Low-frequency noise spectrum of cyclo-stationary random telegraph signals

Transcription

1 Electr Eng (28) 9: DOI.7/s y ORIGINAL PAPER Low-frequency noise spectrum of cyclo-stationary random telegraph signals Gilson Wirth Roberto da Silva Received: 2 September 27 / Accepted: 22 November 27 / Published online: 8 January 28 Springer-Verlag 27 Abstract The noise spectrum of random telegraph signals (RTS) under cyclo-stationary excitation is evaluated through rigorous analytical calculations. First the autocorrelation of the RTS signal is calculated, and then the Wiener Khinchin formula applied, leading to an analytical formulation for the RTS spectrum. The model is valid for any periodic excitation signal. Numerical simulations that corroborate the analytical results and explore the noise behavior under cyclo-stationary excitation are provided. Keywords Low-frequency noise Random telegraph signal Semiconductor devices Introduction Technology scaling made CMOS technology the mainstream choice to implement analog, mixed-signal and RF integrated circuits. However, as device sizes become smaller, the relevance of low-frequency behavior increases. Low-frequency (LF) noise power spectral density is known to increase as device area decreases, and the signal to noise ratio decreases. Furthermore, the /f-noise corner frequency increases with each technology node and poses great challenges to the circuit designers [ 3]. As a consequence, with device G. Wirth (B) Departamento de Engenharia Eletrica, UFRGS, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 3, Porto Alegre, RS, Brazil wirth@ece.ufrgs.br R. da Silva Instituto de Informatica, UFRGS, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 95, Porto Alegre, RS, Brazil rdasilva@inf.ufrgs.br dimensions being scaled down, LF noise plays a larger role in limiting circuit performance. For robust circuit design it is crucial to understand in detail and properly model the device noise behavior in practical operation conditions. As in many practical applications the MOS device is not biased at steady state, but periodically switched, low-frequency noise under cyclo-stationary excitation is of great interest. The switching signal may be a square wave or a sine wave, for instance. Stationary low frequency models cannot properly predict the noise behavior under cyclo-stationary excitation. Low frequency noise is relevant not only in low frequency applications, but also for circuits that operate at high frequencies, since the low frequency noise is up converted and may dominate the noise behavior of widely used circuit blocks, such as oscillators, mixers, modulators and frequency converters. In deep sub-micron technologies, low frequency noise is dominated by random telegraph signals (RTS). Although low frequency noise under cyclo-stationary excitation has received considerable attention [ 6], general models, valid for any excitation frequency and signal, are not yet available. In this work an analytical model that restores the generality of the original derivation by Machlup [7] for the noise spectrum of stationary RTS is derived, and the noise behavior at different excitations frequencies is explored numerically. 2 Analytical model for cyclo-stationary RTS The origin of RTS noise is the capture and subsequent emission of charge carriers at discrete trap levels near the Si SiO 2 interface [ 3]. Figure depicts the cross section of an n-channel MOSFET through the location of the interface trap. The influence of the traps on the electrical current

2 436 Electr Eng (28) 9: I D Gate Oxide Inversion layer Trap Interaction with inversion layer Fig. Schematic cross section of the inversion layer of a MOS transistor through the location of an interface trap. If the trap is electrically charged the inversion layer is disturbed by the trap, affecting the drain current I D. The trap not only affects the number of free carriers in the inversion layer but is also a source of electrical charge carrier scattering The power spectrum of a RTS fluctuation is a Lorentzian, as depicted in Fig. 2. In model derivation, we follow the methodology originally proposed by Machlup [7] for stationary RTS. A RTS is a purely random signal which may be in one of two states, called and. If the signal is in state, the probability of making a transition to in a short time dt is assumed to be dt/σ (t). If the signal is in state, the probability of making a transition to is assumed to be dt/τ(t). For stationary RTS, σ(t) and τ(t) are constants, i.e., independent of time t. For cyclo-stationary RTS, σ(t) and τ(t) are periodic functions of time. Symbols and acronyms used in the text are detailed in Table. In order to derive the LF Noise Spectrum of Cyclo- Stationary RTS, we first calculate the autocorrelation of the RTS, and then apply the Wiener Khinchin formula to obtain the spectrum. Let the