(t)dt = τ1 , (9.33)
and the standard deviation is[∫ ∞
0t2p1(t)dt− τ2
1
]1/2
= τ1 . (9.34)
Equivalent expressions hold for the down state. Thus the standard deviation is equalto the mean time spent in the state. Equation (9.34) can be used as a simple test forexponential behavior.
9.3.2 Power spectrum of RTS: Lorentzian spectrum
Here we shall outline the derivation of the power spectrum of an asymmetric RTS. Initially,we need to evaluate the autocorrelation function of the RTS. It is convenient to choosethe origin of the coordinate system such that state 0 has amplitude x0 = 0, and state1 has amplitude x1 = ∆I. In addition, all statistical properties will be taken to be
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independent of the time origin. The probability that at any given time the RTS is in state1 is τ1/(τ0 + τ1), and similarly for state 0 it is τ0/(τ0 + τ1). Then we have
c(t) = 〈x| (s)x(s + t) |〉=
∑
i
∑
j
xixj × {Prob. thatx(s) = xi}
×{Prob. that x(s + t) = xj , given x(s) = xi} . (9.35)
Since x0 = 0 and x1 = ∆I, we obtain
c(t) = (∆I)2τ1
τ0 + τ1P11(t)
= (∆I)2 × {Prob. that x(s) = ∆I}×{Prob. of even no. of transitions in time t, starting in state1} . (9.36)
If we define P10(t) as the probability of an odd number of transitions in time t, startingin state 1 then we have
P11(t) + P10(t) = 1 . (9.37)
In addition,
P11(t + dt) = P10(t)dt
τ0+ P11(t)
(1− dt
τ1
); (9.38)
that is, the probability of an even number transitions in time t + dt is given by the sumof two mutually exclusive events: first, the probability of an odd number of transitions intime t and one transition in time dt; and secondly, the probability of an even number oftransitions in time t and no transitions in time dt. We can make dt small enough that theprobability of more than one transition is vanishingly small. Substituting from Eq. (9.37)into Eq. (9.38), we obtain the following differential equation for P11(t):
dP11(t)dt
+ P11(t)(
1τ0
+1τ1
)=
1τ0
. (9.39)
This equation can be solved by using exp [∫(1/τ0 + 1/τ1)dt] as an integrating factor:
P11(t) =τ1
τ0 + τ1+
τ0
τ0 + τ1exp
[−
(1τ0
+1τ1
)t
], (9.40)
where P11(t) has been normalized such that P11(0) = 1. Equations (9.40) and (9.36) cannow be used to evaluate the power spectral density S(f):
S(f) = 4∫ ∞
0c(t) cos(2πfτ)dτ =
4(∆I)2
(τ0 + τ1)[(1/τ0 + 1/τ1)2 + (2πf)2]. (9.41)
Here we have used the Wiener-Khintchine theorem. (The d.c. term, which contributesa delta function at f = 0, has been ignored.) For the case of a symmetric RTS, that is,τ0 = τ1 = τ for example, this equation simplifies to
S(f) =2(∆I)2τ
4 + (2πfτ)2. (9.42)
The total power P in the RTS can be obtained by integrating Eq. (9.41) over all frequencies:
P =(∆I)2
(τ0 + τ1)(1/τ0 + 1/τ1). (9.43)
As one would expect, P = (12∆I)2 when τ0 = τ1; P is a maximum under these conditions.
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9.3.3 Occupancy levels and grand partition function
In order to be precise in our meaning, we shall introduce the nomenclature ‘occupancylevel’ E(n + 1/n) to describe the energy level of a defect: E(n + 1/n) marks the Fermilevel EF at which the defect’s occupancy changes from n electrons to n + 1 electrons. Wecan determine the occupancy of the defect using the grand partition function, ZG. Thisis written as
ZG =∑
ASN
exp(−ES −NEF
kT
), (9.44)
where ASN implies the summation is to be carried out over all states S of the system forall numbers of particles N . We have adopted the convention of semiconductor physics andset EF to be equivalent to the temperature-dependent chemical potential. The absoluteprobability that the system will be found in a state (N1, E1) is given by
p(N1, E1) =γ exp[−(E1 −N1EF)/kT ]
ZG, (9.45)
where the state is orbitally (and perhaps also spin) degenerate with degeneracy γ.Consider a defect system that has only two states of charge, n and n + 1, available.
