Direct white noise characterization of short-channel MOSFETs

On-wafer evaluation of white thermal and shot noise in nanoscale MOSFETs is demonstrated by directly sensing the drain current under zeroand nonzsero-drain-bias ( Vd ) conditions for the first time, without recourse to a hot noise source, commonly needed in noise figure measurement. The dependence of white noise intensity on the drain bias clearly shows thermal noise at Vd = 0 V and shot noise at Vd > 0 V with its gatebias-dependent suppression. An empirical expression for the Fano factor (shot-noise suppression factor) that is wellbehaved even at Vd = 0 V exactly and suitable for measurement-based evaluation is proposed. The direct measurement approach could allow more accurate and predictive noise modeling of RF MOSFETs than has conventionally been possible.


I. Introduction
MOSFETs are known to exhibit flicker (or 1⁄ ) noise at low frequencies (LF) and white noise at high frequencies (HF). Physical origins of flicker noise and white noise differ, and therefore, their power spectral densities are independent of each other. White noise can be further classified into thermal noise and shot noise, among others. Unlike the relationship between flicker noise and white noise, thermal noise and shot noise are not completely independent of each other [1][2][3][4][5].
Physics of noise in deep-submicrometer MOSFETs is much more involved than that in long-channel MOSFETs [6][7][8][9][10]. HF noise critically affects operation of very wideband RF circuits, especially millimeter-wave circuits, because the noise integrated over a wide bandwidth contributes to the signal-tonoise ratio (SNR) of a wireless system. Recently, adverse effects of white noise from a local oscillator on wideband millimeter-wave circuits have experimentally been confirmed [11,12]. Accurate noise measurements of transistors over a wide frequency range, including the white noise region, are essential not only for elucidating the physics behind but also for developing predictive device noise models for circuit design.
White noise in MOSFETs is typically characterized by noise figure (NF) measurement at RF, where flicker noise is negligible. In the Y-factor method of NF measurement, a known good "hot" white noise source is required to provide two ("hot" and "cold" (room temperature)) reference noise temperatures [13,14]. The hot or cold white noise is impressed to the device under test (DUT) during the measurement. The method is typically applicable down to 10 MHz at best and usually requires a complicated de-embedding procedure, which adds to measurement uncertainty. LF (flicker) noise measurement, on the other hand, is performed by directly sensing noise generated by the DUT under "cold" dc-biased conditions. Several systems for LF noise measurement are commercially available. Although a typical maximum measurement frequency is a few tens of MHz in product specifications, in practice, frequency roll-off due to parasitic capacitance tends to make the maximum measurable frequency considerably lower, often below 1 MHz. Noise measurement around the MHz range is not very well covered by either method. Since typical MOSFET noise spectra become white only above 1 MHz or higher, it is difficult to apply commercial LF noise measurement systems to white-noise measurement.
There have, nevertheless, been some reports on "cold" direct noise measurement of discrete MOSFETs over frequencies ranging from flicker noise region to white noise region [15,16]. We also chose a "cold" dc-biased approach for direct on-wafer device noise measurement. A proof-of-concept noise probe that extended the maximum measurable frequency to above 100 MHz was demonstrated in [17 ,18]. It has a broadband low-noise amplifier (LNA) built in the probe itself, thereby reducing the parasitic capacitance significantly. At present, an improved version of the noise probe is commercially available [19]. Measurements can be conducted even at and near zero drain bias ( d = 0 V), which is not always straightforward by other means of measurement.

II. MEASUREMENTS
In this study, we demonstrate measurement of MOSFET white noise by utilizing a noise probe, shown in Fig. 1(a). Figure 1(b) shows a schematic diagram of the measurement system. The ac-component (i.e., noise) of dc drain current is amplified by the LNA in the noise probe and its output is read by a spectrum analyzer (N9030A, Agilent Technologies). Each individual noise probe is calibrated by extensive measurements, and that information is stored in a postprocessing software program [19]. It calculates the drain current noise from raw power spectrum reading by the spectrum analyzer, accounting for the noise generated by the LNA. The noise generated by the spectrum analyzer itself is accounted for, in effect, by turning on its Noise Floor Extension option [20]. Measurement results at low frequencies (< 100 kHz), where commercial LF measurement solutions work reliably, correlate well with results from such a system (9812D, ProPlus Design Solutions). In all measurements presented in the following, we employed a semiconductor device analyzer (B1500A, Agilent Technologies), equipped with four high-resolution sourcemeasure units (HR-SMUs), for biasing DUTs. Custom-built low-pass filters were used to filter out noise from the HR-SMUs.

