Abstract
Maxwell’s demon is an imaginary entity that reduces the entropy of a system and generates free energy in the system. About 150 years after its proposal, theoretical studies explained the physical validity of Maxwell’s demon in the context of information thermodynamics, and there have been successful experimental demonstrations of energy generation by the demon. The demon’s next task is to convert the generated free energy to work that acts on the surroundings. Here, we demonstrate that Maxwell’s demon can generate and output electric current and power with individual randomly moving electrons in small transistors. Realtime monitoring of electron motion shows that two transistors functioning as gates that control an electron’s trajectory so that an electron moves directionally. A numerical calculation reveals that power generation is increased by miniaturizing the room in which the electrons are partitioned. These results suggest that evolving transistorminiaturization technology can increase the demon’s power output.
Introduction
One famous example of energy generation by Maxwell’s demon is when it observes individual randomly moving gas particles in a box^{1,2,3}. As feedback responding to the observation, the demon opens and closes a gate separating the box into two areas so that colder and hotter gas particles are sorted out. Thanks to such observation and the following feedback, the temperature in one area differs from that in the other, which means free energy is generated and entropy is reduced in the box. The noteworthy feature of this energy generation is that, without friction, Maxwell’s demon does not consume energy during the feedback. This gives rise to a question regarding a violation of the second law of thermodynamics because the demon looks like it is producing energy from a thermal bath without any dissipation.
This question has been answered by the concept of information thermodynamics^{4,5,6,7,8,9,10,11,12,13}: The generated free energy originates from the demon’s memory. The demon uses energy during a cycle of the memory operation for storing the information about randomly moving objects at the measurement and erasing it after the feedback^{4,5}. Consequently, the energy it generates perfectly follows the second law of thermodynamics: When Maxwell’s demon generates energy, a corresponding dissipation takes place in the demon’s memory.
In addition to this theoretical progress, technical advances allowing access to a small system in which a few small objects are thermally fluctuating has led to experimental demonstrations of Maxwell’s demon^{14,15,16,17,18,19,20,21,22,23}, by using rotaxane molecules^{14} and beads a fewhundred nanometers in diameter^{15}. Using electrons, the demon has also realized a Szilard's engine that generates free energy^{18}. Further, an autonomous Maxwell’s demon that enables automatic observation of and feedback to electrons has been recently demonstrated using a chip integrated with superconductor devices, and the thermodynamics of the demon itself was revealed^{20}. In addition, some previous experiments have succeeded in extracting work^{15,20} and power^{22} with the demon. Here we present a demonstration of one of the most natural possible Maxwell’s demon that outputs electric power by rectifying randomly moving electrons in transistors.
Results
Device structure and the demon’s feedback protocol
The power generation by the demon is carried out on a silicon transistor chip integrating a small box (hereafter referred to as a singleelectron box (SEB)) and a charge sensor that monitors electron motion (Fig. 1a,b, also see Methods section and Supplementary Note 1). The SEB is separated from the source and drain by two transistors, G1 and G2, which function as gates for electrons. The number of electrons n in the SEB is around 100 on average. It fluctuates due to thermal agitation (Fig. 1c), which is monitored in real time with the sensor used for the feedback, which is explained next.
When G1 opens and G2 closes, electrons shuttle randomly between the source and SEB due to thermal fluctuation (hereafter referred to as state A). When n increases by Δn_{thresh} larger than zero from n observed initially at state A, G1 and G2 are closed and open, respectively, and then electrons shuttle randomly between the drain and SEB (state B). When n decreases by Δn_{thresh} from n observed initially at state B, G1 and G2 are again open and closed, respectively (state A). This feedback cycle comprising state A and B transfers electrons from the source to drain (Fig. 1d). In other words, the feedback cycle rectifies the random motion of electrons and converts it into electric current I_{MD} (= × e/Δt_{cyc}, where 〈Δn_{t}〉 is the ensemble average of the number of transferred electrons, e is the elementary charge and Δt_{cyc} is the time taken for one cycle in which the state changes from A to B and back to A). Here, I_{MD} is defined as a current flowing from the drain to source (forward current). As can be easily expected, when Δn_{thresh} is negative, current flows in the opposite direction (backward current). When source–drain bias voltage V_{SD} (= V_{S}−V_{D}) is applied across the source and drain, the electric power the demon generates can be evaluated to be −I_{MD} × V_{SD}, where the minus sign represents the current direction against V_{SD}, or electrons climbing up the chemical potential.