RTS signal be x(t). The autocorrelation is then given by A(s) = x(t) x(t + s) average = P(x(t) = ) P (s) () S(log) where P (s) is the probability of an even number of transitions in time s, given we start in state. P(x(t) = ), the probability of being in state at time t, is given by Current δι d σ τ Time f(log) Fig. 2 Time and frequency domain representation of a stationary random telegraph signal (RT S). In frequency domain, the power spectrum of a RTS is a Lorentzian. In time domain discrete fluctuations are observed in the drain current, where σ is the average time in the high current state, which corresponds to the state where the trap is electrically neutral (empty). τ is the average time in the low current state, which corresponds to the state where the trap is electrically charged. δi d is the amplitude of the current fluctuation flowing through the channel is twofold. On the one hand, the occupation of a trap changes the number of free carriers in the inversion layer. On the other hand, a charged trap state has an influence on the local mobility near to its position due to Coulomb scattering. If the MOSFET biasing is kept constant, a stationary RTS is observed at the terminals of the device as a discrete fluctuation in electrical current, δ I d being the amplitude of the current fluctuation, as shown in the inset of Fig. 2. The average high current time corresponds to the electron capture time constant (σ ), and the average low current time corresponds to the emission time constant (τ). P(x(t) = ) = = T T µ(t)dt T T µ(t)dt + T T λ(t)dt where T is the period of the cyclo-stationary excitation, and µ(t) = /τ(t) and λ(t) = /σ (t). Calculation of P (s) leads to P (s) = e s (λ(y)+µ(y))dy + s x e (λ(y)+µ(y))dy µ(x)dx (2) e s (λ(y)+µ(y)dy. (3) The autocorrelation can then be calculated as A(s) = s x + e (λ(y)+µ(y)dy µ(x)dx e s (λ(y)+µ(y))dy. This formulation for the autocorrelation is a generalization of the Machlup formula, and is valid for any kind of periodic excitation and any frequency. If σ(t) and τ(t) become constant, µ = /τ(t) and λ = /σ (t) also become constants, and (4) becomes equal to (7)in[7]. (4)

3 Electr Eng (28) 9: Table Symbols used in the text and in the equations LF Low frequency MOSFET Metal-oxide-semiconductor field-effect transistors psd Power spectral density RTS Random telegraph signal T Period of the cyclo-stationary excitation signal. Duty cycle, i.e., fraction of the switching period the bias is in the state labeled on σ (t) Mean life-time of the state τ(t) Mean life-time of the state µ Inverse of τ(t), i.e., equal to /τ(t) λ Inverse of σ(t), i.e., equal to /σ (t) σ on Mean life-time of the state during the on cycle, for square wave excitation τ on Mean life-time of the state during the on cycle, for square wave excitation σ off Mean life-time of the state during the off cycle, for square wave excitation τ off Mean life-time of the state during the off cycle, for square wave excitation ω s Angular frequency of the cyclo-stationary excitation signal ω cyc Angular corner frequency of cyclo-stationary RTS psd 3 Approximation for high excitation frequencies If the excitation frequency is high an approximation can be made. This case corresponds to the limit where the transition probabilities dt/σ (t) and dt/τ(t) are much smaller than the period T of the cyclo-stationary excitation signal. In this case we can write, without loss of generality, that s = nt, where n is a positive integer (, 2, 3,...). In this casewehave s o (λ(y) + µ(y))dy = which leads to A(s) = n i= e s() (i+)t it n λ(y)dy + i= (i+)t µ(y)dy = nt( λ(t) +) (5) [ T e T () T + e T (). (6) For small values of T a Taylor expansion may be done, leading to [ A(s) = ] e s() 2 + () 2. (7) This means that, in the case where the transition probabilities dt/σ (t) and dt/τ(t) are much smaller then the excitation period T, the values of µ(t) and λ(t) in the integrals of (4) are equivalent to their time averages and λ(t). ] it The power spectrum S(ω) is the Fourier transform of the autocorrelation S(ω) = 2π A(s)e iωs ds (8) which leads to S(ω) = λ(t) T π ω cyc ωcyc 2. (9) + ω2 The cyclo-stationary noise spectrum is still Lorentzian, with angular corner frequency ω cyc given by ω cyc =. () This takes us to the conclusion that making a RTS signal cyclo-stationary leads to a Lorentzian spectrum with corner frequency equal to the sum of the inverse time average values of the capture and emission times. Please note that for stationary RTS the corner frequency is equal to the sum of the inverse values of the constant capture and emission times (please refer to equation 9 in [7]). The result for this limit is valid for any kind of periodic excitation. This is the limit studied in [] and [6]. However, for this limit, we obtain the same result in a much simpler derivation than in [] and [6], and without making any further assumption or simplification. The single assumption is that the transition probabilities dt/σ (t) and dt/τ(t) are much smaller than the excitation period T. Equation (9) is equivalent to (9) in[] and (37) in [6], except for the pre-factor /π instead of 2. Please note that although in [] and [6] the power spectral density is written as a function of ω, their pre-factors correspond to power spectral density per Hz. The formulation here proposed recovers the correct pre-factor for ω in rad/s, in accordance to the

