Solid-State Electronics 79 (2013) 152–158

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Solid-State Electronics

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Phase noise calculation and variability analysis of RFCMOS LC oscillator basedon physics-based mixed-mode simulation

Sung-Min Hong a, Yongho Oh b, Namhyung Kim b, Jae-Sung Rieh b,⇑a EIT4, Bundeswehr University, 85577 Neubiberg, Germanyb School of Electrical Engineering, Korea University, Seoul 136-713, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 February 2012Received in revised form 16 July 2012Accepted 25 July 2012Available online 3 October 2012

The review of this paper was arranged byProf. S. Cristoloveanu

Keywords:OscillatorPhase noiseSemiconductor device modelingSemiconductor device noise

0038-1101/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.sse.2012.07.022

⇑ Corresponding author. Tel.: +82 2 3290 3257; faxE-mail address: jsrieh@korea.ac.kr (J.-S. Rieh).

A mixed-mode technology computer-aided design framework, which can evaluate the periodic steady-state solution of the oscillator efficiently, has been applied to an RFCMOS LC oscillator. Physics-basedsimulation of active devices makes it possible to link the internal parameters inside the devices andthe performance of the oscillator directly. The phase noise of the oscillator is simulated with physics-based device simulation and the results are compared with the experimental data. Moreover, the statis-tical effect of the random dopant fluctuation on the oscillation frequency is investigated.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In modern communication systems, oscillators are the keybuilding blocks that provide the local oscillation in both receiversand transmitters, by themselves or inside phase-locked loops.Their performance, especially the phase noise, significantly affectsthe integrity of the signals received or generated [1]. Therefore, theaccurate simulation of oscillator circuits is important to preciselypredict the performance. An ideal oscillator would have localizedtones at the harmonics of the oscillation frequency and this canbe easily analyzed by the circuit simulation, as much as the neces-sary information for the technology is provided. Moreover, if theoscillator circuits which are actually fabricated are identical, theirperformance can be well represented by a single simulation result.

However, any imperfection inside the oscillator introduces thedeviation from this ideal behavior, and the simulation should beable to predict the amount of the deviation. The deviation can beobserved either in the time domain for a given oscillator or be-tween different oscillators in the same fabrication lot. In the timedomain, the corrupting temporal noise source spreads these puretones, resulting in sideband powers around the harmonics [2],and this phenomenon is well known as the phase noise. On theother hand, since the statistical variability of the device technologybecomes increasingly important as the transistor size is rapidly

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shrinking, it is expected that the dispersion of the oscillator perfor-mance will be an important design issue. Although these two prob-lems are completely different from each other, they can becommonly described by a small deviation from the representativebehavior due to the imperfection inside the semiconductor devices.When the perturbative approach is employed, these problems canbe tackled in a unified manner.

Since these problems – the temporal noise and the statisticalvariation – are closely related with the device technology, thephysics-based device simulation using the technology computer-aided design framework would provide a great insight on these is-sues. Although considerable efforts have been made to addressthese subjects at the circuit simulation level (for example [1,3–5]for the phase noise calculation), at the device simulation levelthe noise and statistical variability problems are usually treatedwithin a single device only [6,7]. In principle, the mixed-mode sim-ulator which couples the device simulator and the circuit simulator[8–11] can be used to improve this situation.

In reality, however, the application of the mixed-mode simula-tor to the perturbative analysis of the realistic circuits under thelarge-signal operation can be rarely found. Even in the relativelysimple case of circuits under the forced large-signal operation suchas mixers, only a few examples can be found from the literature[12,13]. In a more complicated case of oscillators due to its auton-omous nature, only a single contribution [14] has been devoted tothis subject [15], and the application to realistic oscillator circuitsis still lacking. One major reason of this situation might be the

Nomenclature

c proportional constant (s)f0 oscillation frequency (Hz)Kn,n magnitude of the electron diffusion noise source (A2 s/

m)Kp,p magnitude of the hole diffusion noise source (A2 s/m)L device gate length (m)nf number of fingersDN+ dopant fluctuation (/m3)Nþk average doping density at kth node (/m3)~Nþk actual number of net dopant at kth nodeSz power spectrum ((unit of z)2/Hz)T0 oscillation period (s)v1(t) perturbation projection vector (a.u.)

v1,n PPV for electron continuity equation (/A)v1,p PPV for hole continuity equation (/A)v1,W PPV for Poisson equationVtune tuning voltage of VCO (V)W device gate width (m)x(t) perturbed solution (a.u.)xs(t) unperturbed solution (a.u.)y(t) noise term (a.u.)z a state variable (a.u.)Zi i-th Fourier coefficient of z (unit of z)a(t) phase deviation (s)X volume of the device (m3)

S.-M. Hong et al. / Solid-State Electronics 79 (2013) 152–158 153

expensive computational cost for the periodic steady-state simula-tion in the case of the mixed-mode simulation, which is always re-quired in order to perform the perturbative analysis. With theadvent of computers (especially affordable computer memories),this approach is getting more and more feasible.

In this paper we present the mixed-mode simulation for anoscillator circuit. Deviations from the ideal characteristics – thephase noise and the statistical variation of the oscillation fre-quency – are numerically investigated. To our knowledge, thiswork is the first attempt to simulate these issues with a realisticoscillator circuit based on a commercial foundry technology usingthe mixed-mode simulator, and further compare the results withactual experimental data. This paper is organized as follows: InSection 2, the underlying theory is described briefly. In Section 3,the numerical application to an RFCMOS LC oscillator is presented.The conclusion is given in Section 4.

2. Theory

For the perturbative analysis of oscillators, it is known that theconventional linearization fails because of their autonomous nat-ure [2]. Instead, the ‘‘nonlinear perturbation theory’’ shown in [2]should be used. Although it has been mainly proposed and appliedto the phase noise calculation, where the perturbative source is arandom time-varying signal, it can be also applied to the time-independent perturbative source as shown below (also in [2]).Since our aim is the application of the nonlinear perturbation the-ory to the mixed-mode simulation, complicated mathematical der-ivations will be avoided. More detailed discussion for theapplication to the phase noise calculation using the mixed-modesimulation can be found in [14].

2.1. Phase noise evaluation

In the noise simulation of a system under the forced large-signaloperation, we assume that a solution x(t) perturbed from theunperturbed solution can be written as a sum of the unperturbedsolution xs(t) and the noise term y(t). T0 is used to denote the oscil-lation period. This assumption is valid only when the perfect timereference is provided from the externally applied periodic voltagesource and the system is synchronized to this time reference. Inthe free-running oscillator, however, there is no perfect time refer-ence. Therefore, for the noise simulation of the free-running oscil-lator, an additional ‘‘phase deviation’’ term a(t) should beintroduced. Although there can be various ways to determine thisadditional quantity, it is widely accepted that only persistent eigenmode contributes to the phase deviation [2]. In this work, the orbi-tal deviation [16] is not considered.

Since the spectral intensities of noise sources inside semicon-ductor devices are well known as shown in [17], the main quantityrequired for the phase noise calculation in the device simulation le-vel is the transfer function which establishes the quantitative rela-tion between the noise sources and the phase deviation. Theperturbation projection vector (PPV) v1(t), which is a T0-periodicreal vector, represents the influence of the noise source on thephase deviation. Note that when the device simulation is per-formed the semiconductor equations – for example, continuityequations for mobile carriers and the Poisson equation – are solvedinside the device. Therefore, the PPV is indexed with the equationwhere the noise source is imposed, and it depends on the realspace. With the help of an augmented linearized system [18] thisvector can be numerically calculated.

The power spectra of the state variables in the oscillator areused for characterization of the phase noise. The power spectrumof a state variable z (for example, the voltage of an output node),which is perturbed by the noise sources, is given by [18]:

Szðf Þ ¼ 2X1

i¼�1ZiZ

�i

f 20 i2c

p2f 40 i4c2 þ ðf þ if0Þ2

; ð1Þ

where f0 is the oscillation frequency, Zi is the i-th Fourier coefficientof z, and c is a constant which is defined below. Since the oscillationfrequency f0 and the Fourier coefficient Zi are calculated from theperiodic steady-state simulation without considering the noisesources, the constant c solely represents the amount of the phaseinstability, originated from the noise sources. Also the total amountof c can be obtained by summing contribution from individual de-vices (and passive elements).

When the diffusion noise sources, which is the origin of thethermal noise, are considered, the relevant PPV is the one for theelectron (or hole) continuity equation. It represents the phase devi-ation originated from additionally injected electron (or hole) insidethe device. In this case, the proportional constant c can be calcu-lated [14]:

c ¼ 1T0

Z T0

0

ZXrrm1;nðr; tÞ�� ��2Kn;nðr; tÞdr

� �dt ¼ 1

T0

Z T0

0cðtÞdt

¼Z

XcðrÞdr; ð2Þ

where X is the volume of the device, v1,n (or v1,p) is the PPV for theelectron (or hole) continuity equation, and Kn,n (or Kp,p) representsthe magnitude of the electron (or hole) diffusion noise source. c(t)and c(r) are interpreted as the temporal and spatial distribution ofc, respectively. Note that Kn,n depends on the (time-varying) elec-tron density and the diffusion constant [17]. Following to theunderlying assumption of the conventional drift–diffusion model,

Fig. 1. Schematic of a cross-coupled CMOS LC VCO.

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which is the basis of our device simulation, the Einstein relation isused when relating the carrier mobility and the diffusion constant[19].

2.2. Statistical analysis

The device simulation is typically performed with an idealizeddoping profile. Such an ideal doping profile is obtained either bysmoothening the experimental doping profile or by the numericalsimulation for the fabrication process. Calculated results representthe electrical properties of the nominal device. However, due tounavoidable imperfection inside the device, the doping profilecan show local deviation from its nominal value, and it will affectthe performance of the oscillator consisting of those fluctuating de-vices. There can be many different realizations of a certain oscilla-tor circuit design, and the statistical analysis is required in order tocharacterize their deviations from the nominal performance.

Let us consider a particular oscillator consisting of several de-vices whose doping profiles are slightly different from their nomi-nal profiles. The effect of the dopant fluctuation can be modeled asa small deviation of the space charge in the device. Since this devi-ation is time-independent for a given realization of the oscillatorcircuit, the change of the oscillator performance is deterministic,as much as the perturbation due to the dopant fluctuation is con-sidered. In this work, the deviation of the oscillation frequencydue to the random dopant fluctuation is numerically investigated.

Since the origin of the perturbation is the electric charge of ion-ized impurities, the relevant PPV in this case is the one for the Pois-son equation in the device. When the PPV for the Poisson equationin a certain device is denoted as v1,W (r, t), the additional phasedeviation during one period, Da will be given by:

Da ¼Z T0

0

ZXm1;wðr; tÞDNþðrÞdr

� �dt

¼ T0

ZX

�m1;wðr; tÞDNþðrÞdr; ð3Þ

where DN+ is the dopant fluctuation at the position r and �v1;w is thetime-averaged PPV for the Poisson equation. Compared with theoscillator free from the dopant fluctuation, the change of the oscil-lation period as large as �Da will be observed.

For a given discretized k-th node with a control volume of Xk,the average number of the net dopant is given by XkNþk ; whereNþk is the average doping density. The actual value of the net dop-ant assigned with the k-th node, denoted as ~Nþk , follows the Poissondistribution. Then the probability of an event of ~Nþk ¼ N is given by:

Pð~Nþk ¼ NÞ ¼ ðXkNþk ÞN

N!expð�XkNþk Þ: ð4Þ

In the simulation, by generating the random number, the valueof ~Nþk is determined. Finally, the value of the dopant fluctuation atthe discretize k-th node is given by:

DNþðrkÞ ¼~NþkXk� Nþk : ð5Þ

When both the acceptor and the donor contribute to the net doping,they are treated separately.

3. Numerical results

The simulation facilities for simulating the phase noise and thestatistical variation, described in the previous section, have beenimplemented in our in-house device-circuit mixed-mode simula-tor Circuit LEvel SImulation Code (CLESICO) [13,14,20]. By applyingthese simulation techniques to realistic oscillator circuits, it is ex-pected that impacts of temporal noise sources and random dopant

fluctuations on the oscillator performance can be analyzed at thedevice simulation level.

Fig. 1 shows a voltage-controlled oscillator (VCO) considered inthis work. It adopts the CMOS-based conventional LC cross-cou-pled topology including a tail transistor driven by a current mirrorand common-source output buffers, aimed at operation around20 GHz. The VCO was designed and fabricated based on a commer-cial 0.13 lm RFCMOS technology provided by Dongbu Hitek. Itadopts a triple well process for improved isolation and back-end-of-the-line (BEOL) based on 1 poly and 8 metals for interconnec-tion and passive devices. An ultra-thick metal (UTM) layer of3.3 lm is used for the top metal to provide a low loss interconnectsas well as high-Q passive devices. The NMOSFETs in the circuit em-ploy a unit finger length and width of 0.13 lm (drawn) and 2 lm,respectively, with the total number of fingers (nf) varying between4 and 10 depending on the device. A typical device shows fT/fmax ofaround 75 GHz/70 GHz at the optimal bias point. The accumula-tion-mode NMOS varactors adopt a unit finger length and widthof 0.3 lm and 2 lm, respectively, with nf of 8, which results in acapacitance of 70 fF at the accumulation state. The details of thedevice dimensions are given in the Table 1.

A process simulator ATHENA [21] has been used in order to gen-erate the two-dimensional doping profiles for transistors and varac-tors. The gate electrode resistance, which cannot be properlyincluded in the two-dimensional device simulation, is modeled asan additional lumped resistance connected to the gate terminal, fol-lowing the approach shown in [22]. All of transistors and varactorsshown in Fig. 1, except for those for the current mirror (M6 and M7),are included in the mixed-mode simulation. Each of transistors andvaractors is discretized with 6 265 and 2 058 grid points, respec-tively. Since seven two-dimensional semiconductor devices(M1–M5, VAR1, and VAR2) are included in the simulation, the totalnumber of unknown variables for the DC calculation is 63377.

In addition to the physics-based modeling of MOS devices, accu-rate modeling of the inductors and the interconnect lines is man-datory in order to have reasonable simulation results. Basedupon the results of the electromagnetic wave simulation withADS MOMENTUM [23], the inductors and the interconnect linesare modeled as networks of lumped elements. The equivalentmodel shown in Fig. 2 and similar variants were used for modelingof the inductors and interconnect lines.

3.1. Periodic steady-state solution

Either the phase noise calculation or the statistical analysis ofthe oscillation frequency requires the periodic steady-state solu-

Table 1Device parameters.

Element Parameters

M1, M2, M3, M4 nf = 4, L/W = 0.13 lm/2 lmM5 nf = 10, L/W = 0.13 lm/2 lmVAR1, VAR2 nf = 8, L/W = 0.3 lm/2 lmL1, L2 radius = 60 lmL1, L2 width = 9 lmL1, L2 # of turns = 1

Fig. 2. Equivalent circuit for inductor modeling.

Fig. 3. Typical convergence behavior of the periodic steady-state simulation.

Fig. 4. Simulated waveforms of Vout+ and Vout�. The tuning voltage is 0 V.

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tion, which is free of temporal noise sources or the spatial dopantfluctuation, because they are based on the perturbation approach.

A lumped circuit element can be fully characterized by its ter-minal voltages and currents. The semiconductor equations (thePoisson equation and continuity equations for electrons and holes)for each device in the circuit are discretized as in the conventionaldevice simulators. Then the state variables for the system are theelectric potential, the electron density, and the hole density atevery node of the spatially discretized devices, and the terminalvoltages and current of the lumped elements. To obtain the peri-odic steady-state solution, several sampling time points are as-signed in a single period. At each sampling time points, the stateequations for the entire oscillator system are written in the timedomain. The resultant system of equations can be solved by usingthe periodic steady-state analysis methods, such as the finite-dif-ference method [24,25], the harmonic balance method [26], orthe shooting-Newton method [14,27]. In this work, the finite-dif-ference method is chosen in order to enable a straightforwardimplementation. This results in 6214377 unknown variables with97 sampling time points. Such a large set of linearized systemequations is solved with ILUPACK, which is an efficient sparse ma-trix solver [28].

Since the oscillation frequency cannot be determined before thesimulation, the oscillation period is added as an additional un-known variable and an additional equation is added that constrainsthe phase of the computed solution. The initial guess for the oscil-lation frequency is obtained by performing a sufficiently long tran-sient simulation. Fig. 3 shows a typical convergence behavior of theperiodic steady-state simulation. Errors of the oscillation periodand the state variables decrease rapidly when the Newton iterationnumber is increased. The last iteration shows a huge decrease ofthe maximum error which is characteristic of a quadratic conver-gence. The quadratic convergence can be observed when the line-arization of the system equations is performed consistently. It isparticularly important for our purpose, because the perturbativeapproach is used for the analysis. The output voltages, which aresimulated with the tuning voltage of 0 V, are shown in Fig. 4. Thesimulated oscillation frequency is 18.89 GHz. The voltage swing

of an output voltage is about 300 mV peak-to-peak, and the totaloutput power is �6.42 dBm.

3.2. Phase noise

Based upon the periodic steady-state solution, the PPV vector isnumerically calculated. Using Eq. (2), the constant c, which is anindicator of the phase noise level, is calculated. Table 2 shows con-tribution of each component to the overall c, expressed in per cent.Two different values of the tuning voltage are considered. Whenthe tuning voltage is 0 V, c and the period jitter are found to be1.26 � 10�19 s and 2.58 fsec, respectively. The calculated value ofthe period jitter is about 0.0049% of the oscillation period. The big-gest contribution to c comes from two cross-coupled MOSFETs, M1and M2, while the inductor losses also contribute considerably.Note that this ratio can be used as a figure of merit representingthe contribution of each component to phase noise or timing jitter,as pointed out in [2]. The contribution of two biggest contributors,M1 and L1, to the overall c at a given time point, c(t), is shown inFig. 5. Note that the results shown in Table 2 and Fig. 5 can be ob-tained also from conventional circuit simulators based on the com-pact models, but, in this work, they are obtained from the physics-based device simulation.

Table 2Contribution of each element to c.

Element Contribution (%)(Vtune = 0 V)

Contribution (%)(Vtune = 1.5 V)

M1, M2 70.62 73.62M3, M4 1.17 1.26M5 6.66 5.84VAR1, VAR2 5.37 3.47L1, L2 14.30 13.77Interconnect and

others1.88 2.04

Fig. 5. c(t)’s for M1 and L1 as a function of time for one cycle. The tuning voltage is0 V.

Fig. 6. c(r) for the M1 device. The tuning voltage is 0 V. The zero point in the lateralposition is the midpoint of the gate. The source contact is located at a negativelateral position. Dashed lines are representing the metallurgical junctions. Values atSi–SiO2 interface are shown.

Fig. 7. Power spectral density of the output variable up to about the third harmonic.One thousand points are uniformly assigned for the frequency range of theoscillation frequency.

156 S.-M. Hong et al. / Solid-State Electronics 79 (2013) 152–158

Since the magnitude of the temporal noise source at a givenpoint is known, the spatial distribution of the contribution to theoverall c inside the each device can be obtained. Fig. 6 shows c(r)for M1. Only values at Si–SiO2 interface are shown, because c(r) rap-idly decreases when the distance from the interface is increased. Asshown in the figure, the spatial distribution has the peak value at aposition closer to the source contact rather than the drain contact.Therefore, it is apparent that the dominant contribution to c comesfrom the noise sources in the source side of the channel. The mainreason of this behavior is the asymmetric distribution of inversioncharge along the channel direction, which results in the highermagnitude of the noise sources near the source side.

From the calculated value of c, the power spectrum of the out-put variable can be readily evaluated using Eq. (1). In Fig. 7, it isshown up to about the third harmonic. One thousand points areuniformly assigned for the frequency range of the oscillation fre-quency, which yields the smallest offset frequency becomes about9.4 MHz. Therefore, the sharp increase of the power spectrumaround the oscillation frequency (about �35 dBm/Hz at the firstharmonic) is not shown in the figure. The power spectral densityof the output variable, around the first harmonic, is compared withthe measurement data in Fig. 8. The simulation predicts a Lorentz-ian shape of the phase noise spectrum, therefore, the 1/f2 depen-dence of the phase noise spectrum is obtained for sidebandfrequencies well above 100 Hz. At 1 MHz offset frequency, thephase noise of �103.46 dBc/Hz is obtained. It shows a close agree-ment with the experimentally measured value of �105.51 dBc/Hz.

3.3. Random dopant fluctuation

In this subsection, the random dopant fluctuation of the aboveoscillator circuit is numerically calculated. Due to the limited num-

ber of the available samples, the variability data could not be ob-tained experimentally. However, this numerical study canprovide the theoretical lower limit of the variability. When com-bined with the massive experimental data, it can be used for theoptimization of device and circuit design for improved oscillatorperformance.

Fig. 9 shows the time-averaged PPV for the Poisson equation forthe M1 device. It represents the relative sensitivity of the oscilla-tion period with respect to the additional local charge due to thedopant fluctuation. As expected, the oscillation period is most af-fected by an additional charge in the channel region. Additionally,the time-averaged PPV for the Poisson equation has non-negligiblevalues at the junction region between the substrate and the drainregion. It is because that the drain regions of M1 and M2 are sen-sitive regions due to their direct topological connection to theinductors and the varactors. Recall that the noise sources in thesource side of the channel of the M1 and M2 transistors contributeto the phase noise dominantly. Therefore, in the case of the phasenoise calculation, the source side of the channel is the sensitive re-gion, which is completely opposite to the random dopant fluctua-

Fig. 8. Power spectral density of the output variable around the first harmonic. The measurement data is shown for comparison.

Fig. 9. Time-averaged PPV for the Poisson equation for the M1 device. Themetallurgical junctions are shown on top of the figure.

Fig. 10. Simulated deviation of the normalized oscillation frequency due to therandom dopant fluctuation of the M1 device. 10000 realizations are generated and20 bins are used.

S.-M. Hong et al. / Solid-State Electronics 79 (2013) 152–158 157

tion. It is due to the different forms of relevant perturbativesources (the diffusion noise source versus the fluctuation of thespace charge).

For the statistical analysis through simulations, we take 10000realizations of the oscillator circuit as the simulation samples.Fig. 10 shows the simulated deviation of the normalized oscillationfrequency due to the random dopant fluctuations of the M1 device.Twenty bins are used in order to make the histogram. The standarddeviation of the simulated data is about 0.0036% of the oscillationfrequency, whose value is comparable with the timing jitter. Sincethe internal quantities of devices are already available, other mech-anisms causing the statistical variability, such as the line edgeroughness, can be investigated using this simulation framework.

4. Conclusion

In this work, we have presented the application of a mixed-mode device-circuit simulator to a realistic oscillator circuit basedon RFCMOS technology. For this purpose, care has been spent bothfor the semiconductor devices (MOSFETs and varactors) and thepassive elements (inductors and interconnect lines). With the helpof the nonlinear perturbation theory [2], two important modelingissues, the phase noise evaluation and the statistical variation ofthe oscillation frequency due to the random dopant fluctuation,

has been addressed at the device simulation level. The results ob-tained in this work provide an insight to the origin of phase noiseand statistical variations in RF oscillators from the device level per-spective. The results can also be practically used for the optimiza-tion of device and circuit design for improved oscillatorperformance.

Acknowledgments

The authors would like to thank Prof. C. Jungemann of Bundes-wehr University (Germany) for supporting the computing re-sources. The authors also appreciate the careful review andprecious suggestions made by the reviewers. This work was partlysupported by the NRF grant funded by the Korea government(MEST) (No. 2011-0020128).

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