Our silicon chip has two unique features that affect the demon’s performance. The first feature is that the transport of electrons between the SEB and source or drain shows directionally asymmetric characteristics^{24} because the transport is dominated by thermal hopping over the energy barrier induced by the transistor’s gate (Fig. 2a). The transition rates for this thermal hopping is tunable by the gate voltage and proportional to exp(−ΔE/k_{B}T), where k_{B} is Boltzmann’s constant, T is temperature and ΔE is the height of the energy barrier the electrons have to surmount. For electrons in the source and drain, ΔE is the difference between the top of the energy barrier and chemical potential μ_{S} and μ_{D} of the source and drain, respectively. Therefore, the transition rate for the electrons to enter the SEB is constant because voltages applied to the transistors’ gates, source and drain are constant during the measurement of electron motion. On the other hand, for electrons in the SEB, ΔE is the difference between the top of the energy barrier and the energy level of an electron in the SEB. This energy level is given by μ_{SEB}+(k−0.5)e/C, where μ_{SEB} and C are the chemical potential and capacitance of the SEB, respectively, and k is the deviation of n from the integer part of the average of thermally fluctuating n. Since the energy level varies according to k, the rate at which an electron leaves the SEB varies with k (Fig. 2b, the details are explained in Supplementary Note 3)^{24}. Therefore, electrons move in a directionally asymmetric way, which is beneficial for power generation by the demon as explained later.
The second unique feature of our chip is that undesirable work applied to the system during the feedback process can be eliminated. One of the most important conditions for the feedback from the demon is that no work be applied to observed objects, but it is difficult to meet this condition in experiments because of the difficulty in partitioning the objects by a gate. Indeed, in previous reports^{15,18,19,20}, work was applied to the objects and then carefully considered as a net value to discuss informationoriginated free energy. In our chip, since transistors partition electrons using gates as mentioned above, the demon can, in principle, perform the feedback process without work exertion. However, in reality, since the transistors tuned by applied voltage V_{G1} and V_{G2} are capacitively coupled to the SEB, the change in V_{G1} and V_{G2} at the switch between state A and B shifts the SEB’s chemical potential μ_{SEB}, and thus an electron in the SEB can gain or lose energy. This energy shift, Δμ_{SEB}, generates current even without the feedback by the mechanism of a singleelectron ratchet^{25} (the details are explained in Supplementary Note 4). To eliminate Δμ_{SEB}, in our operation scheme, we cover the whole area with another gate (hereafter referred to as an upper gate (UG)). The UG is capacitively coupled to the SEB, which enables us to control μ_{SEB} at state A and B separately. When state A and B are alternately switched periodically without the feedback^{26}, even when voltage between the source and drain is zero, current flows due to the ratchet mechanism. This current can be tuned by tuning the UGvoltage difference ΔV_{UG} between state A and B, that is, ΔV_{UG}=V_{UG,A}−V_{UG,B}, where V_{UG,A(B)} is UG voltage at the state A(B) (Fig. 2c). With the difference ΔV_{UG}=0.4 V, no current is generated with the change between state A and B without feedback, which means that the UG enables us to eliminate undesirable work in the demon’s feedback and monitor pure current generated by the demon.
Power generation by Maxwell’s demon
All measurements were carried out at room temperature. The time interval Δt_{m} of each measurement was set to 60 ms, which is short enough to monitor electron motion or k. More importantly, from the viewpoint of the demon’s performance, Δt_{m} was a bit shorter than (or comparable to) the average time intervals for n to increase by Δn_{thresh} at state A and decrease by Δn_{thresh} at state B, which leads to high power generation and relatively high efficiency for informationtoenergy conversion as discussed later. Note that k and Δn_{thresh} are different: k is the deviation in n from the integer part of the average of n (k=n−n_{ave}, where n_{ave} is the integer part of the average of n) and Δn_{thresh} is the deviation in n from the initially observed n in the state and the threshold value to perform feedback. In our experiment, Δn_{thresh} is determined by threshold value of I_{det}. We set the threshold as 1.5δI_{det}, where δI_{det} is the change in I_{det} induced by the motion of an electron. In this work, since I_{det} showed discrete steps (Fig. 1c), the change in I_{det} became larger than 1.5δI_{det} when Δn_{thresh}=2.
