

Feature Articles: Frontier Research on Lowdimensional Semiconductor Physics Noise in Nanometerscale Electronic DevicesAbstractDownsizing electronic devices in integrated circuits increases the noiserelated degradation of circuit performance, and thus, it is becoming more important to analyze noise with singleelectron resolution. We discuss here the use of a nanometerscale transistor and capacitor in analyzing thermal noise, one of the most fundamental types of noise in electronic devices, with singleelectron resolution. When the capacitor size is substantially reduced, the wellknown model of thermal noise is no longer valid, and voltage noise is squeezed. Keywords: noise, transistor, single electron 1. IntroductionIntegrated circuits comprising silicon transistors and other electronic devices are extremely important components in our information technology society. The transistor, one of the most wellknown electrical devices, controls the flow of electricity, that is, electrons, for data processing in the integrated circuits. The performance of such integrated circuits has advanced with the downsizing of transistors; for example, the 14nm generation of transistors will be reached in 2015. However, downsizing also increases the noiserelated degradation of circuit performance, and therefore, a microscopic analysis of noise and the development of countermeasures to such noise have become more important than ever before. Indeed, random telegraph noise originating from singleelectron trapping in a transistor becomes prominent in small transistors and gives rise to errors in memory circuits. Shot noise originating from the discrete nature of an electron is also expected to become a serious issue in highspeed integrated circuits. Therefore, the downscaling of the transistors increases not only their performance but also the importance of understanding noise microscopically, ultimately with singleelectron resolution, in small transistors as well as in other electronic devices. In this article, we discuss the analysis of thermal noise, one of the most fundamental types of noise in electronic devices, with singleelectron resolution using dynamic random access memory (DRAM) composed of a nanometerscale transistor and capacitor [1]. Because the analysis was done by using a DRAM, one of the most common devices, we believe that our results are relevant to all electronic devices. 2. Techniques for analyzing thermal noiseOne of the simplest and most wellknown techniques for analyzing thermal noise is to measure voltage noise in a circuit composed of one resistor and one capacitor, as shown in Fig. 1. In this circuit, a massive number of electrons enter and exit the capacitor through the resistor in a random manner with the assistance of thermal energy. This random motion of electrons is thermal noise, and it exists absolutely at a finite temperature. When voltage noise at a node between the resistor and capacitor is measured with an oscilloscope or voltage meter, we can obtain certain information. For example, we can get a histogram of voltagenoise amplitude based on a Gaussian distribution, whose average is voltage V applied to the resistor. We can also get the variance V_{var}^{2} of the distribution given by k_{B}T/C, where k_{B} is the Boltzmann’s constant, T is temperature, and C is the capacitance of the capacitor. The fact that V_{var}^{2} is proportional to T means that the random motion of electrons originates from thermal energy. In contrast, V_{var}^{2} increases as C decreases, regardless of the resistor. Therefore, the downsizing of electronic devices in circuits such as memory and analog circuits reduces C and thus increases noise, which leads to an undesirable hindrance of device downsizing. These features can be explained by the wellknown JohnsonNyquist model proposed in 1928 [2, 3].
In our technique for analyzing thermal noise with singleelectron resolution, we use instead of a resistor a nanometerscale transistor for transferring the electrons to the capacitor, as shown in Fig. 2(a), because electrical control of the resistance of the transistor helps us to monitor the motion of individual electrons precisely in real time. In a common circuit, time intervals for the electrons to enter and exit the capacitor are too short for the electrons to be monitored: current I of 1 mA corresponds to the time interval of 1.6 × 10^{−16} (=e/I) seconds, which cannot be measured by any measurement system. However, increasing the resistance of the transistor when it transfers electrons prolongs the interval so that the electron motion can be monitored in real time. These mechanisms and the structure are the same as in a DRAM. Since the charge of an electron is too small (1.6 × 10^{−19} C) to be detected by any charge sensor, the tiny signal is amplified by using another transistor [4]: The transistor for the charge amplifier has an extremely small channel (~10 nm) and is integrated with the capacitor, as shown in Fig. 2(b). Consequently, the electrons in the capacitor can be detected precisely with singleelectron resolution even at room temperature. This success is supported by wellestablished fabrication techniques for silicon transistors. Using these features, we can monitor the random motion of individual electrons entering and exiting the capacitor through the transistor in real time, as shown in Fig. 2(c): An output signal from the chargeamplifier transistor changes among discrete values, and one gap between these discrete values corresponds to the charge from one electron. To slow down the electron motion so that it can be monitored, the resistance of the transistor used for the electron transfer is adjusted to be around 10^{20} ¦¸. This extremely high resistance, which cannot be achieved in conventional resistors and transistors, is possible thanks to our highquality nanometerscale transistor.
3. Thermal noise with singleelectron resolutionIn analyzing the thermal noise with singleelectron resolution, we first discuss the fluctuation, that is, the deviation from the average, of the number of electrons in the capacitor. Because singleelectron injection into the capacitor increases the voltage of the capacitor by e/C (= dV), the fluctuation of the number of electrons in the capacitor can be converted into a voltage fluctuation, that is, voltage noise, at the capacitor. The dV can be evaluated from the change in the output signal originating from one electron, as shown in Fig. 2(c) [4]. A histogram of the number of electrons in the capacitor is shown in Fig. 3(a). The histogram follows a Gaussian distribution (solid line), as in the case shown in Fig. 1. This histogram gives another piece of information: dV^{2} multiplied by the variance of the distribution shown in Fig. 3(a) corresponds to V_{var}^{2}, and this V_{var}^{2} follows k_{B}T/C, as shown in Fig. 3(b), which is the same as the case where thermal noise originates from the fluctuation of a massive number of electrons, as explained above and as shown in Fig. 1. Consequently, we can conclude that the wellknown JohnsonNyquist model for thermal noise is adaptable to the random motion of single electrons.
However, when charging energy E_{C} (= e^{2}/2C) for injecting one electron into the capacitor is larger than thermal energy k_{B}T, the JohnsonNyquist model is no longer valid. A histogram of the number of electrons in the capacitor at E_{C} > k_{B}T is shown in Fig. 4(a). Since the distribution (solid line) becomes sharper, and the available number of electrons becomes smaller than in the case of E_{C} < k_{B}T (Fig. 3(a)), the discrepancy between the experimental results (bars) and expected values (solid line) becomes larger in the form of quantization errors in digital circuits. Indeed, V_{var}^{2} (1.08 × 10^{−4} V^{2}) evaluated from the variance of the experimental distribution shown in Fig. 4(a) is smaller than k_{B}T/C (1.38 × 10^{−4} V^{2}). This deviation from the JohnsonNyquist model is similar to the case of electromagnetic radiation from a black body. Energy of the electromagnetic radiation is quantized by multiples of hν, where h is Planck’s constant and ν is the frequency of the electromagnetic radiation. Therefore, when a higher ν makes hν larger than thermal energy k_{B}T, the thermal energy cannot assist the emission of electromagnetic waves from the black body. This idea, based on energy quantization proposed by Max Planck, overcomes the ultraviolet catastrophe in the lowfrequency region of the spectrum of blackbody radiation. Our results for electrons can also be explained qualitatively by the same model as that for the blackbody radiation. Energy for injecting N electrons into the capacitor is given by (Ne)^{2}/2C = N^{2}E_{C} and quantized because e is a unit charge. Therefore, as in the case of blackbody radiation, when E_{C} is comparable or larger than k_{B}T, the thermal energy cannot assist electrons in entering and exiting the capacitor, which suppresses electron random motion and thus makes V_{var}^{2} smaller than the thermal energy. However, we can observe a unique feature in electron motion unlike in the case of blackbody radiation. The average of the number of electrons in the capacitor is given by CV/e, where V is the voltage applied to the transistor as shown in Fig. 2(a). However, for example, when the average number is 0.5, the number of electrons in the capacitor switches between 0 and 1 with the same probability because an electron cannot be divided in half. As a result, the variance of the electron number becomes larger than 0.25, regardless of E_{C} and k_{B}T. In this sense, the fluctuation in the number of electrons depends on the average number of electrons. When the average number is an integer, the fluctuation becomes minimum; when a fractional part of the average number is 0.5, the fluctuation becomes maximum. The change in voltage noise V_{var}^{2} at the capacitor as a function of C is shown in Fig. 4(b). When E_{C} < k_{B}T at larger C, V_{var}^{2} follows k_{B}T/C (open circles) and increases as C decreases. When E_{C} > k_{B}T at smaller C, V_{var}^{2} deviates from k_{B}T/C, and the reduction in C increases the discrepancy between V_{var}^{2} and k_{B}T/C, as indicated by the shaded area depicting the possible values of V_{var}^{2}. It should be noted that the minimum border of the shaded area in Fig. 4(b) decreases with C, behavior that is opposite to that of the conventional k_{B}TClimited case depicted by the open circles. In the conventional case valid at E_{C} < k_{B}T, the reduction in C increases noise, which is a serious issue that hinders the downsizing of electronic devices. However, the case where E > k_{B}T at much smaller C is preferable to the reduction in C due to noise reduction, which accelerates the downsizing of electronic devices.
4. ConclusionAs the downsizing of electronic devices accelerates, thermal noise will continue to deviate from its wellknown behavior. However, this does not mean that any new phenomena will appear. Instead, it means that all of the results observed for single electrons follow wellestablished thermodynamics, for example, a Boltzmann distribution, and are thus valid for all electronic devices. Therefore, we believe that our results are important in the downsizing of devices to achieve a reduction in the number of electrons in the device. Indeed, using nanometerscale transistors, we have developed memory and data information circuits that use one electron as one bit of information. In these circuits, errors caused by thermal noise represent one of the most serious issues. However, a wider view is that the minimum energy consumed for computation by data information circuits is governed by the thermal energy. Therefore, the analysis of thermal noise is very important for realizing lowpowerconsumption circuits. References