4 438 Electr Eng (28) 9: original derivation by Machlup [7].For ω in rad/s, equations in [] and [6] over estimate S(ω) by a factor of 2π. Equations (7) and (9) are a generalization of the Machlup formulation for cyclo-stationary RTS with high excitation frequency. The Machlup equations for the auto-correlation and power spectrum are recovered if we consider µ = /τ and λ = /σ, i.e., constant, not time dependent values. 4 Case studies An important case study for cyclo-stationary RTS is square wave excitation. Besides being of interest for many practical applications, for this case experimental and numerical results are available in the literature [ 5,9,]. If the cyclo-stationary excitation waveform is known, autocorrelation may be explicitly calculated starting from (4), and the frequency domain spectrum derived applying the Wiener Khinchin formula. We start without making at this point the approximation for high excitation frequency. After the full derivation of the power spectrum is performed, the limit for high excitation frequency can be calculated. For the sake of clarity, in the model equations and numerical results here shown the dc alias and harmonics of the modulating frequency are omitted. At this point, we derive the equation without making the assumption that the transition probabilities dt/σ (t) and dt/τ(t) are much smaller than the excitation period T.In square wave excitation, the bias voltage abruptly alternates between two states, called on and off. Please note that the state names on and off do not mean that the device has necessarily to be periodically turned on and off. The names may refer, for instance, to states with high and low gate bias. Hence, σ(t) may assume two values, σ on in the on state, and σ off in the off state. Similarly, τ(t) assumes the value τ on in the on state, and τ off in the off state. The fraction of the period T in which the device is in the on state (i.e., the duty cycle) is. During the on state the RTS behavior is governed by σ on and τ on. During the off state the RTS behavior is governed by σ off and τ off. After evaluation of the autocorrelation, the power spectrum can be evaluated, leading to S(ω) = π τ on + ( ) [ τ off ωcyc 2 + ω2 2 2 σ on σ on +τ on (e T ω on ) e T (ω on+( )ω off ) σ off σ off +τ off e T (ω on ω off ) [ e T ω off e T ω ] off e T (ω on+( )ω off ) σ on σ on +τ on e ( )T (ω off ω on ) [ e T ω on e ( )T ω ] on 2 e T (ω on+( )ω off ) σ off σ off +τ off (e ( )T ω ] off ) 2 e T (ω on+( )ω off ) () where ω cyc = ω on + ( )ω off = (2) ω on = σ on + τ on = µ on + λ on and τ on σ on ω off = σ off + τ off = µ off + λ off. (3) σ off τ off Equation () above is valid only for small values of T. If T is made large compared to the mean capture and emission times (i.e., ω s <ω on,ω off ), basic modulation theory applies, and the Spectrum becomes simple the superposition of two Lorentzian Spectra [8]. One Lorentzian corresponding to the RTS signal with time constants λ on = /σ on and µ on = /τ on, and corner frequency ω on = λ on + µ on.the other Lorentzian corresponding to the RTS signal with time constants λ off = /σ off and µ off = /τ off, and corner frequency ω off = λ off + µ off. Applying l Hôpital s rule to equation () above, the limit for small values of T (i.e., ω s ω on,ω off ) is calculated as being S(ω) = τ on + ( ) ( τ off τ T π ωcyc 2 + on + ) τ off ω2 ω cyc = π ωcyc 2 + ω2 ( ω cyc ). (4) For square wave excitation, equation (4) is equivalent to (9). This means that (4) is the particular case of (9) for square wave excitation. To explore the behavior of cyclo-stationary RTS and validate the model, we studied square wave excitation in detail, through numerical simulations and comparison to experimental results available in the literature. There are experimental results available for the limit ω s ω on,ω off [ 5,9,]. Furthermore, in this limit the model here presented is equivalent to the models previously published in [] and [6], where only this limit (ω s ω on,ω off ) is studied. To perform numerical cyclo-stationary noise simulations, we first generate the time domain RTS using Monte Carlo techniques, in an approach as described in [2], except by the fact that here the RTS is not assumed to vanish in the off state of the switching signal. The Fourier Transform is then numerically evaluated, from which we can obtain the noise psd. To simulate the dependence of the mean life times σ and τ on bias point, a factor m is introduced [2]. Experimental results show that as the gate voltage is decreased σ increases and τ decreases, i.e., the probability of a trap becoming filled decreases and the probability of a trap becoming empty increases (in n-channel MOSFETS). Hence, in the off state σ is multiplied by m, while τ is divided by m. The factor m is assumed to be greater than one. This simple and insightful model changes the effective time constants of the RTS signal as it is switched between the on and off states. For

