Experimental Demonstration of
<i>g<sub>m</sub></i>/<i>D<sub>D</sub></i>
Based Noise Analysis

Jack Ou, Pietro M. Ferreira, Pietro M. Ferreira, Jui-Chu Lee

► To cite this version:

Jack Ou, Pietro M. Ferreira, Pietro M. Ferreira, Jui-Chu Lee. Experimental Demonstration of <i>g<sub>m</sub></i>/<i>D<sub>D</sub></i> Based Noise Analysis. Circuits and Systems, 2014, 05 (04), pp.69-75. 10.4236/cs.2014.54009. [hal-01222130v2](https://hal.science/hal-01222130v2)

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Experimental Demonstration of $g_m/I_D$
Based Noise Analysis

Jack Ou<fnref n="1" />, Pietro M. Ferreira<fnref n="2" />, Jui-Chu Lee<fnref n="3" />

Email: jack.ou@sonoma.edu, maris@ieee.org, juichu@us.ibm.com

Received 20 January 2014; revised 20 February 2014; accepted 28 February 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Open Access

Abstract

Recent studies using BSIM3 models have suggested that noise depends on the transconductance-to-drain ratio ($g_m/I_D$) of a transistor. However, to the best of our knowledge, no experimental result demonstrating $g_m/I_D$ dependent noise previously observed in simulation is available in the literature. This paper examines the underlying principles that make it possible to analyze noise using $g_m/I_D$ based noise analysis. Qualitative discussion of normalized noise is presented along with experimental results from a 130 nm CMOS process. A close examination of the experimental results reveals that the device noise is width independent from 1 Hz to 10 kHz. Moreover, noise increases as $g_m/I_D$ is reduced. The experiment observation that noise is width independent makes it possible for circuit designers to generate normalized parameters that are used to study noise intuitively and accurately.

Keywords

$g_m/I_D$ Design Methodology; Noise Analysis; Flicker Noise

1. Introduction

In integrated circuits, noise phenomena are caused by small current and voltage fluctuations that are generated within the devices themselves. At low frequencies, the noise spectrum is dominated by the flicker noise. At high frequencies, the noise spectrum is dominated by the thermal noise. The study of noise is important because it ultimately determines the smallest signal that a circuit can amplified without significant deterioration in signal quality [1].

<footnote n="1">¹Department of Engineering Science, Sonoma State University, Rohert Park, USA</footnote>

<footnote n="2">²IEMN, UMR CNRS 8520 Department of DHS, University of Lille, Lille, France</footnote>

<footnote n="3">³IBM Microelectronics Essex Junction, New York, USA</footnote>
From a designer's perspective, noise is a property of the circuit that must be designed carefully along with other circuit parameters such as gain, power dissipation, speed, and linearity. Noise analysis is particularly challenging in sub-micron CMOS circuit design because it involves parameters that depend on the bias condition of a transistor as well as the geometry of a transistor. In the absence of an easy-to-use model for accurate back-of-the-envelope noise calculation, designers often explore design space using arduous circuit simulations. Over-reliance on circuit simulator can be problematic, potentially luring inexperienced designers to dive into simulation without understanding basic trade-offs in properly optimized circuits.

## 1.1. A Brief Overview of Related Work

In 1996, Silveira et al. proposed a powerful transconductance-to-drain current ($g_m/I_D$) technique that has since become the basis of many later developments in structured analog circuit design [2]. The so-called “$g_m/I_D$ design approach” was originally developed to help designers to size up transistors quickly with good accuracy and to calculate parameters such as small signal gain and bandwidth. Recently, it has found applications in large signal behavior of a power amplifier [3], phase noise optimization of an LC oscillator [4], MOSFET nonlinearity characterization [5], MOSFET variability and ageing degradations [6]. A simple $g_m/I_D$ CAD tool has also been developed recently to optimize analog circuits without lengthy simulations [7]. A book dedicated to $g_m/I_D$ methodology has also been published [8].

A $g_m/I_D$ based noise analysis was reported in 2011 [9]. Bias dependent thermal noise coefficient ($\gamma$) and device noise corner frequency ($f_{co}$) were used to characterize MOSFET noise. In 2012, Alvarez and Abusleme published a formulation of $g_m/I_D$ based noise analysis using normalized noise power instead of $\gamma$ and $f_{co}$ [10]. In [10], noise curves for a set of transistors are pre-computed by means of SPICE simulations, scaled for the appropriate device parameters using the $g_m/I_D$ technique, and finally, noise is computed using interpolations within the curves. The work in [10] was applied in the context of charge amplifiers in [11].