With the feedback, even when V_{SD}=0 V, I_{MD} of about 20 zA flows from the drain to source (forward current) (Fig. 3). On the other hand, no current is generated without the feedback (Fig. 2c). Current in the opposite direction (backward current) can also be generated when Δn_{thresh}=−2. In addition, even when negative (positive) V_{SD} is applied to the drain, forward (backward) current counter flows from the drain (source) to source (drain). These results indicate that the demon generates current by rectifying randomly moving electrons. Also noteworthy is that MonteCarlo simulation with consideration of the directionally asymmetric electron shuttling mentioned above (Supplementary Note 2) can reproduce the experimental results well, which means that the directionally asymmetric electron shuttling actually takes place in our silicon chip.
Generated power is given by −I_{MD} × V_{SD}. When V_{SD} is changed from 0 V in a negative direction, the generated power increases because of the increase in V_{SD} and then decreases because of the reduction in I_{MD} (Fig. 4a). This behaviour can also be reproduced by MonteCarlo simulations. In our experiments, maximum power of about 0.5 zW was generated. Such small power and current can be evaluated thanks to realtime electron counting by the highsensitivity charge sensor, which leads to this proofofconcept demonstration of Maxwell’s demon. On the other hand, the generated power can be increased, in principle, by reducing Δt_{m} and the gatetunable transition time for electrons to hop over the energy barrier. However, from the technical perspective, Δt_{m} is limited by the time resolution of the charge sensor. Highspeed charge sensing, for example, using a reflectometry method based on radiofrequency signals^{27} or a faster feedback process using an onchip circuit, can reduce Δt_{m} and thus increase I_{MD} and the power.
Efficiency of the demon’s power generation
The efficiency is defined as the ratio of the generated energy to the mutual information, which represents conversion efficiency from information to energy. The efficiency of informationtoenergy conversion is given by , where ΔF is the change in the free energy and I is the mutual information, and depicts the mean per one measurement. When Δn_{t} electrons are transferred from the source to drain, ΔF becomes Δn_{t}eV_{D}. When the number of measurement points for this electron transfer is N_{t}, is given by . When there is no error in the measurements of n, I equals Shannon entropy H given by , where P_{i} is the probability for the ith event. In our experiments, we use information about n: n is either increased by Δn_{thresh} (decreased by Δn_{thresh}) or not at state A (state B). From the probabilities experimentally obtained, Shannon entropies H_{A} and H_{B} of the measurement outcome at state A and B, respectively, are estimated. When the number of the measurement points at state A and B are N_{A} and N_{B}, respectively, H is given by (N_{A}H_{A}+N_{B}H_{B})/(N_{A}+N_{B}). We assume I=H, or neglect errors in measurement of n. Consequently, since , we can estimate the lower bound of the efficiency based on electron counting statistics in the experiment (Supplementary Note 6).
The efficiency increases with V_{SD} and reaches about 18% at the V_{SD} that provides the maximum power (Fig. 4b). This is because the increase in V_{SD} increases the energy gain of an electron transferred from the source to drain. The efficiency also depends on Δt_{m}: When Δt_{m} becomes longer, the efficiency increases and then saturates (Fig. 4c). The point of this behaviour is that it reflects how much of the information obtained by each measurement is used for the feedback: Measurements in the absence of the feedback waste the obtained information and reduce the efficiency. When Δt_{m} becomes shorter (longer) than the time taken for n to increase or decrease by Δn at state A or B, respectively, the number of measurements without the feedback increases (saturates), which leads to a decrease (saturation) in efficiency (Fig. 4c; the details are explained in Supplementary Note 5). Compared to the simulated efficiency (24%) at Δt_{m} of 60 ms (Fig. 4c), the efficiency of 18% obtained in our experiment at V_{SD}=−34 mV is reasonably large. The further increase of V_{SD} also increases the number of measurements in the absence of the feedback, which leads to reduced efficiency. Due to these features, the efficiency cannot reach 100% like it can in an ideal Szilard’s engine utilizing all of the information. On the other hand, the power decreases with increasing Δt_{m} because the time interval for carrying out the feedback increases with Δt_{m}, which reduces I_{MD}. Therefore, there is a tradeoff between the efficiency and power against Δt_{m} and, in this work, we adjusted it to 60 ms to ensure the reduction in the power is small and the efficiency is relatively large. Note that the origin of the power reduction is the missed detection of fast electron transitions within Δt_{m}. MonteCarlo simulation indicates that the power reduction caused by detection errors is <10% in this work.