Let the energy zero of the system correspond to the defect occupied by n electrons. Then
ZG = γ(n) exp(
nEF
kT
)+ γ(n + 1) exp
[−E(n + 1/n)− (n + 1)EF
kT
], (9.46)
where γ(n) and γ(n + 1) are the degeneracies of the n- and (n + 1)-electron states. Thenthe probability of finding the defect in the (n + 1)-electron state is
f = p(n + 1) ={
1 + g exp[E(n + 1/n)− EF
kT
]}−1
, (9.47)
whereg = γ(n)/γ(n + 1) . (9.48)
This looks like a Fermi-Dirac distribution with a degeneracy factor g. In addition, we canwrite
p(n + 1)p(n)
=γ(n + 1)
γ(n)exp
[−E(n + 1/n)− EF
kT
]. (9.49)
That is, when the Fermi level crosses the level E(n + 1/n), the (n + 1)-electron statedominates over the n-electron state.
For an individual RTS generated by a trap with occupancy level E(n + 1/n) and withmean capture and emission times τc and τe, we have
f =τe
τc + τe=
{1 + g exp
[E(n + 1/n)− EF
kT
]}−1
, (9.50)
τe =τc
gexp
[−E(n + 1/n)− EF
kT
], (9.51)
where g is given by Eq. (9.48).
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9.4 Random Telegraph Signals and Single-Electron Switch-ing Events
We begin the discussion of RTS’s with an overview of the diverse range of electronic devicesand systems in which discrete-switching behavior in the resistance has been observed.The specific example of the small-area silicon MOSFET is then taken. We show thatthe experimental data is consistent with an RTS being generated through the captureand emission of a single electron at an individual defect residing in the oxide close to theSi/SiO2 interface.
9.4.1 Overview of RTS phenomena
The so-called burst noise in reverse-biased p-n junctions and bipolar transistors provides uswith our first example of discrete switching behavior in electronic devices. Figure 9.4 showshe temporal variation in current through a reverse-biased germanium p-n junction[6]. Itis found that the times in the high- and low-current states obeyed Poisson statistics. Inaddition, they noted that the switching rate was thermally activated, with an activationenergy larger than the germanium band gap. Although first observed nearly thirty yearsago, the origins of burst noise still remain uncertain; dislocations, metal precipitates andthe switching on and off of surface conduction channels have all been implicated in its pro-duction. The switching behavior in double-gated silicon JFETs; is shown in Fig. 9.5[7].
Figure 9.4: Waveform of burst noise in a reverse-biased (7.5 V) germaniumjunction. The horizontal scale is 20 ms div−1 and the vertical scale 20 nAdiv−1.
The RTSs were generated through the charging and discharging of single defects in theDebye region. between the channel and fully depleted region. In its more negative charge
222
state and for n-channel devices, the defect produced a constriction of the channel; on elec-tron emission, the channel resistance was lowered and the source-drain current increased.The extra gate allowed the channel to be moved in a direction normal to the current flow,and thus for defects to be moved into and out of the active region.
Figure 9.5: Current switching in silicon JFET, XS01: (a) T = 83 K, VD = 6V, ID = 16 µA; (b) T = 166 K, VD = 2 V, ID = 17 µA. VG= + 0.44V.
The fluctuation phenomena in very small MOSFETs (dimensions 0.1 µm × 1.0 µm)operating at cryogenic temperatures was also studied. They noted that as the devicearea as scaled down, the total number of Si/SiO2 interface defects was correspondinglyreduced. In small enough devices it is quite likely that only a handful of traps will haveenergy levels within kT or so of the surface Fermi level and thus will be fluctuating inoccupancy. This is borne out in 9.6, which shows some of their results for several gatevoltages and temperatures[8]. The observed resistance changes are consistent with a singleelectron being removed from the channel and captured in a localized defect state. Notethat as the gate voltage changes the mark-space ratio changes as the separation of thetrap energy level and surface Fermi level is altered. It is also quite clear that the switchingrate is a sensitive function of temperature. In addition, one can see that at elevatedtemperatures, where several RTSs are active, the resistance fluctuations are beginning toresemble the trace one would observe for a l/f noise source.
223
Figure 9.6: Resistance switching observed in a small (0.15 µm × 1.0 µm) siliconMOSFET.
A deep-level transient spectroscopy (DLTS) was applied to the study of small MOSFETs[9].First they applied a voltage pulse of several volts to the gate for a few minutes to fill the in-terface traps. They then pulsed the device into weak inversion and observed the emissionof carriers from filled interface states and single-electron switching in the drain-voltagetransient (fig. 9.7). This technique allows the simultaneous measurement of all the trapsin the accessible range of EF, but is rather hard to analyze.
The switching phenomena in the tunnelling current through large-area (180 µm2) GaAsn-i-n diodes was observed, where the intrinsic region consists of a linearly-graded bandgap of AlGaAs: see: Fig. 9.8[10]. It is clearly worth noting that their results bear astrong resemblance to the a forementioned burst-noise work. They concluded that theswitching was controlled by a single defect in the barrier region. Close to turn-on, the I-Vcharacteristic will be dominated by inhomogeneities in the barrier. Any defect residingclose to such an inhomogeneity will have a dramatic effect as it charges and discharges,leading to filamentary current transport and a measurable switching effect.