Direct white noise characterization of short-channel MOSFETs Kenji Ohmori and Shuhei Amakawa
Device Lab Inc., Tsukuba, Ibaraki, Japan (e-mail: ohmori@devicelab.co.jp) We characterized N-MOSFETs with gate length/width of 120 nm/10 µm, which were fabricated by using a 0.13-µm CMOS technology. Each DUT has two sets of ground-signalground (GSG) pads: one connected to the drain and the source, and the other connected to the gate and the source. The gate was biased through an RF probe and the drain was biased through and probed by the noise probe. All measurements were conducted at room temperature between 24 and 26°C. Figure 2 shows dd curves of an N-MOSFET at gate voltages, g , of 0.5, 0.6, 0.,, and 0.8 V. The symbols on the curves represent bias points where noise measurements were carried out. The drain current increases even in the saturation region owing to the short channel length ( = 120 nm). The threshold voltage estimated from an dg curve (not shown) at d of 30 mV is 0.44 V. Figure 3(a) shows drain-current noise power spectral density, Id , for d = 0 V. The spiky peaks observed in the frequency range from 20 to 400 kHz were induced by gate biasing. The white noise seen above 1 MHz can be regarded as thermal noise associated with the differential resistance dif = d / d [21]. As g increases, dif decreases, resulting in higher white noise levels. We extracted Id values at 500 MHz and compared them with theoretical values in Fig. 3(b). The horizontal axis is the theoretical thermal current noise spectral density th = 4 B / dif at d = 0 V, where B and are the Boltzmann constant and the DUT absolute temperature, respectively. A very good agreement was obtained from 7 × 10 −24 to 2 × 10 −22 A 2 /Hz, corresponding to dif values from 16,0 to 83.5 Ω, respectively.

