ABSTRACT

Recent studies using BSIM3 models have suggested that noise depends on the transconductanceto-drain ratio of a transistor. However, to the best of our knowledge, no experimental result demonstrating 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 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 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: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] .

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 backof-the-envelope noise calculation, designers often explore design space using arduous circuit simulations. Overreliance 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 technique that has since become the basis of many later developments in structured analog circuit design [2] . The so-called “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 CAD tool has also been developed recently to optimize analog circuits without lengthy simulations [7] . A book dedicated to methodology has also been published [8] .

A based noise analysis was reported in 2011 [9] . Bias dependent thermal noise coefficient and device noise corner frequency were used to characterize MOSFET noise. In 2012, Alvarez and Abusleme published a formulation of based noise analysis using normalized noise power instead of and [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 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 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 subcircuit 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 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 is reduced.

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

2. Formulation of Noise Measurement

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

2.1. Fundamentals of Analysis

The principle is applicable to parameters which are independent of the width of a transistor. Figure 1 shows a transistor with a transconductance, a drain-to-source conductance, and a current

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

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] ,

(1)

where is the Boltzmann constant, is the temperature, is the transconductance, and is the biasdependent noise parameter. According to this model, the approaches 1 when the drain-to-source voltage approaches zero and decreases to when the device enters the saturation regime. The saturation value of is valid for long-channel MOS devices built on lightly doped substrates. Early studies have reported values between and 4 [14] , but recent studies have shown that by accounting for parasitic resistances, is approximately for channel lengths equal or greater than 100 nm [15] .

The thermal noise at the drain terminal of a MOS transistor is

(2)

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

(3)

where is a function of and, as well as, and consequently a function of transistor’s

. Since is proportional to (see Equation (3)), is proportional to according Equation (2). Since is proportional to, becomes independent of. Equation (2) also shows that is inversely proportional to L². Even though the transconductance of a transistor also depends on the L, it is not inversely proportional to L². Therefore, is a function of L. Once of a transistor for a given is characterized over a range of; of the transistor can be generalized to a transistor of arbitrary as long as L and the are constant. We will verify this observation in Section 3. This width-independent property is the crucial link to the 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

(4)

where is a process dependent constant and is the oxide capacitance. Equation (4) can be rewritten to explicitly show its dependence on parameters as

(5)

If we divide by, we find

(6)

Thus, we have in a parameter which depends exclusively on the of a transistor. It should be pointed out that (the current density) is also a function of. Furthermore, Equation (6) implies that any two transistors, regardless of their widths (), theoretically exhibits the same if they are biased at the same. This is a useful property to compare measured noise of different transistors biased at the same.

3. Experimental Results

Section 2 identifies as a function of parameter satisfying all characteristics described in Section 2.1. The 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. We present the noise as a function of and as a function of the frequency. Both and are determined experimentally without curve fitting.

3.1. Noise as a Function of

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 and ten-finger data sets biased at the same voltages.

The average, and 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:. The total width of each transistor is obtained by multiplying by. 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, ,. The and are approximately the same since, and are proportional to the width of a transistor. We conclude that dependent parameters (e.g.,) are independent of.

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

The measured at 100 Hz is dominated by flicker noise. Equation (5) suggests that is proportional to and. the current density is inversely proportional to. Therefore, increases as 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 are width independent, and therefore dependent as stated in Section 2.3.

3.2. Noise as a Function of Frequency

The 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

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 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 based noise analysis. Fundamentals of 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 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.

References