Our final example comes from the scanning tunnelling microscope (STM) to study thin(l.5-2.0 nm) layers of thermally grown SiO2[11]. With the STM operating in the constant-current mode, they found regions of the surface where current transients were faster thanthe response time of the feedback loop. These regions were further investigated and, withthe feedback loop set to maintain an average current of 1 nA, the instantaneous currentwas found to switch between levels at 2 nA and close to zero. Figure 9.9 shows the tunnelcurrent against time for various values of the potential on the tip of the probe and asthe tip is moved away from the site of bistable-current production. It was concluded that
224
Figure 9.7: DLTS transients of electron re-emission in an n-channel MOSFET(1.2 µm × 1.5 µm) after complete filling of interface traps by a gate voltage of4.4 V for 5 min. VG = 0.85 V, ID = 20 nA.
their data were consistent with electrons filling and leaving localized states on the surface.They also estimated the area surrounding the defect in which the tunnelling current wasperturbed by the presence of the trapped charge. They found sufficiently close agreementbetween theory and experiment to conclude that the fluctuations were brought about bysingle-electron capture and emission events.
9.4.2 Single-electron switching in small-area MOSFETs
Using the silicon MOSFET as an example, we shall now show that in this system single-electron trapping at interface defects provides a consistent explanation for the measuredproperties of the RTSs.
A. Gate-voltage dependence of RTSs
The dependence on gate voltage of an RTS measured in a 0.4 µm2 n-channel MOSFETat room temperature is shown in Fig. 9.10 This figure shows that as the gate voltage isincreased, the time in the high-current state is reduced dramatically, while the time in
225
Figure 9.8: Current against forward bias voltage at 4.2 K. The solid lines areexperimental traces at magnetic fields of 0, 5 and 8 T. The dashed line isthe calculated response. The upper inset shows a schematic diagram of thestructure. The lower inset shows the current as a function of time for twoconstant voltages.
the low-current state appears to be largely unaffected. Figure 9.11 shows a schematicrepresentation of the band bending in a (small-area) MOSFET in which there is only onedefect energy level ET within kBθ of the surface Fermi level EF; the effect of a positiveincrement in gate voltage is shown by the dotted line. The important point to note isthat the energy separation ET − EF becomes less positive (or, with ET below EF, morenegative) as VG increases. For the linear regime of MOSFET operation the fractionaloccupancy of the defect is governed by Eq. (9.50)
τc
τe= g exp
(ET − EF
kBθ
). (9.52)
Figure 9.10 in conjunction with Eq. (9.51) allows one to identify the times in the upstate with electron capture and the down times with emission. On electron capture into alocalized electronic state, it would appear that the negative electrostatic potential set up bythe trapped charge is responsible for a localized increase in channel resistance. Figure 9.12shows the measured distributions of down (emission) times from a single RTS. For thisparticular RTS the distribution of up (capture) times was virtually identical. The captureand emission times were exponentially distributed, and thus the switching is governed bythe two attempt rates 1/τc and 1/τe. The means and standard deviations were also in
226
Figure 9.9: (a)-(d) STM tunnel current through thin SiO2 layer as a functionof time for four values of Vtip. The tip is positioned over peak of noise source.(e) The tunnel current against time for Vtip at 0.340 V and the tip positioned3 nm away from the peak of the noise source.
close agreement. A slight departure of the data from the theoretical curve at very smalltimes is expected owing to the limited resolution of the experimental sampling rate.
Let us consider the consequences of the RTSs being due to multi-electron capture intoa single defect rather than just single-electron capture. For the purposes of simplicitywe shall consider two-electron capture, although this can obviously be generalized. Onthis basis, the time in the high-current state corresponds to the (rate-limiting) capture ofthe first electron. This is then followed by the (unseen) very fast capture of the secondelectron. Thus the time spent in the one-electron level is below the experimental resolutionlimit.