5 Electr Eng (28) 9: Table 2 Parameters used in the case study numerical simulations for square excitation, where device bias is periodically cycled between two phases, labeled on and off Clock phase on off Mean life-time of the state σ on = σ σ off = σ m Mean life-time of the state τ on = τ τ off = τ/m Fraction of the period T Noise PSD [Arbitrary Units] ω [rad/s] Stationary (m=) m= m=5 Fig. 3 Effect of making a symmetric (i.e., σ/τ = ) RTS cyclo-stationary. Numerical simulation results: (filled circle) correspond to m =, i.e., steady state bias (RTS not modulated); (pluse symbol) correspond to m = ; (multi symbole) correspond to m = 5. Full lines: respective results for S(ω) as given by model equation (4). Duty cycle is equal to.5 the parameters of the numerical simulations, please refer to Table 2. The data in Figs. 3 and 4 show the results from numerical simulation and analytical equation (4). As can be seen in Fig. 3, making a symmetric RTS cyclo-stationary (i.e., σ/τ = ) always leads to a reduction in noise power. Figure 4 shows that making an asymmetric RTS (i.e., σ/τ = ) cyclo-stationary may lead to either increase or reduction in noise power. This data is in line with the numerical simulations and experimental results presented in [2]. Further results comparing analytical equations to numerical simulation and exploring noise behavior are shown in Figs. 5, 6 and 7. These results are also in agreement to previous observations [ 5,9,]. Figures 6 and 7 show that under square wave excitation the integrated psd of cyclo-stationary noise depends on duty cycle. For a RTS that is asymmetric when stationary, the integrated noise psd may increase or decrease under cyclostationary excitation, depending on duty cycle.an equal to one corresponds to the stationary condition. The duty cycle affects the averages and λ(t). If the ratio / λ(t) is equal to one (i.e., = λ(t) ), the Noise PSD [Arbitrary Units] Stationary (m=) m= m= ω [rad/s] Fig. 4 Effect of making an asymmetric (i.e., σ/τ = ) RTS cyclostationary. Numerical simulation results: (filled circle) correspond to m =, i.e., steady state bias (RTS not modulated); (plus symbol) correspond to m = ; (multi symbol) correspond to m = 5. Full lines respective results for S(ω) as given by model equation (4). Duty cycle is equal to.5 Angular Frequency [rad/s] 2 - Analytical Numerical m Fig. 5 Angular corner frequency ω cyc as a function of m, forarts that is symmetric (i.e., σ/τ = ) at steady state bias. m = corresponds to steady state bias. Full line ω cyc as given by equation (2); filled circles numerical simulation. For this relation between σ and τ in the on and off states, the ω cyc always increases as the RTS is made cyclo-stationary. For asymmetric RTS, making it cyclo-stationary may increase or decrease ω cyc noise power is maximized. The cyclo-stationary RTS time constants and λ(t) as a function of varying dutycycle have been studied, for instance, in [9]. For a RTS that is asymmetric under steady state, there may be a value of that leads to = λ(t), maximizing noise power. For instance, see Fig. in [9]. If the period T of the excitation signal is not much shorter than capture and emission times, low frequency noise behavior is no longer described by equations (9), () and (4). In the case where the period of the excitation signal is not much shorter than capture and emission times, the RTS spectrum becomes a superposition of the spectrum in the on and off

6 44 Electr Eng (28) 9: Integrated Noise PSD [Arbitrary Units] ,,2,4,6,8, Noise PSD [Arbitrary Units] -5-6 Small T Large T Fig. 6 Noise power S(ω) as given by equ. (4) integrated from ω = to ω as a function of duty cycle, for a RTS that has σ = τ =.6 (i.e., is symmetric) in the on state. In the off state σ =.6 5and τ =.6/5, i.e., m = 5. Note that for decreasing from to the integrated noise power steadily decreases Integrated Noise PSD [Arbitrary Units] ,,2,4,6,8, Fig. 7 Noise power S(ω) as given by equ. (4)integratedfromω = to ω as a function of duty cycle, for a RTS that has σ =.4 and τ =.8 (i.e., is asymmetric) in the on state. In the off state σ =.4 5 and τ =.8/5, i.e., m = 5. In this case, for decreasing from to, the integrated noise power shows a maximum at =.82 states. Fig. 8 depicts the low frequency noise for