III. RESULTS AND DISCUSSION
When a nonzero d is applied, a dc drain current flows. Figure 4(a) shows the Id spectra of six MOSFETs with the same dimensions under the same bias condition ( d = 0.2 V, g = 0.7 V). Flicker noise is considered to result from a large number of traps located near the channel [5]. Each trap exhibits a Lorentzian power spectrum with a certain time constant, also known as the burst noise [21]. In nanoscale MOSFETs, the number of traps in a device is small and some Lorentzian components become visible as in Fig. 4(a) [22,23]. Note that Lorentzian components below 10 MHz do not usually affect the shape of Id ( ) above 100 MHz because each of them drops with 1 2 ⁄ . Therefore, Id ( ) above 100 MHz can be regarded as the sum of white noise and residual flicker noise. The noise spectral density above 100 MHz can, therefore, be approximated as   where − (≈ −1) is the slope of flicker noise power spectrum on a log-log plot, w is the white noise level, and ( w ) ⁄ 1/ is the corner frequency (onset frequency of the white noise region). Figure 4(b) shows Id ( ) at g = 0.8 V with d ranging from 0 to 0.8 V. It clearly shows that predominant noise changes from flicker noise (1⁄ ) to white noise (1 0 ⁄ ) as frequency becomes higher. As d becomes larger, flicker noise power increases, resulting in a higher corner frequency. Although the slope of Id ( ) becomes very small at high frequencies (> 100 MHz), the slope for large d values is not quite zero. Note that the absolute value of the slope of (1) above the corner frequency is smaller than (≈ 1). Figure 5 shows dependence of Id on d and g . Open circles and diamonds show measured Id at 300 and 500 MHz, respectively. As seen in Fig. 4(b), these values are nearly the same in the lowd region ( d < 0.1 V). Meanwhile, as d increases, Id at 500 MHz becomes lower than that at 300 MHz due to higher levels of residual flicker noise (Fig. 4(b)). The solid squares in Fig. 5 show estimated white noise w , calculated by using Eq. (1) with = 1 and measured values of Id at 300 and 500 MHz, from which , too, can be determined. Using measured dd curves (Fig. 2), we calculated the full shot noise intensity, 2 d , shown by the thick solid (orange) lines in Fig. 5. Theoretical d -dependent thermal noise intensity is shown in Fig. 5 by the dashed (green) lines, given by [21,24]   where ch ( d ) = d / d is the chord resistance [25]. Note that ch ( d ) ≠ dif ( d ) unless d = 0 V. Note also that we used (2) instead of the better-known integral expression [10,21], because the latter requires more a priori knowledge about the DUT, making its purely measurement-based evaluation difficult.
The w values at d = 0 V, shown in Fig. 5, agree well with th , consistent with Fig. 3(b). Given the fact that experimentally observed shot noise is usually lower than the full shot noise [26], w should lie somewhere between th and th + 2 d . The latter is shown in Fig. 5 by the dot-dashed (blue) lines. As d increases, w increases similarly to 2 d . The d value at the point of intersection of 2 d and th could be regarded as the onset point of shot noise dominance. Notably, w of Fig. 5(a) ( g = 0.5 V, moderate inversion), assumes a minimum value at d ≈ 0.04 V. Although the difference between the observed minimum and the w value at d = 0 V is small, most of our DUTs showed similar behavior at g ≲ 0.5 V. It is known that, in weak inversion of a long-channel device, the following white noise power expression (3) can be derived whether assuming a thermal noise origin or a shot noise origin [4,5].
where sat is the saturation current corresponding to the given gate voltage. (3) implies that at d = 0 V, w,wi = 4 sat = th . Although our devices are short-channel and we have no weakinversion data because of the limited device width of = 10 µm, the observed minimum in w at d ≈ 0.04 V could be related to the second term of (3), originating from electrons flowing in the reverse direction (from drain to source). Theoretically expected decrease in w (versus d ) in strong inversion due to channel pinch-off [5] is not clearly observed in Fig. 5(d).
Fano considered a theoretical limit for the statistical fluctuation in the number of traveling particles [2,]. The full shot noise corresponds to the situation where statistical fluctuation is maximized. The so-called Fano factor (≤ 1) [28] is a coefficient for shot noise suppression and defined as the ratio of the actual shot noise power to the full shot noise power 2 d . has successfully been employed for describing white noise of nanoscale MOSFETs when d is sufficiently high [26]. However, the conventional expression, = Id / [2 d coth( d /2 B )] [26], is not suitable for measurementbased evaluation of near d = 0 V. To find an expression suitable for measurement-based evaluation, it should be defined such that it equals unity at d = 0 V in weak inversion, considering (3), which includes shot noise contributions from currents flowing in both directions [4], and Fig. 3(b). We herein introduce an empirical expression for the Fano factor that exhibits the desired behavior, as follows: where d is the measured drain current, th is given by (2), and w is the measured (estimated) white noise. The denominator of (4) corresponds to the dot-dashed (blue) lines in Fig. 5. The th in the denominator makes up for the "lost" full shot noise when d is below a few times the thermal voltage, B / ≈ 0.026 V. Figure 6 shows the suppression factor as a function of d for g = 0.5, 0.6, 0.7, and 0.8 V. decreases as g increases presumably due to the reduction of the potential barrier near the source [29]. Shot noise is generally considered to be generated primarily due to the presence of this potential barrier, although quasi-ballistic transport in the pinched-off region near the drain may also play a role in short-channel devices. More precise discussion will require physical modeling and simulation including quasi-ballistic transport and Coulomb interaction [6,30].
IV. CONCLUSIONS In summary, we have developed a methodology for conveniently characterizing white noise of dc-biased nanoscale MOSFETs through wafer probing without a hot noise source or complicated de-embedding procedures. Using the approach developed, we have successfully measured white noise in shortchannel MOSFETs under various bias conditions, including d = 0 V. We also proposed an empirical expression for the Fano factor (shot-noise suppression factor) , which is wellbehaved at d → 0 V and suitable for purely measurementbased, modeling-free evaluation.
In general, various factors should affect , such as device structures and dimensions, impurity profiles, and choice of materials. Therefore, actual measurement of devices will be important for understanding of device physics. Note also that in circuits like an FET resistive mixer, which boasts the best lownoise performance among FET mixers [31], FETs are used under a zero-drain-bias condition. Accurate noise modeling, including near d = 0 V, therefore is very important. Actual measurement of white noise will be essential for developing a predictive transistor noise model valid in all regions of operation.
ACKNOWLEDGMENT KO would like to thank Prof. N. Sano of University of Tsukuba for critical comments on an early version of the manuscript.