We shall denote the high-current level as level 0, the (unseen) middle level as level 1,and the low-current level as level 2. Then, using Eq. (9.45) we find the following relative
227
Figure 9.10: Random telegraph signals in small MOSFET measured at theindicated gate voltages. Active device area is 0.4 µm2, VD = 4 mV, T = 293K.
probabilities of occupation (all degeneracies are taken to be equal):
p(1)p(0)
= exp[−E(1/0)− EF
kT
], (9.53)
p(2)p(1)
= exp[−E(2/1)− EF
kT
], (9.54)
p(2)p(0)
= exp[−E(1/0) + E(2/1)− 2EF
kT
]. (9.55)
Since p(1) ≈ 0, this implies that E(1/0) − EF is large and positive and E(2/1) − EF islarge and negative with respect to kT . Further, since p(2) ≈ p(0), this requires E(1/0) andE(2/1) to be roughly equidistant from the Fermi level. Thus the occupancy level E(1/0)must be ≥ 10kT above the Fermi level and E(2/1) ≥ 10kT below. Overall, the defect mustexhibit very strong negative-U properties. Such a scenario is by no means impossible, butlet us now investigate how consistent this model is with further interpretation of the data.
We shall now outline how the behavior of the mark-space ratio of the RTS can be usedto estimate the distance of the trap into the oxide. Taking the logarithm of both sides
228
Figure 9.11: Band bending in n-channel silicon MOSFET. The dotted linesshow the changes accompanying a positive increment in gate voltage δVG. δφs
is the change in surface potential. ET and E′T denote the trap energy-level
positions before and after changing gate voltage. φb denotes the potential ofthe bulk Fermi level EF with respect to the intrinsic level Ei.
of Eq. (9.52) and differentiating with respect to gate voltage, we find for the one-electroncase
d
dVG(∆ETF) =
kT
q
d
dVG(ln τc − ln τe) (eV V−1) , (9.56)
where ∆ETF = ET − EF. Thus, for a given increment in gate voltage, the measuredchanges in τc and τe allow one to estimate the change in separation of the trap energylevel and Fermi level: see Fig. 9.11. The change in surface potential δφs can be estimatedfrom standard MOSFET analysis[13]. The distance of the trap into the oxide is then givenby the relation
q(δVG − δφs)tox
=δ(∆ETF)− qδφs
d, (9.57)
where tox is the thickness of the gate oxide. The values of d were found up to 2 nm.If the capture process involves two-electron capture then
d
dVG(2∆E
′TF) =
kT
q
d
dVG(ln τc − ln τe) (eV V−1) , (9.58)
where ∆E′TF = 1
2 [E(1/0)+E(2/1)]−EF. Using Eq. (9.58) for devices operating in stronginversion, we find the potential change at the trap is usually about half the change at thesurface. Since the potential at the inversion-layer charge centroid moves at half the rateof the surface potential, this places the trap in the middle of the inversion layer, that is,in the silicon rather than the oxide.
229
Figure 9.12: Distribution of 4425 emission times for device H6 at 95 K andVG = 1.15 V, showing that the time is distributed exponentially. τe = 0.0528 s,standard deviation 0.0505 s. The inset shows a portion of the I-t characteristic;the down time corresponds to emission.
B. The amplitude of RTSs
With reference to Fig. 9.10, we shall take our initial working hypothesis to be thefollowing: after an electron is trapped into an Si/SiO2 interface defect state, the reductionin source-drain current comes about through a reduction in the number of free carriersin the channel. The strong-inversion case is simplest to consider first. In this regime thescreening of the trapped charge is carried out by the inversion-layer electrons. We thereforeexpect on electron capture a reduction of unity in the total number of free carriers in theinversion layer, Ninv,tot, and hence (ignoring scattering) ∆ID/ID = 1/Ninv,tot. As thegate voltage is reduced to threshold and below, the screening of the trapped charge is nowprincipally carried out by the depletion region and the gate; the estimation of the reductionin total carrier number becomes accordingly more complex. We shall now outline a simpletheory for the expected variation in ∆ID/ID from weak to strong inversion.
Figure 9.13(a) depicts the location of charge in an MOS structure. We define QG, Qit, QD, Qn
and Qt to be the charge density (per unit area) associated with the gate, interface traps,depletion region, inversion-layer and the oxide traps respectively. When the trapped chargeQt fluctuates at fixed VG the charge conservation in the structure is such that
δQG + δQit + δQD + δQn + δQt = 0 . (9.59)
These fluctuations can be related to the change in surface potential δφs via the followingrelations:
δQG = −Coxδφs , (9.60)
230
δQit = −Citδφs , (9.61)
δQD = −CDδφs , (9.62)
δQn = −Cnδφs . (9.63)
Cox, Cit, CD and Cn are the capacitances (per unit area) associated with the oxide, interfacetraps, depletion region and channel respectively. Now
Qn = Q0 exp(
qαφs
kT
), (9.64)
and thereforeδQn/δφs = βαQn , (9.65)
where β = q/kT, Q0 is a constant, and α has a value of 1 in weak inversion falling to 0.5in strong inversion[14].
Equations (9.59) and (9.60 – 9.63) allow us to write
δQt/δφs = Cox + Cit + CD + Cn . (9.66)
Using the relation δQn/δQt = (δQn/δφs)δQs/δQt and Eqs. (9.63) and (9.66), we obtain
δQn
δQt=
δQn/δφs
Cox + Cit + CD − δQn/δφs. (9.67)
In the limit of strong inversion the terms in the denominator are dominated by δQn/δφs,and we find δQn/δQt = −1; if the trap gains one electron then the inversion layer loses oneelectron, and vice versa. As weak inversion is approached |δQn/δQt| < 1, correspondingto charge sharing between the gate and the inversion and depletion layers.
If we assume that all changes in the charge distributions on electron capture into anoxide defect are located within the small area ∆a = ∆X∆Y as shown in Fig. 9.13(b) thenwe can write
∆a δQt = −q , (9.68)
a single electronic charge. Further, we assume that within this area ∆a
δQn = −Qn . (9.69)
Thus the model is based on total exclusion of inversion-layer charge. In practice, thecharge does not fall to zero. Integrating Poisson’s equation with a trapped charge at theSi/SiO2 interface along a line perpendicular to the surface above the trap, we found gavea reduction to 22% of the value without the trap for the device biased in weak inversion.Thus our model is oversimplified and is not realistic, but it is nevertheless a useful firstapproximation. We estimate ∆a as follows. Equations (9.65) and (9.67) can be combinedto give
δQn
Qn=
αβ δQt
Cox + Cit + CD − αβ Qn. (9.70)
Multiplying the top and bottom of the right-hand side of Eq. (9.70) by area ∆a andconfining all changes in charge density to this area, as well as noting the relationshipsEqs. (9.68) and (9.69), we find
∆a =αβq
Cox + Cit + CD − αβ Qn. (9.71)
231
Figure 9.13: (a) Charge structure in a MOSFET. Qt, Qit, Qn, QD and QG arethe charge densities (per unit area) associated with oxide traps, interface states,the inversion layer, the depletion region and the gate respectively. (b) Planview showing small area of channel cored out by electron trapped in oxidedefect.
232
Since we have assumed that within area ∆a all inversion-layer charge is excluded (i.e. itsresistance is infinite) it is a straightforward matter to show that
∆ID
ID=
∆R
R=
∆a
A=
αβq
A(Cox + Cit + CD − αβ Qn), (9.72)
where R is the channel resistance and A is the device area.In Fig. 9.14 we show the measured values of ∆ID/ID recovered from a survey of small-
area (0.375 µm2) MOSFETs from the same wafer. The data were obtained by choosingseven values of the source-drain current (at fixed VD) and at each value the amplitudes ofthe RTSs that were visible were measured. The time window in which the measurementswere carried out was 10−3 − 102 s. At each current the number of distinct traps seenwas 51, 30, 70, 89, 56, 60 and 58 in 27, 12, 28, 18, 11, 10 and 12 devices, in order ofincreasing current magnitude. For a given current, the distribution of amplitudes wassorted into percentiles. For example, the 50th percentile is the value below which 50% ofthe distribution lies. On Fig. 9.14 we have marked the 0, 25, 50, 75 and 100th percentiles.The theoretical values of ∆ID/ID from Eq. (9.72) have also been plotted. Apart fromthe data points around 10−6 S −1–which represent only 30 traps–the theoretical curveessentially passes through the 50th percentile of the data.
The data presented in Fig. 9.14 suggest that the average behavior is reasonably rep-resented by our simple theory. However, it is somewhat disconcerting to see that a largenumber of RTS amplitudes appear to correspond to significantly less than or significantlygreater than one electron trapped.
9.5 Capture and Emission Kinetics of Individual Defect States
Random telegraph signals (RTSs) generated through the fluctuating occupancy of indi-vidual defects provide a unique probe into the trapping dynamics of single defects. Inthe following subsections we shall review microscopic models of the capture and emissionkinetics of individual defects at the Si/SiO2 interface and defects in the insulator of MIMtunnel junctions.
9.5.1 Si/SiO2 interface
The capture and emission kinetics of single defects at the Si/SiO2 interface can be inves-tigated by analyzing RTSs as a function of temperature and gate voltage. First of all, weshall consider the situation in an n-channel MOSFET with a trap level lying close to theFermi level, as shown in Fig. 9.11. Our initial aim is to determine the factors governing thebehavior of the mean capture and emission times of an RTS as a function of temperature.
A. Temperature dependence of RTSs in silicon MOSFETs (I): Theory
Formally, we can write the carrier capture rate for an interface defect as
1τc
=∫ ∞
Ec
r(E)dE , (9.73)
233
Figure 9.14: Measured values of RTS amplitudes recovered from a survey ofsmall-area (0.375 µm2) MOSFETs, from the same wafer, as a function of chan-nel conductivity G = IDl/VDw (Siemens/ ). The dashed lines show the be-havior of the 0, 25, 50, 75 and 100th percentiles (see the text). The solid lineshows the theoretical value of ∆ID/ID obtained from Eq. (9.72).
where r(E) is the transition rate (per unit energy) at energy E in the inversion layer. r(E)can be written as the product of a particle flux and cross-section:
1τc
=∫ ∞
Ec
n(E)υ(E)σ(E)dE . (9.74)
n(E) is the inversion-layer number density (per unit volume per unit energy) at energy E,and υ(E) is the carrier velocity at E. We shall use Eq. (9.74) in the following simplifiedform:
1/τc = nυσ . (9.75)
We have made the major assumption that the particle density (per unit volume) in theinversion layer can be represented by the constant value n. υ is the average thermalvelocity of the carriers and σ the average capture cross-section.
The relationship between the capture and emission times for a defect which is fluctu-ating between its n- and (n + 1)-electron states is given by Eq. (9.51). Further, we writethe cross-section in the form
σ = σ0 exp(−∆EB
kT
). (9.76)
234
In Eq. (9.76) we are anticipating the experimental findings and taking the cross-sectionto be thermally activated, with a barrier for capture ∆EB and a cross-section pre-factorσ0[14]. In fact, the proposed mechanism of charge-carrier capture and emission is througha process directly analogous to the multiphonon mechanism familiar from deep levels inbulk semiconductors. A schematic configuration-coordinate diagram showing the changesin total energy of the system as an electron is transferred from the inversion layer into aninterface defect is shown in Fig. 9.15. The energy zero in this figure corresponds to theempty trap with an electron at the Fermi level. The dashed curve shows the variation intotal energy as the empty defect distorts (plotted against a single normal coordinate). Thefull curve marked with an open circle shows the same with the electron in the conductionband. The full curve marked with the full circle depicts the variation in total energyof the trap after it has captured the electron. At the cross-over there is strong mixingbetween the inversion-layer state and the defect state. The non-radiative transition isinduced by off-diagonal elements in the Hamiltonian, which induce transitions betweenvibronic states that differ in electronic energy but have the same total energy. On electroncapture, the detect state is well away from equilibrium and the excess energy is dissipated
Figure 9.15: Configuration coordinate diagram: elastic + electronic energiesagainst single normal coordinate. The energy zero of the system correspondsto the empty defect with the electron at the Fermi level. This is shown asthe dashed curve. ◦ labels the empty trap plus a free electron in the inversionlayer. • marks the filled trap.
by multiphonon emission.
235
Substituting Eq. (9.76) into Eq. (9.75), we obtain
τc =exp(∆EB/kT )
σ0υn. (9.77)
Equations (9.51) and (9.77) contain all of the physics that we require in order te investigatethe temperature dependence of the mean capture and emission times of a single RTS.
Equation (9.77) can be further expanded by noting that
υ = (8kT/πm∗)1/2 . (9.78)
where m∗ is the average mass of a carrier in the inversion layer. Previously we haveused the RMS velocity; however, the difference between υRMS and υ is insignificant. Thetemperature dependence of the number density n can be obtained from the variation ofthe drain current I(T ). Since all of the measurements that we shall report on were carriedout in the linear regime where the device is operating in a simple resistive fashion, we canwrite
I(T ) = n(T )qµ(T )VDt(T )w/l , (9.79)
where µ(T ) is the temperature-dependent electron mobility; VD is the applied drain volt-age, and t(T ), w and l are the channel thickness, width and length respectively. Thethickness was estimated from a classical integration of the surface charge. The tempera-ture dependence of the mobility and inversion-layer thickness can be expressed as[15]
µ(T ) = µ0T−3/2 (9.80)
t(T ) = t0T . (9.81)
Combining Equaitons (9.79)–(9.81), we obtain the temperature variation of the averageinversion-layer number density:
n(T ) =IT 1/2
qµ0 VDt0(w/l). (9.82)
Incorporating Equations (9.78) and (9.82) into Eq. (9.77), we find
τc =qµ0 VDt0(w/l) exp(∆EB/kT )
σ0(8kT/πm∗)1/2I(T )T 1/2, (9.83)
and thereforeI(T )T τc =
exp(∆EB/kT )σ0χ
, (9.84)
where χ is a constant and is given by
χ =(8k/πm∗)l/2
qµ0 VDt0(w/l). (9.85)
Equations (9.51), (9.77) and (9.78) can be combined to give
τe =exp(∆EB/kT ) exp{−[E(n + 1/n)− EF]/kT}
σ0g(8kT/πm∗)1/2n(T ). (9.86)
236
In expanding Eq. (9.86), we shall choose to write the mean inversion-layer number densityas
n(T ) = NC exp(−EC − EF
kT
), (9.87)
where EC is the energy of the conduction band at the inversion-layer charge centroid (notat the surface) and NC is the effective density of states. Thus
τe =exp(∆EB/kT ) exp{[EC −E(n + 1/n)]/kT}
σ0g(8kT/πm∗)1/2NC. (9.88)
Now NC can be expressed as NC0T3/2, and therefore Eq. (9.88) becomes
T 2τe =exp[∆EB + ∆ECT/kT ]
σ0gη, (9.89)
whereη = NC0(8k/πm∗)1/2 , (9.90)
∆ECT = EC − E(n + 1/n) . (9.91)
Figure 9.16: Temperature dependence of capture time (◦) and emission time(•) for device H21. VG = 3 V, VD = 50 mV.