## 1.2. Main Contributions of This Paper

To the best of our knowledge, what is currently known in the literature with respect to $g_m/I_D$ based noise analysis is derived from either HSPICE analysis [10] or BSIM simulation ([9] and [11]). According to a study conducted by Rhayem et al. [12], each noise model has a different accuracy with respect to measured data. For example, the SPICE noise model is not accurate for all regions of operation for both NMOS and PMOS. The HSPICE noise model cannot predict noise accurately for PMOS. For NMOS, the model can be used if a sub-circuit includes access resistances and their excess noise sources. Even though the BSIM3v3 noise model is more accurate than either the SPICE noise model or HSPICE noise model, the most accurate portrayal of a device's noise is through measurement.

This paper presents an experimental study of $g_m/I_D$ based noise. In particular, we present wafer measurement data from a 130 nm CMOS process utilizing both NMOS and PMOS transistors. Two experimental results are presented. First, we present the experimental data to show that the normalized noise is independent of a transistor's gate width. Second, we present the experimental data to show that the normalized noise increases as a transistor's $g_m/I_D$ is reduced.

The organization of this paper is as follows: Section 2 provides a review of $g_m/I_D$ analysis, followed by a discussion of thermal noise and flicker noise in the context of $g_m/I_D$ based noise analysis. Section 3 shows the experimental results on $g_m/I_D$ noise. Finally, we present our conclusions.

# 2. Formulation of Noise Measurement

We begin this section with a review of fundamentals of $g_m/I_D$ analysis. We point out in this section that noise current ($i_n^2$) of an MOS transistor is width dependent, it is therefore not a valid $g_m/I_D$ parameter. In order to make the noise current a valid $g_m/I_D$ parameter, we normalize the noise by dividing $i_n^2$ by $g_m$. The details are shown below.

## 2.1. Fundamentals of $g_m/I_D$ Analysis

The $g_m/I_D$ principle is applicable to parameters which are independent of the width ($W$) of a transistor. Figure 1 shows a transistor with a transconductance ($g_m$), a drain-to-source conductance $g_{ds}$, and a current
<figure><img src="image_3.png" /><figcaption>Figure 1. Transistors biased at the same $g_m/I_D$.</figcaption></figure>

$(I_D)$ biased at a gate-to-source voltage $(V_{GS})$ and a drain-to-source $(V_{DS})$. If an identical device is connected in parallel so both devices are biased at the same $V_{GS}$ and $V_{DS}$, both devices have the same $g_m$, $g_{ds}$ and the same $I_D$. Since the devices are connected in parallel, they can be treated as one device with an aspect ratio of $2W/L$. The effective transconductance over current ratio is $g_m/I_D$ for both the merged device and the stand alone device because $g_m$ and $I_D$ are doubled. The drain-to-source conductance is also doubled for the merged device. Therefore the intrinsic gain $(g_m/g_{ds})$ is identical for both the stand alone device and the merged device. As long as transistors are biased at the same $g_m/I_D$, they will have the same $g_m/g_{ds}$. This observation is true for any two parameters whose ratio depend solely on the $g_m/I_D$ and not on the width of a transistor. Once a transistor of a given $W$ is characterized over a range of $g_m/I_D$, the $g_m/I_D$ based parameters can be generalized to a transistor of an arbitrary $W$, assuming that $L$ remains constant. The $g_m/I_D$ methodology will hold as long as the parameter of interest scales with $W$.

## 2.2. Thermal Noise

The MOSFET noise arises from thermal noise fluctuations in the channel. It can be shown that the thermal noise at the drain terminal is [13],

$$ \overline{i_{n,id}^2} = 4kT\gamma g_m, \qquad (1) $$

where $k$ is the Boltzmann constant, $T$ is the temperature, $g_m$ is the transconductance, and $\gamma$ is the bias-dependent noise parameter. According to this model, the $\gamma$ approaches 1 when the drain-to-source voltage $(V_{DS})$ approaches zero and decreases to $2/3$ when the device enters the saturation regime. The saturation value of $\gamma = 2/3$ is valid for long-channel MOS devices built on lightly doped substrates. Early studies have reported $\gamma$ values between $2/3$ and $4$ [14], but recent studies have shown that by accounting for parasitic resistances, $\gamma$ is approximately $2/3$ for channel lengths equal or greater than 100 nm [15].