Discussion
Numerical calculations indicate another way to increase the power generated by the demon (details are explained in Supplementary Note 2). For simplicity, we assume infinitely fast sensing and feedback and Δt_{m}=0. In the case of Szilard’s engine, the generated energy is given by k_{B}Tln2, which is proportional to temperature. In our feedback with transistors, in which electrons are transported by thermal hopping, the generated power depends on another factor, which is the charging energy (E_{C}=e^{2}/2C) of the SEB. The power increases monotonically with E_{C} (Fig. 4c,d). There are mainly two reasons for this behaviour of the generated power. The first is that the electron can gain energy multiples of 2E_{C} when it enters the SEB from the source, which allows the electron to reach the drain highly biased and thus to gain larger energy. The second is related to how often this energy gain happens at larger E_{C}. As explained above, the transition rate for electrons in the source to enter the SEB is independent of E_{C}. This is in contrast to an SEB sandwiched by tunneling barriers: Larger E_{C} reduces this transition rate due to the Coulombblockade effect, which leads to a reduction of power in spite of larger energy gain. In other words, Maxwell’s demon exploits transistors’ benefits as an increase in power by increasing E_{C}, that is, decreasing C, which can be achieved by miniaturization technology for current transistors. We should note that large E_{C} shortens the time that an electron stays in the SEB and tends to require faster sensing and feedback. At Δt_{m}=60 ms, E_{C} should be smaller than 35 meV to monitor the desired electron motion and perform the feedback before the electron’s departure from the SEB; otherwise, n changes before the feedback is completed.
The transistors provide a benefit to the generated power in this work and also have other merits. Since transistors can act as gates partitioning electrons, which have not been achieved with other systems, the procedure for the demon’s power output is relatively simpler than in previous work. In addition, with an electrical approach, the transistors can not only control the chemical potential and size of the box^{28,29}, in which an electron is confined, but also form tunnel barriers^{30} and a couple of dots^{31,32}. Operation in a wide range of temperatures, including room temperature as in this work, also deepens the analysis and applications of Maxwell’s demon. These features promise a experimental platform on which the demon plays active roles, such as a quantum Szilard engine^{33}, noise squeezer^{21,34} and nanoscale heat engine^{35,36,37,38}. Transistors also have another perspective supported by nanotechnology, which continues to make commercially available transistors smaller and smaller. Further shrinkage of transistors will further improve the controllability of electron motion and charge sensitivity and increase E_{C}, all of which will lead to improvement in the demon’s performance. Therefore, we believe that Maxwell’s demon in transistors continue to improve in performance and find new applications.
Methods
Device fabrication and structure
The device is fabricated from a silicononinsulator (SOI) wafer. First, silicon nanowire channels for a SEB, source, drain and sensor are formed on an SOI layer with boron concentration of 10^{15} cm^{−3}, followed by thermal oxidation. The width and thickness of SOI channels for the SEB and sensor are around 30 and 20 nm, respectively. Oxide thickness is 38 nm. Then, two gates, G1 and G2, composed of polycrystalline silicon are formed on the SOI channel between the SEB and source/drain, followed by oxidation. G2 is designed to be larger than G1 to reduce capacitive coupling between the SEB and drain and suppress changes in the chemical potential at the SEB when voltage applied to the drain is changed. However, there is a risk that small electron traps are unintentionally formed under a large gate due to the structural fluctuation of the transistor channel. Therefore, we use a small G1 to mitigate the risk and a larger G2 to reduce the potential change. The width of G1 and G2 are 30 and 200 nm, respectively. Then the whole area is covered with a 50nmthick oxide interlayer formed by a chemical vapour deposition. Finally, another gate (UG) is formed on the whole area. The UG is used to control the chemical potential of the SEB, induce electrons in the source and drain, and to control current flowing through the channel of the sensor.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Chida, K. et al. Power generator driven by Maxwell’s demon. Nat. Commun. 8, 15301 doi: 10.1038/ncomms15301 (2017).
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Acknowledgements
We appreciate comments on this manuscript from I. Mahboob of NTT Basic Research Laboratories.
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NTT Basic Research Laboratories, NTT Corporation, 31 Morinosato Wakamiya, Atsugi, Kanagawa 2430198, Japan
 Kensaku Chida
 , Samarth Desai
 , Katsuhiko Nishiguchi
 & Akira Fujiwara
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Contributions
K.C. and K.N. conceived the experiments, analysed the data, performed the simulations and prepared the manuscript. K.C. and S.D. performed the measurements. K.N. fabricated the devices and performed the numerical calculations. All authors discussed the results and commented on the manuscript. A.F. supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Kensaku Chida.
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