B. Temperature dependence of RTSs in silicon MOSFETs (II): Results
The two equations to be used in investigating the temperature dependence of the meancapture and emission times are Eqs. (9.84) and (9.89). From plots of ln[I(T )T τe] and of
237
ln[T 2τe] against 1/T (see Fig. 9.16) the energies ∆EB and ∆ECT can be obtained and fromthe intercepts two independent estimates of σ0 (assuming a nominal value for g of unity)recovered. We found two notable discrepancies using this approach: first the independentestimates of σ0 differed by typically two to three orders of magnitude; and secondly, thevalue of ∆ECT was in general around 0.2 eV greater than estimates of EC−EF, placing thetrap occupancy level E(n+1/n) well below the surface Fermi level. However, if the energylevel were at this position then the defect’s occupancy would not fluctuate on accessibletime scales. Note also that this result cannot be explained by two-electron capture, sinceit would place the average energy level 1
2 [E(2/1) + E(1/0)] well below the Fermi level andthe defect would remain in its two-electron state.
Engstrom and Alm noted that in thermal experiments it is Gibbs free-energy changesthat are measured, and the energy ∆ECT should be split into its corresponding enthalpy∆HCT and entropy ∆S components, namely[16]
∆ECT = ∆HCT − T∆S . (9.92)
An alternative way of viewing this equation is to say that the trap energy level is temperature-dependent. We have used a first-order expansion and assumed that, within the smalltemperature range accessed, ∆HCT is constant. Substituting Eq. (9.92) into Eq. (9.89),we find that
T 2τe =exp(−∆S/k) exp[(∆EB + ∆HCT)/kT ]
σ0η, (9.93)
where g has been incorporated into the exp(−∆S/k) term. Thus, from the intercept andgradient of the plot of Eq. (9.93), the change in trap entropy and the enthalpy of ionizationcan be determined.
C. Temperature dependence of RTSs in silicon MOSFETs (III): Discussion
There are two features of the experimental results the wide range, 10−20 − 10−15 cm2,of cross-section pre-factors σ0; and the wide range, 0.19- 0.65 eV, of energy barriers ∆EB.The range of activation energies is consistent with capture into defects in an amorphousmaterial, with its consequent continuous distribution of trap environments. The pre-factorσ0 reflects the nature (in terms of symmetry, degeneracy and so on) and overlap of theinitial- and final-state wave-functions. Thus the distribution of values of σ0 is compatiblewith the traps being distributed over a range of distances into the oxide.
A particularly interesting aspect of the results is the determination of the entropiesof ionization ∆S of individual Si/SiO2 interface defects; a representative value for ∆S isabout 5k. In order to place these results in context, it is useful to consider, as an illustrativeexample, the temperature dependence of the band gap of silicon. The forbidden gap ∆ECV
of a semiconductor corresponds to the increase in Gibbs free energy upon increasing thenumber of electron-hole pairs np by one at constant temperature and pressure:
∆ECV ≡ ∂G
∂nP
∣∣∣∣T,P
.
∆ECV may be decomposed into standard enthalpy and entropy components. It is anexperimental fact that the band gap of silicon decreases with increasing temperature:
238
the entropy change has a surprisingly large value of 2.9 k at room temperature. Thereason for the entropy change lies in the fact that on creation of an electron-hole pairan electron is removed from a bonding valence-band state and placed in an anti-bondingconduction-band state. This process softens the lattice, leading to a decrease in phonon-mode frequencies. In particular,
∆SCV(T ) ≈ k∑
i
ln
(ωi
ω′i
)for kT > hωi ,
where ωi and ω′i represent the lattice vibration frequencies before and after creation of an
electron-hole pair. It is concluded that the large value of ∆S for silicon was the result ofeach n-p pair softening the lattice vibrations by an amount equivalent to the removal ofseveral bonds.
D. The gate-voltage dependence of trapping into individual oxide defects inMOSFETs (revisited)
We should now like to turn our attention again to the behavior of RTSs as gate bias isvaried. We discussed the behavior of the mark-space ratio of the RTS with gate voltage andconsidered the arguments for and against the standard two-level RTS being due to multi-electron trapping. Here, we wish to take the single-electron capture model and addressthe following three areas: the gate-voltage dependence of the capture time; estimates ofthe trap depth into the oxide for the device operating around threshold; and the behaviorof the emission time with gate voltage.
On the basis of Eqs. (9.74) and (9.75), it would be expected that the change in capturetime on increasing gate voltage would come about predominantly from changes in n, theinversion-layer number density (per unit volume). This assumes an invariant cross-section.Equation (9.79) predicts that as the gate bias increases, n is roughly proportional to thecurrent I. Figure 9.17 shows for one particular RTS that as the gate voltage is changedfrom 2.25 to 3.5 V the current increases by a factor of 1.8, while there is a correspondingdecrease in τc by a factor of 1.57. Thus the capture time is decreasing much more rapidlythan predicted by Eqs. (9.75) and (9.79). This is quite a general phenomenon and is byno means restricted to the RTS of Fig. 9.17. This particular RTS was singled out fordetailed investigation because it showed a more marked effect than most. To accountfor the data of Fig. 9.17 on the basis of Eqs. (9.75) and (9.79), it is required that theproduct µt decrease by a factor of nine over the gate-voltage range 2.25-3.5 V. However,the inversion-layer thickness changes by no more than a factor of two and the mobility isfound to change by 8%. Thus variations in µ and t alone are unable to account for thedata presented in Fig. 9.17.
In a bulk solid, where the free carriers uniformly bathe a defect, the definition τc =(συn)−1 is perfectly sensible. In the case of a defect in the oxide of an MOS structurethere are the following two problems: first, the inversion-layer charge is displaced fromthe defect site: and second, as the gate voltage changes so does the electric field in theinversion layer. As the electric field strength increases, the inversion-layer charge-densitypeak moves closer to the interface, thus increasing wavefunction overlap. Our originalapproximation of a uniform inversion-layer charge density becomes less valid. In addition,the changing oxide field strength also lowers the tunneling barrier. The net result is an
239
Figure 9.17: Gate-voltage dependence of the capture (©) and emission (4)times for device H21 at T = 320 K. ( ) indicates the drain current.
increase in the value of σ0. To quantify this explanation at room temperature will bea very difficult task, but it may be possible from the results of low-temperature studies,where only one sub-band is filled and for which model calculations of the electronic statesare more tractable.
9.5.2 Metal-insulator-metal tunnel junction
We described the decomposition of 1/f noise into its individual fluctuating RTS compo-nents. Here we wish to describe the nature of the dynamics of these fluctuations.
In sufficiently small-area devices the capture of a single electron in the insulator givesrise to a discrete and measurable change in the tunnel resistance. As for the case ofcarrier trapping at Si/SiO2 interface states, the process can be described by the two timeconstants τc and τe representing the time for electron capture (trap filling) and electronemission (trap emptying) respectively. In the frequency domain the power spectrum of aresistance change ∆R is given by (see Eq. (9.41))
SR(f) =4(∆R)2
(τe + τc)[(1/τe + 1/τc)2 + (2πf)2]. (9.94)
The total integrated power S1 is then
S1 =(∆R)2
(τe + τc)[1/τe + 1/τc). (9.95)
240
Writing (∆R)2 = S0 and1
τeff=
1τe
+1τc
(9.96)
so as to follow the notation of Rogers and Buhrman[17], we find
S1 =S0τeff
τe + τc
=S0τeτc
(τe + τc)2. (9.97)
¿From the roll-off frequency of the Lorentzian power spectrum of the fluctuations,Rogers and Buhrman were able to determine τeff for a given set of operating conditions.Above temperatures of about 15 K, they found 1/τeff to increase in a thermally activatedmanner; at the same time S1 varied much more slowly with temperature: see Fig. 9.18.The behavior is consistent with both τe and τc being thermally activated, see Eqs. (9.95)and (9.97). They chose to write the temperature dependence of τc and τe in the form
1τi
=1τ0i
exp(−EBi
kT
). (9.98)
Using Eqs. (9.96)–(9.98), they obtained a fit to the data shown as the solid lines in Fig. 9.18.From the fits, they recovered values for the activation energies EBi, attempt frequenciesτ0i and the resistance change S0.
241
Figure 9.18: Plot of log log τeff and log S1 against 1/T , showing thermallyactivated behavior. Solid lines are fits assuming Eqs. (9.97) and (9.98). Theinset shows a two-well model describing the rate-limiting kinetics.
242
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