high and low excitation frequencies. As can be seen in Figure 8, for low excitation frequency the power spectrum is no longer a pure Lorentzian, but simply the superposition (sum) of two Lorenzians, each one corresponding to the power spectrum of the RTS signals in the on and off states. The condition that delimits both situations is the relation between ω s and ω on and ω off.ifω s is much greater than both ω on and ω off, the resulting noise spectrum is a pure Lorentzian as given by equation (4). If ω s is lower than both ω on and ω off,the resulting noise spectrum is simply a superposition of the two Lorentzians, as predicted by simple modulation theory [8]. For the Monte Carlo simulations shown in Fig. 8, the corner frequency ω cyc is approximately 3,6 rad/s, ω on is approximately 2,2 rad/s, and ω off is approximately ω [rad/s] Fig. 8 Effect of the angular frequency ω s of the cyclo-stationary excitation signal on the low frequency noise behavior. The excitation signal is a square wave with duty cycle equal to.5. (Filled circle) corresponds to the Monte Carlo simulation results for high excitation frequency ω s = 6 rad/s, with full line showing the results for S(ω) as given by model equ. (4). In this case the psd is a pure Lorentzian. (plus symbol) corresponds to the Monte Carlo simulation results for low excitation frequency ω s = rad/s, with full lines showing the results evaluated using basic modulation theory, i.e., computing the superposition (sum) of two Lorentzians. In this case two inflection points can be seen, one corresponding to the corner frequency of the first Lorentzian (at ω 2, 2 rad/s), and the other one corresponding to the corner frequency of the second Lorentzian (at ω 25, rad/s) 25, rad/s. T large corresponds to an excitation frequency ω s of rad/s, while T small corresponds to an excitation frequency ω s of 6 rad/s. Statistical analysis of noise behavior as described by the model equations here presented leads to the conclusion that average LF noise power will not decrease if σ and τ are log-normal distributed and trap density is uniform over the bandgap. In this case, noise performance remains practically unchanged under periodic excitation. However, if trap densities are U shaped in energy, cyclo-stationary behavior here described will lead to a decrease in the average LF noise for MOSFETs under periodic excitation, in agreement to previous observations [ 5,]. The reason for the LF-noise decrease under cyclo-stationary excitation is twofold: (i) the traps that contribute most to the low-frequency noise are the ones close to the center of the bandgap, where trap density is smaller; (ii) the decreased contribution of traps close to the bandgap edge, where trap density is higher []. 5 Conclusion A rigorous analytical analysis of the noise spectrum of RTS under cyclo-stationary excitation is provided. Numerical simulations to validate the model are performed. The work

7 Electr Eng (28) 9: restores the generality of the original derivation by Machlup [7] for the noise spectrum of stationary RTS. The derived formulation is valid for arbitrary periodic excitation signals. The behavior of the RTS power spectrum at different excitation frequencies is numerically explored. References. van der Wel A, Klumperink EAM, Hoekstra E, Nauta B (25) Relating random telegraph signal noise in metal-oxidesemiconductor transistors to interface trap energy distribution. Appl Phys Lett 87: van der Wel A, Klumperink EAM, Vandamme LKJ, Nauta B (23) Modeling random telegraph noise under switched bias conditions using cyclostationary RTS noise. IEEE Trans Electron Dev 5: Brederlow R, Koh J, Thewes R (25) A physics-based low frequency noise model for MOSFETs under periodic large signal excitation. In: Proc. of the European solid-state dev res conf pp Bloom I, Nemirovsky Y (99) / f noise reduction of metal-oxidesemiconductor transistors by cycling from inversion to accumulation. Appl Phys Lett 58: Dierickx B, Simoen E (992) The decrease of random telegraph signal noise in metal-oxide-semiconductor field-effect transistors when cycled from inversion to accumulation. J Appl Phys 7: Roy A, Enz C (26) Analytical modeling of large signal cyclo-stationary low frequency noise for arbitrary periodic input. In: Proc. of the European solid-state dev res conf pp Machlup S (954) Noise in semiconductors: spectrum of a twoparameter random signal. J Appl Phys 35: Taub H, Schilling DL (986) Principles of communication systems. McGraw-Hill, New York 9. Kolhatkar JS, Hoekstra E, Salm C, van der Wel AP, Klumperink EAM, Schmitz J, Wallinga H (24) Modeling of RTS noise in MOSFETs under steady-state and large-signal excitation. In: IEEE electron devices meeting IEDM, pp Brederlow R, Koh J, Thewes R (26) A physics-based low frequency noise model for MOSFETs under periodic large signal excitation. Solid State Electronics 5: