Low-Frequency Noise Analyzer

1. Theory

• Definition: A low-frequency noise analyzer biases a device with very quiet sources, amplifies the fluctuating current or voltage and computes its power spectral density, and records random telegraph noise in the time domain (Keysight; Balogh et al.).
• Flicker noise: Spectral density falling roughly as $1/f$, explained by carrier-number fluctuations from oxide traps (McWhorter) or mobility fluctuations (Hooge) (Jomaah et al.; Liu et al.).
• Random telegraph noise: In small devices, single traps produce two-level switching with a Lorentzian spectrum (Jomaah et al.).
• Defect probe: Because number-fluctuation noise scales with oxide trap density, noise is a sensitive gate-oxide quality monitor (Magnone et al.; Jomaah et al.).

2. Working Principle

• Quiet bias: Low-noise sourcing and measurement units set the operating point, and cross-correlating two parallel amplifiers suppresses amplifier noise (Balogh et al.).
• Amplification: A voltage or transimpedance amplifier with an input noise near 0.56 $nV/\sqrt{\mathrm{Hz}}$ raises the signal above the analyzer floor (Keysight; computed).
• Spectrum: Sampled records are windowed, Fourier transformed and averaged (Tektronix/Keithley; NI; Balogh et al.).
• Shielding and proximity: The device, probe and conditioning circuits are shielded and placed close together (Keysight).

3. Applications

• Compact models: Extraction of BSIM flicker-noise parameters for circuit design (Liu et al.; Keysight).
• Gate-oxide quality: Oxide trap density in logic and power MOSFETs (Magnone et al.).
• Degradation monitoring: Low-frequency noise has been used to detect degradation in interconnects (Balandin).

Noise Models

Name	Formula	Related Formulae	Variables/Constants
Hooge mobility-fluctuation model	$\frac{S_{I_d}}{I_d^2}=\frac{q{\alpha }_H}{WL{Q}_{i}f}$	$\frac{S_I}{I^2}=\frac{{\alpha }_H}{Nf}(\text{homogeneous sample, }N\text{carriers})$	$\begin{array}{r}{S}_{I_d}:drain-currentnoisepowerspectral\\ density\end{array}$ $I_d:draincurrent$ ${\alpha }_H:Hoogeparameter$ $W,L:channelwidthandlength$ $Q_i:inversionchargeperarea$ $f:frequency$ $\begin{array}{r}N:totalnumberofcarriers;Jomaahet\\ al.:theMOSFETformwith{\alpha }_{H}ofabout1\times 10^{-4}\\ to1\times 10^{-6};Liuetal.:Hoogeparametersof\\ 2.42\times 10^{-5}and4\times 10^{-4}at65\mathrm{nm},PMOS\\ strong-inversionnoisefollowingHooge;\\ Balandin:Hoog{e}^{\prime }sempiricalformula,Nthe\\ numberofcarriers(equationgarbledin\\ extraction)\end{array}$ $\begin{array}{r}(assumed:W=1\mu \mathrm{m},L=0.1\mu \mathrm{m},Q_i=7.5\times 10^{-7}\\ C/c{\mathrm{m}}^2,{\alpha }_H=1\times 10^{-5};computed:S_{I_d}/I_d^2=2.1\times 10^{-10}\\ \mathrm{H}{\mathrm{z}}^{-1}at10\mathrm{Hz})\end{array}$
Carrier-number fluctuation (McWhorter)	$S_{V_{fb}}=\frac{q^2{k}_{B}T\lambda N_t}{WL{C}_{ox}^2{f}^{\gamma }}$	$\frac{S_{I_d}}{I_d^2}={\left(\frac{g_m}{I_d}\right)}^2{S}_{V_{fb}}{\left(1+{\alpha }_c{\mu }_{eff}{C}_{ox}\frac{I_d}{g_m}\right)}^2$	$S_{V_{fb}}:flat-bandvoltagenoise$ $\begin{array}{r}\lambda :tunnellingattenuationlength(about\\ 0.1\mathrm{nm})\end{array}$ $N_t:volumetricoxidetrapdensitypereV$ $C_{ox}:oxidecapacitanceperarea$ $\gamma :frequencyexponentnear1$ $g_m:transconductance$ ${\alpha }_c:Coulombscatteringcoefficient$ $\begin{array}{r}{\mu }_{eff}:effectivemobility;Jomaahetal.:S_{V_{fb}}\\ equationanditsinterface-trapvariant;\\ Liuetal.:NMOSnoisefollowing\\ McWhorter,trapdensities3.94\times 10^{17}and\\ 3.56\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}\mathrm{e}{\mathrm{V}}^{-1};Omura:(g_m/I_d{)}^{2}scaling\\ withthecorrelatedmobilityterm,{\alpha }_c\\ empirical\end{array}$ $\begin{array}{r}(assumed:\lambda =0.1\mathrm{nm},C_{ox}=1.5\times 10^{-6}F/c{\mathrm{m}}^2,\\ sameWandL;computed:N_t=3.94\times 10^{17}\\ \mathrm{c}{\mathrm{m}}^{-3}\mathrm{e}{\mathrm{V}}^{-1}gives{S}_{V_{fb}}=1.2\times 10^{-10}{\mathrm{V}}^2/Hzat10\\ \mathrm{Hz},or10.8\mu V/\sqrt{\mathrm{Hz}})\end{array}$
Random telegraph noise	$S_{I_d}=\frac{4A\Delta I_d^2\tau }{1+{\omega }^2{\tau }^2},\tau ={\left(\frac{1}{{\tau }_c}+\frac{1}{{\tau }_e}\right)}^{-1},A=f_t\left(1-f_t\right)$	$\frac{\Delta I_d}{I_d}=\frac{g_m}{I_d}\frac{q}{WL{C}_{ox}}\left(1-\frac{x_t}{t_{ox}}\right),{\tau }_c=\frac{1}{\sigma n_s{v}_{th}}$	$\Delta I_d:switchingamplitude$ ${\tau }_c,{\tau }_e:captureandemissiontimes$ $f_t:trapoccupancy$ $\omega :angularfrequency$ $x_t:trapdepthfromtheinterface$ $t_{ox}:oxidethickness$ $\sigma :capturecrosssection$ $n_s:surfacecarrierdensity$ $\begin{array}{r}{v}_{th}:thermalvelocity;Jomaahetal.:these\\ equations,{\tau }_e=1/(\sigma n_1{v}_{th});Keysight:RTN\\ measuredinthetimedomainalongside\\ spectra\end{array}$ $\begin{array}{r}(assumed:g_m/I_d=10{\mathrm{V}}^{-1},trapatthe\\ interface;computed:\Delta I_d/I_d=0.11per\\ centfor1by0.1\mu \mathrm{m};\tau =1m/sputsthe\\ Lorentziancornerat159\mathrm{Hz})\end{array}$

Measurement Chain

Name	Formula	Related Formulae	Variables/Constants
Instrument noise floor	$S_{V,floor}=-185\mathrm{dB}{\mathrm{V}}^2/Hz=3.2\times 10^{-19}{\mathrm{V}}^2/Hz$	$S_{V,R}=4k_{B}TR,S_{I,R}=\frac{4k_{B}T}{R}$	$S_{V,floor}:amplifierinputvoltage-noisefloor$ $\begin{array}{r}{S}_{V,R},S_{I,R}:thermalvoltageandcurrent\\ noiseofaresistorR;Keysight:-185\\ \mathrm{dB}{\mathrm{V}}^2/Hzsensitivity,upto200\mathrm{V},currents\\ upto1\mathrm{A},measurementatultra-low\\ frequenciesstatedas30MHz(seethe\\ Q/A),signalconditioningclosetothe\\ device,turnkeyproberdrivers;Baloghet\\ al.:S_V=4k_{B}TR,with{S}_I=S_V/R^2(computed)\end{array}$ $\begin{array}{r}(computed:-185\mathrm{dB}{\mathrm{V}}^2/Hzis0.56nV/\sqrt{\mathrm{Hz}},the\\ thermalnoiseofa19\Omega resistorat300\mathrm{K};\\ a1\mathrm{M\Omega }loadadds1.7\times 10^{-26}{\mathrm{A}}^2/Hz)\end{array}$
Spectrum estimation	$S\left(f_k\right)=\frac{2}{N{f}_s}{\left|{\sum }_{n=0}^{N-1}{x}_n{w}_n{e}^{-j2\pi kn/N}\right|}^2(\text{one-sided, unit-power window})$	$\frac{\delta S}{S}\approx \frac{1}{\sqrt{M}},f_{min}\approx \frac{1}{T_{rec}}$	$S(f_k):powerspectraldensityatbink$ $x_n:sampledcurrentorvoltage$ $w_n:window$ $f_s:samplingrate$ $N:pointsperrecord$ $M:numberofaveragedrecords$ $T_{rec}:recordlength$ $\begin{array}{r}{f}_{min}:lowestresolvablefrequency;NI:Welch\\ averagingofwindowedperiodogramsreduces\\ variance;Baloghetal.:FFTaveragedover\\ subsequenttimetraces;one-sided\\ normalizationfromParseva{l}^{\prime }stheorem\\ (computed),1/\sqrt{M}scatterand{f}_{min}\approx 1/T_{rec}(my\\ reading);Tektronix/Keithley:SMUandPMU\\ samplingwithbuilt-inFFTfor1/fnoise\\ onthe4200A-SCS(introductiononly)\end{array}$ $\begin{array}{r}(computed:100averagesgive10percent\\ scatter;reaching30\mathrm{mHz}needsrecordsof\\ about33\mathrm{s})\end{array}$

Parameter Extraction

Name	Formula	Related Formulae	Variables/Constants
BSIM flicker-noise parameters	$NOIB\propto \frac{1}{V_{GS}-V_T}$	$NOIA=\text{constant}(\text{both 65 nm devices})$	$\begin{array}{r}NOIA,NOIB:BSIMunifiedflicker-noise\\ parameters,V_{GS}-V_T:effectivegatevoltage;\\ Liuetal.:NOIAconstant,NOIB=9.31\times 10^{-4}\\ {\mathrm{V}}^{-1}forPMOSand1.53\times 10^{-2}{\mathrm{V}}^{-1}forNMOSat65\\ \mathrm{nm},inverselyproportionaltoeffective\\ gatevoltage;Keysight:DC,flickernoise\\ andRTNdataexportedtomodellingtools\end{array}$ $\begin{array}{r}(computed:NOIBdiffersbyafactorof16\\ betweenthetwodevices)\end{array}$
Defect density from power-device noise	$N_t=\frac{WL{C}_{ox}^{2}f{S}_{V_{fb}}}{q^2{k}_{B}T\lambda }$	$S_{V_{fb}}=\frac{S_{I_d}}{g_m^2}(\text{number fluctuation dominant})$	$\begin{array}{r}Jomaahetal{.}^{\prime }s{S}_{V_{fb}}relationrearranged\\ (computed);Magnoneetal.:alow-noise\\ set-upmeasuringdrain-currentflicker\\ noiseoftrenchpowerMOSFETs,the\\ fluctuationmodelchosenempiricallyand\\ thedefectdensityextractedfromit,in\\ linewithvaluesexpectedforthe\\ technology\end{array}$ $\begin{array}{r}(computed:therearrangementinvertsthe\\ S_{V_{fb}}example,returning3.94\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}\mathrm{e}{\mathrm{V}}^{-1})\end{array}$

Important Questions

Ques: What does a low-frequency noise analyzer measure?

The spectrum and time behaviour of tiny current or voltage fluctuations in a biased device. The device is biased from very quiet sources, its current fluctuation is amplified, and the sampled signal is Fourier transformed into a power spectral density, averaged over many records (NI; Balogh et al.); with 100 averages the scatter is about 10 per cent if it falls as one over the square root of the number of averages (my reading). Keysight’s description includes flicker-noise spectra and random telegraph noise in the time domain, at package and wafer level, up to 200 $\mathrm{V}$ and 1 $\mathrm{A}$, with a sensitivity of -185 $\mathrm{dB}{\mathrm{V}}^2/Hz$, 0.56 $nV/\sqrt{\mathrm{Hz}}$ (computed). It recommends keeping external noise out and placing the probe and signal conditioning close to the device. Results feed compact models and oxide-quality studies (Keysight; Magnone et al.). Reference: NI, ‘The Welch Method’, LabVIEW Advanced Signal Processing Toolkit manual: variance of the periodogram reduced by averaging windowed periodograms of overlapping segments, window length trading bias against variance (fetch-tool summary); Z. Balogh et al., ‘1/f noise spectroscopy and noise tailoring of nanoelectronic devices’, arXiv:2106.02683: S_V = 4k_{B}TR, low-noise sourcing and measurement units, cross-correlation of two parallel amplifiers, FFT averaged over subsequent time traces (fetch-tool summary); Keysight, ‘How to Measure Low-Frequency Noise Accurately’ solution page (fetch-tool summary); P. Magnone, P. A. Traverso, G. Barletta and C. Fiegna, ‘Low-Frequency Noise Measurements in Silicon Power MOSFETs as a Tool to Experimentally Investigate the Defectiveness of the Gate Oxide’, IMEKO TC4 Symposium 2013 (abstract) (fetch-tool summary).

Ques: How are number and mobility fluctuations told apart?

By how the normalized noise scales with bias. In the number-fluctuation model, trapping in the oxide modulates the flat-band voltage, so $S_{I_d}/I_d^2=(g_m/I_d{)}^2{S}_{V_{fb}}$, and the normalized noise tracks $(g_m/I_d{)}^2$ from weak to strong inversion (Jomaah et al.; Omura). In the Hooge model, $S_{I_d}/I_d^2=q{\alpha }_H/(WL{Q}_{i}f)$ falls as $1/Q_i$, roughly as one over the gate overdrive (Jomaah et al.; Balandin gives Hooge’s general form in terms of the number of carriers). Plotting both against drain current shows which fits; a correlated mobility term adds a strong-inversion rise. Liu et al. found 65 $\mathrm{nm}$ NMOS noise followed carrier-number fluctuation and PMOS strong-inversion noise followed Hooge’s model, a common pattern for surface-channel n- and p-MOSFETs (my reading). Reference: J. Jomaah, F. Balestra and G. Ghibaudo, ‘Low frequency noise in advanced Si bulk and SOI MOSFETs’, Journal of Telecommunications and Information Technology (year not stated, about 2005) (fetch-tool summary); Y. Omura, ‘Empirical and Theoretical Modeling of Low-Frequency Noise Behavior of Ultrathin SOI MOSFETs…’, Micromachines 10(1), 5, 2018: S_ID/I_D^2 = (1 +- alpha_c mu_eff C_OX I_D/g_m)^2 (g_m/I_D)^2 S_VFB, alpha_c empirical; S_VFB = q^2 k_{B}T lambda N_t/(W L C_OX^2 f^gamma) (fetch-tool summary); A. A. Balandin, ‘Review of the Low-Frequency 1/f Noise in Graphene Devices’, arXiv:1307.4797, 2013: Hooge parameter based on his empirical formula (Eq. B3, garbled in extraction), N the number of charge carriers, S_R/R^2 = S_I/I^2 = S_V/V^2, noise used to detect degradation in interconnects (fetch-tool summary); Y. Liu et al., ‘Low-frequency noise characteristics in the MOSFETs processed in 65 nm technology’, J. Semicond. 37, 064012 (2016) (abstract) (fetch-tool summary).

Ques: How is the oxide trap density extracted from noise?

From the flat-band voltage noise. Dividing the drain-current noise by $g_m^2$ gives $S_{V_{fb}}$, and the McWhorter relation $S_{V_{fb}}=q^2{k}_{B}T\lambda N_t/(WL{C}_{ox}^{2}f)$ gives the trap density per unit volume and energy (Jomaah et al.). With an assumed tunnelling length of 0.1 $\mathrm{nm}$ and $1.5\times 10^{-6}$ $F/c{\mathrm{m}}^2$ oxide capacitance, a 1 by 0.1 $\mu \mathrm{m}$ device with $3.94\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}\mathrm{e}{\mathrm{V}}^{-1}$, the value Liu et al. report at 65 $\mathrm{nm}$, has $S_{V_{fb}}=1.2$e-10 ${\mathrm{V}}^2/Hz$ at 10 $\mathrm{Hz}$, 10.8 $\mu V/\sqrt{\mathrm{Hz}}$ (computed). Because the noise scales as $1/(WL)$, small devices are noisier, and the extracted density is only as good as the assumed tunnelling length (my reading). Magnone et al. applied the same approach to trench power MOSFETs and found defect densities consistent with that technology. Reference: J. Jomaah, F. Balestra and G. Ghibaudo, ‘Low frequency noise in advanced Si bulk and SOI MOSFETs’, Journal of Telecommunications and Information Technology (year not stated, about 2005) (fetch-tool summary); Y. Liu et al., ‘Low-frequency noise characteristics in the MOSFETs processed in 65 nm technology’, J. Semicond. 37, 064012 (2016) (abstract) (fetch-tool summary); P. Magnone, P. A. Traverso, G. Barletta and C. Fiegna, ‘Low-Frequency Noise Measurements in Silicon Power MOSFETs as a Tool to Experimentally Investigate the Defectiveness of the Gate Oxide’, IMEKO TC4 Symposium 2013 (abstract) (fetch-tool summary).

Ques: Is a Hooge parameter of $4\times 10^{-4}$ unusually high?

Jomaah et al. give typical MOSFET values of about $1\times 10^{-4}$ to $1\times 10^{-6}$, while Liu et al. report $2.42\times 10^{-5}$ and $4\times 10^{-4}$ for 65 $\mathrm{nm}$ devices, the larger apparently for the PMOS. The higher value lies four times above the top of the typical range (computed). Both are plausible: the typical range reflects mature, well-annealed oxides, while a high apparent Hooge parameter in a scaled device can include some number-fluctuation noise fitted with the Hooge formula, or interface damage from processing (my reading). The $2.42\times 10^{-5}$ value is the more representative of a good interface, and $4\times 10^{-4}$ should be read as a property of that particular technology rather than a general value (my reading). Liu et al.’s abstract also assigns its trap densities ambiguously between devices, so the original paper should be checked before using them. Reference: J. Jomaah, F. Balestra and G. Ghibaudo, ‘Low frequency noise in advanced Si bulk and SOI MOSFETs’, Journal of Telecommunications and Information Technology (year not stated, about 2005) (fetch-tool summary); Y. Liu et al., ‘Low-frequency noise characteristics in the MOSFETs processed in 65 nm technology’, J. Semicond. 37, 064012 (2016) (abstract) (fetch-tool summary).

Ques: What is random telegraph noise, and how big is it?

Switching of the drain current between two levels as a single oxide trap captures and emits a carrier. Its spectrum is a Lorentzian, $4A\Delta I_d^2\tau /(1+{\omega }^2{\tau }^2)$, flat below the corner $1/(2\pi \tau )$ and falling as $1/f^2$ above, with $\tau$ combining the capture and emission times (Jomaah et al.); a 1 $m/s$ time constant puts the corner at 159 $\mathrm{Hz}$ (computed). The relative amplitude is $(g_m/I_d)q/(WL{C}_{ox})(1-x_t/t_{ox})$, so a trap at the interface of a 1 by 0.1 $\mu \mathrm{m}$ device at $g_m/I_d=10$ ${\mathrm{V}}^{-1}$ switches the current by 0.11 per cent (assumed values; computed). In large devices many such Lorentzians add to the $1/f$ spectrum. Keysight measures RTN in the time domain for this reason. Reference: J. Jomaah, F. Balestra and G. Ghibaudo, ‘Low frequency noise in advanced Si bulk and SOI MOSFETs’, Journal of Telecommunications and Information Technology (year not stated, about 2005) (fetch-tool summary); Keysight, ‘How to Measure Low-Frequency Noise Accurately’ solution page (fetch-tool summary).

Ques: Is the lowest frequency on Keysight’s page 30 MHz or 30 mHz?

Keysight’s page states measurements at ultra-low frequencies as low as 30 MHz, which contradicts itself, since 30 MHz is a radio frequency. 30 $\mathrm{mHz}$ is far more likely, because the page is about ultra-low frequencies and flicker noise, for which a megahertz limit has no meaning (my reading). The difference matters for planning: resolving 30 $\mathrm{mHz}$ needs sampling records of about 33 $\mathrm{s}$, and averaging 100 of them takes nearly an hour per bias point (computed), so ultra-low-frequency data are slow and sensitive to drift. A 30 MHz limit would instead be a high-frequency bound, unrelated to flicker noise. The vendor’s datasheet should be checked before quoting the specification. Reference: Keysight, ‘How to Measure Low-Frequency Noise Accurately’ solution page (fetch-tool summary).

Ques: How does noise data become a compact model?

Through flicker-noise parameters such as those in BSIM. Liu et al. extracted the unified flicker-noise parameters for 65 $\mathrm{nm}$ devices: NOIA was constant for both, while NOIB was $9.31\times 10^{-4}$ ${\mathrm{V}}^{-1}$ for PMOS and $1.53\times 10^{-2}$ ${\mathrm{V}}^{-1}$ for NMOS, a factor of 16 apart (computed), and inversely proportional to the effective gate voltage. Because the two devices followed different mechanisms in their data, carrier-number fluctuation for NMOS and Hooge-type mobility fluctuation for PMOS in strong inversion, one parameter set cannot describe both polarities, and each is extracted separately (my reading). Keysight emphasizes measuring DC characteristics, flicker noise and RTN together and exporting them to modelling tools. Fitted models are then checked against spectra across bias and geometry (my reading). Reference: Y. Liu et al., ‘Low-frequency noise characteristics in the MOSFETs processed in 65 nm technology’, J. Semicond. 37, 064012 (2016) (abstract) (fetch-tool summary); Keysight, ‘How to Measure Low-Frequency Noise Accurately’ solution page (fetch-tool summary).

References

1. Keysight Technologies, “How to Measure Low-Frequency Noise Accurately,” solution page. Available: https://go.keysight.com/hk/en/solutions/measure-low-frequency-noise-accurately.html
2. P. Magnone, P. A. Traverso, G. Barletta and C. Fiegna, “Low-Frequency Noise Measurements in Silicon Power MOSFETs as a Tool to Experimentally Investigate the Defectiveness of the Gate Oxide,” in Proc. IMEKO TC4 Symposium, Barcelona, 2013. Available: https://imeko.org/index.php/proceedings/4462-low-frequency-noise-measurements-in-silicon-power-mosfets-as-a-tool-to-experimentally-investigate-the-defectiveness-of-the-gate-oxide
3. J. Jomaah, F. Balestra and G. Ghibaudo, “Low frequency noise in advanced Si bulk and SOI MOSFETs,” Journal of Telecommunications and Information Technology (year not stated in the copy read). Available: https://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-article-BAT3-0022-0005/c/httpwww_itl_waw_plczasopismajtit2005124.pdf
4. Y. Liu, Y. Liu, Y. He, B. Li, Y. En and W. Fang, “Low-frequency noise characteristics in the MOSFETs processed in 65 nm technology,” Journal of Semiconductors, vol. 37, no. 6, 064012, 2016. Available: https://www.jos.ac.cn/en/article/doi/10.1088/1674-4926/37/6/064012
5. A. A. Balandin, “Review of the low-frequency 1/f noise in graphene devices,” arXiv:1307.4797, 2013. Available: https://arxiv.org/abs/1307.4797
6. Y. Omura, “Empirical and theoretical modeling of low-frequency noise behavior of ultrathin silicon-on-insulator MOSFETs aiming at low-voltage and low-energy regime,” Micromachines, vol. 10, no. 1, 5, 2018. Available: https://pmc.ncbi.nlm.nih.gov/articles/PMC6356238
7. Z. Balogh, G. Mezei, L. Posa, B. Santa, A. Magyarkuti and A. Halbritter, “1/f noise spectroscopy and noise tailoring of nanoelectronic devices,” arXiv:2106.02683. Available: https://arxiv.org/abs/2106.02683
8. NI, “The Welch Method,” LabVIEW Advanced Signal Processing Toolkit User Manual. Available: https://www.ni.com/docs/en-CC/bundle/labview-advanced-signal-processing-toolkit/page/the-welch-method.html

Related pages in this vault: Charge Pumping, SMU and Probe Station, Deep-Level Transient Spectroscopy (DLTS), MOSFET Noise Models, LCR Meter and C-V, Vector Network Analyzer (VNA).