The thermal noise $\overline{i_{n,id}^2}$ at the drain terminal of a MOS transistor is

$$ \overline{i_{n,id}^2} = 4kT \frac{\mu}{L^2} (-Q_l), \qquad (2) $$

where $Q_l$ is the total inversion layer charge underneath the gate oxide, $L$ is the length of the transistor, and $\mu$ is the mobility. Equation (2) is valid for all regions of operation [13]. The total inversion layer charge $(Q_l)$ is obtained by integrating the inversion charge per unit length $(Q'_l(x))$ over the length of the channel,

$$ Q_l = \int_0^L Q'_l(x) W dx. \qquad (3) $$

where $Q'_l(x)$ is a function of $V_{GS}$ and $V_{DS}$, as well as $V_{SB}$, and consequently a function of transistor's $g_m/I_D$. Since $Q'_l$ is proportional to $W$ (see Equation (3)), $\overline{i_{n,id}^2}$ is proportional to $W$ according Equation (2). Since $g_m$ is proportional to $W$, $\overline{i_{n,id}^2}/g_m$ becomes independent of $W$. Equation (2) also shows that $\overline{i_{n,id}^2}$ is inversely proportional to $L^2$. Even though the transconductance of a transistor also depends on the $L$, it is
not inversely proportional to $L^2$. Therefore, $\overline{i_{n,id}^2}/g_m$ is a function of L. Once $\overline{i_{n,id}^2}/g_m$ of a transistor for a given (W/L) is characterized over a range of $g_m/I_D$; $\overline{i_{n,id}^2}/g_m$ of the transistor can be generalized to a transistor of arbitrary W as long as L and the $g_m/I_D$ are constant. We will verify this observation in Section 3. This width-independent property is the crucial link to the $g_m/I_D$ design methodology described earlier.

## 2.3. Flicker Noise

Since MOSFETs are surface-conduction devices, flicker noise is important at low frequencies. The flicker noise present at a MOSFET can be expressed into a noise current at the drain terminal of a transistor with

$$ \overline{i_{n,fn}^2} = \frac{K_f}{C_{ox}WL} \frac{g_m^2}{f}, \qquad (4) $$

where $K_f$ is a process dependent constant and $C_{ox}$ is the oxide capacitance. Equation (4) can be rewritten to explicitly show its dependence on $g_m/I_D$ parameters as

$$ \overline{i_{n,fn}^2} = \frac{K_f}{C_{ox}Lf} \frac{g_m}{I_D} \frac{I_D}{W} g_m. \qquad (5) $$

If we divide $\overline{i_{n,fn}^2}$ by $g_m$, we find

$$ \frac{\overline{i_{n,fn}^2}}{g_m} = \frac{K_f}{C_{ox}Lf} \frac{g_m}{I_D} \frac{I_D}{W}. \qquad (6) $$

Thus, we have in $\overline{i_{n,fn}^2}/g_m$ a parameter which depends exclusively on the $g_m/I_D$ of a transistor. It should be pointed out that $I_D/W$ (the current density) is also a function of $g_m/I_D$. Furthermore, Equation (6) implies that any two transistors, regardless of their widths (W), theoretically exhibits the same $\overline{i_{n,fn}^2}/g_m$ if they are biased at the same $g_m/I_D$. This is a useful property to compare measured noise of different transistors biased at the same $g_m/I_D$.

## 3. Experimental Results

Section 2 identifies $\overline{i_n^2}/g_m$ as a function of $g_m/I_D$ parameter satisfying all characteristics described in Section 2.1. The $\overline{i_n^2}/g_m$ is used here as a parameter to compare noise characteristics of different transistors. In order to experimentally prove such formulations, ten NMOS transistors from a CMOS 130 nm process are chosen from three different wafers. Each transistor has a length of 0.12 µm and a width of 10 µm. The gate of each transistor is biased at 0.45 V and the drain of each transistor is biased at 1.0 V. The source terminal is grounded ($V_{SB} = 0$). We present the noise as a function of $g_m/I_D$ and as a function of the frequency. Both $\overline{i_n^2}$ and $g_m$ are determined experimentally without curve fitting.

### 3.1. Noise as a Function of $g_m/I_D$

The single-finger and ten-finger NMOS transistors from a 0.13 µm CMOS process are measured, and separated into two sets of data. The noise current at the drain terminal is measured comparing a single-finger ($W = 10$ µm) and ten-finger ($W = 100$ µm) data sets biased at the same voltages.

The average $g_m$, $g_{ds}$ and $I_D$ of both sets of transistors are shown in Table 1. The dimensions of the transistors are shown in the first row of the table using the format: width(W)×length(L)×fingers(Nf). The total width of each transistor is obtained by multiplying W by Nf. The transistors shown in the fourth column have ten times the width than the transistors in the third column, and hence have ten times the $g_m$, $g_{ds}$, $I_{ds}$. The $g_m/I_D$ and $g_m/g_{ds}$ are approximately the same since $g_m$, $I_D$ and $g_{ds}$ are proportional to the width of a transistor. We conclude that $g_m/I_D$ dependent parameters (e.g., $g_m/g_{ds}$) are independent of W.

Having verified that transistors are biased correctly, we investigate $\overline{i_n^2}/g_m$ as a function of $g_m/I_D$. The $V_{GS}$ is varied as $V_{DS}$ is held constant in order to change $g_m/I_D$ of a transistor. The drain-current density is measured at 100 Hz as $V_{GS}$ is changed. The $\overline{i_n^2}/g_m$ of a 10 µm device is compared to the $\overline{i_n^2}/g_m$ of a 100
µm device. In Figure 2, the average $\overline{i_n^2}/g_m$ (represented with marks) and the standard deviation (represented with error bars) of ten samples are shown.

The $\overline{i_n^2}/g_m$ measured at 100 Hz is dominated by flicker noise. Equation (5) suggests that $\overline{i_n^2}/g_{g_i}$ is proportional to $g_m/I_D$ and $I_D/W$. The current density $I_D/W$ is inversely proportional to $(g_m/I_D)^2$. Therefore, $\overline{i_n^2}/g_m$ increases as $g_m/I_D$ is reduced. The 10 µm device exhibits more process variation than the 100 µm device as expected. The close correlation of the 10 µm device and the 100 µm device measurements (see Figure 2) demonstrate that $\overline{i_n^2}/g_m$ are width independent, and therefore $g_m/I_D$ dependent as stated in Section 2.3.

## 3.2. Noise as a Function of Frequency

The $\overline{i_n^2}/g_m$ of the transistors measured from 1 Hz to 100 kHz are shown in Figure 3. The PMOS devices exhibit less noise than the NMOS devices as previously observed and theoretically expected [13]. Good correlation

<table><caption>Table 1. Comparison of $g_m/I_D$ parameters. NFETs are biased at $V_{GS} = 0.45$ V and $V_{DS} = 1.0$ V; PFETs are biased at $V_{SG} = 0.41$ V and $V_{SD} = 1$ V. Device dimensions: $W(\mu m) \times L(\mu m) \times$ Fingers.</caption><thead><tr><th>Device</th><th>Parameter</th><th>10 × 0.12 × 1</th><th>10 × 0.12 × 10</th></tr></thead><tbody><tr><td rowspan="5">NFET</td><td>$g_m$ (mS)</td><td>2.253</td><td>24.190</td></tr><tr><td>$i_{ds}$ (mA)</td><td>0.150</td><td>1.679</td></tr><tr><td>$g_{ds}$ (mS)</td><td>0.100</td><td>1.143</td></tr><tr><td>$g_m/I_D$ (S/A)</td><td>13.929</td><td>14.347</td></tr><tr><td>$g_m/g_{ds}$</td><td>22.077</td><td>20.924</td></tr><tr><td rowspan="5">PFET</td><td>$g_m$ (mS)</td><td>0.383</td><td>3.981</td></tr><tr><td>$i_{ds}$ (mA)</td><td>0.023</td><td>0.242</td></tr><tr><td>$g_{ds}$ (mS)</td><td>0.015</td><td>0.165</td></tr><tr><td>$g_m/I_D$ (S/A)</td><td>16.764</td><td>16.398</td></tr><tr><td>$g_m/g_{ds}$</td><td>25.693</td><td>23.963</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Figure 2. The $\overline{i_n^2}/g_m$ comparison over a range of $g_m/I_D$ for NFET $V_{DS} = 1.0$ V, and PFET $V_{SD} = 1.0$ V.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3. The $\overline{i_n^2}/g_m$ noise compared from 1 Hz to 10 kHz for: NFET $V_{DS} = 0.45$ V, PFET $V_{SD} = 0.41$ V.</figcaption></figure>

between the 10 µm devices and 100 µm devices is presented from 1 Hz to 1 kHz. Our calculation shows that the noise flattening at frequencies higher than 10 kHz is attributed to the noise introduced by the instrumentation amplifier used in the measurement set-up. The close correlation of $i_n^2/g_m$ from 1 Hz to 1 kHz demonstrates that device noise is width independent.

## 4. Conclusion

This paper examines the underlying principles that make it possible to study noise using $g_m/I_D$ based noise analysis. Fundamentals of $g_m/I_D$ analysis were presented, followed by a discussion of normalized thermal noise and normalized flicker noise. Experimental results were shown to demonstrate that noise is indeed width independent from 1 Hz to 10 kHz. Furthermore, noise increases as $g_m/I_D$ is reduced. The experiment observation that noise is width independent makes it possible for circuit designers to generate normalized parameters that are used to study noise intuitively and accurately.

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