Nanoelectromechanical Systems
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 6 december 2006 om 15.00 uur door
Omar Azim USMANI
Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. G. E. W. Bauer
Samenstelling van de promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. ir. G. E. W. Bauer Technische Universiteit Delft, promotor Dr. Ya. M. Blanter, Technische Universiteit Delft
Prof. dr. Yu. V. Nazarov, Technische Universiteit Delft Prof. dr. H. S. J. van der Zant, Technische Universiteit Delft Prof. dr. P. J. French, Technische Universiteit Delft
Prof. dr. ir. B. J. van Wees, Rijksuniversiteit Groningen, Nederland Prof. dr. K. Flensberg, University of Copenhagen, Denemarken
Dr. Ya. M. Blanter heeft als begeleider in belangrijke mate aan de totstand-koming van het proefschrift bijgedragen.
Printed by: Ponsen & Looijen b.v., Wageningen, The Netherlands Casimir PhD Series, Delft-Leiden, 2006-16
ISBN-10: 90-8593-023-5 ISBN-13: 978-90-8593-023-5
Copyright c 2006 by Omar Usmani
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or me-chanical, including photocopying, recording or by any information storage and retrieval system, without permission from the publisher.
Welcome to the only part that most of you will read (apart from the proposi-tions). I know your expectations are high. You of course hope to be mentioned and have some nice and funny words about you. If you are disappointed because of your absence, I apologise. There are mainly two reasons for that: either I don’t like you or I forgot you. The other possible source of disappointment is that some of you may expect some creative and funny writing. Inspiration may come to visit me before I finish these acknowledgements, otherwise it will come later in the form of a goodbye message. Anyway, this part is to thank all the people that made these four years so great.
The first person I will mention is my advisor, Yaroslav. I actually met Yaroslav quite some time ago, in my first year of studies at the university of Geneva, where he was a teaching assistant. In the following years, he was a teaching assistant for various courses I followed. I will always be amazed by the depth of his knowledge as well as his ability to explain things clearly. Of course his interests and knowledge go beyond physics. His knowledge about culture is really impressive (sorry for taking your DVD hostage for so long). Even more impressive is his ability to learn languages. I thought I was good at languages before meeting him. I must even admit to my shame that he knows more French physics vocabulary than I do. I will always be grateful for everything.
It was a great chance to have Gerrit as a promotor, despite not having many conversation about actual physics. His knowledge of the politics of academics have been a great lesson that should be useful in the future. His permanent push for the group to be a nice and fun place was also very much appreciated and contributed greatly to the nice atmosphere I enjoyed here.
Yuli’s guidance that contributed to Chapters II and IV of this thesis was incredibly useful. His advice also would come as a mix of very clear and easy to understand things, mixed with more cryptic ones. After I would leave his office, sure of how sound and important his advice was, I would have to spend some time to understand the full extent of what he had said. Then I would truly understand how deep his knowledge of the problem was and how right what he had said was. For this, he truly deserves the title of Delft Oracle.
some normality among the craziness of this group. The great cheerfulness of Miriam for whom everything is great and fantastic was very much appreciated. I would like to thank Yvonne for helping me through the jungles of various ad-ministrations and for making sure that people would remember their birthdays and bring cake. I would also like to thank Henk, Kok, Dima B.,Inanc, Michael, Max, Hatami, Alessandro,Izak, Youjin, Hong Duo, Udo and Henri. I wish I had spent more time with you so that I could say something nice and funny about you.
Now comes the main part, which goes about the people I spent the most time with. It is natural to start with the people that shared my room, the infamous F328 people.
First comes Sijmen. We actually started the same day, but he was smarter than me in scamming FOM for more money and will enjoy the life of a Ph.D. student for a few months more than me. This also means that any jokes made during my graduation party may be the source of a revenge (this should not be a deterrent, though: please go ahead with any ideas you may have). He was one of the main sources from which I learned Dutch. He was patient enough to endure the torture I made to his language in the beginning. He is also probably one of the most normal persons I met in a physicist’s disguise. It was a great pleasure to have serious, actual conversations rather than fights about some secondary details, as well as less serious ones. It was great to have someone with cultural references and taste that would match mine, since I felt a little bit lost among people only praising so called higher forms of culture. He was the only one could actually discuss the greatness of Dutch television without having a scorning look in return. I am sure he will enjoy a great life, both on the personal level and on the professional one, whether he chooses to become a rich manager or a stay at home dad.
Fabian was also a great roommate. It’s too bad that he only was here for the last year of this thesis. It was always great fun to speak with him in the office, or outside. It was great to be able to discuss some of the subtleties of Swiss and world politics as well as sharing some great and funny discussions. He was also by far the most active guy outside of it and it was always a pleasure to go out with him. His knowledge of languages as well as his willingness to use them was most impressive. I was really impressed by the extent of his family that seems to cover the entire world as well as an incredible array of professions. This is probably the reason why he is the person for whom it is the hardest to predict a future. He may end up as a barman on a South American beach, a Swiss politician, a soap opera actor, a very rich businessman, a scam artist, a racing sailor on Alinghi, a multilingual tourist guide, or anything that might go trough his mind. In any case, I am sure he will have a lot of fun.
left to Regensburg.
Gabriele was the other person I spoke to extensively when applying here. Despite the fact that he is a Jazz music and Macintosh lover, I really had some nice moments with him.
Markus was unfortunately not present very often, as he preferred spending his time in Leiden. This had the great advantage of blocking one desk and preventing undesired people from invading F328.
The last two occupants of F328 I would like to mention are Dima F. whose ranting about many things, as well as his passion for destruction were entertain-ing and Richard. They both chose to play with money instead of continuentertain-ing in the paths of physics. I am sure they enjoy their choice very much.
Of course, there were many other people of the group I spent quite some time with, often invading their offices. The first I would like to thank is Rachid. I had the privilege of being his assistant supervisor for his bachelor and master projects. His work is integrated in Chapter III. It was a great satisfaction for me to be able to help him obtain his degree. Outside of physics, I really appreciated his passion for football and that he made me discover and enjoy Moroccan food. Since I am on the subject of mentioning French-speaking master students, I also would like to thank Catherine who was always cheerful and a great addition to the atmosphere of the group. Her willingness to speak French with me was greatly appreciated.
One of the great things that made me appreciate my time here is that many of the Ph.D.s started at relatively close times. This brought a feeling of closeness and made us share lots of time both at the university and outside of it.
Among these people is Oleg, who would wait until it was late at night, until the building was almost empty, until it was dark, until I would rest my eyes from a hard day of work to sneak in my office and shout ”Food, Girls !”. This was a great test of my cardiac resistance. Maybe one day I will understand what that joke is about. His liveliness and energy have been most noted after he left, as the theory group suddenly became as silent as a cemetery library.
Discussions with Jens ranged from very reasonable (about physics, even) to somehow extremely strange, which was a great pleasure. His obsessions ranged from dancing Salsa while walking from one office to another, trying to assassinate people with volleyballs (I would like to thank him to let me hang out with his team at PUNCH parties, as my own team mates from BL were not there very often), taking and showing pictures of Japan and Japanese people, to eating extremely strange Scandinavian foods.
different translations I needed in this thesis.
I had many great moments with Alex, sharing extremely crazy conversations. He is truly one extreme philosopher. The numerous activities we had outside the university were also great. I wish him and Elvin a great time in Texas (chasing cows on a horse with a shotgun and driving SUVs should be fun) and in the rest of his career.
I was impressed by Freek’s idea of no compromise. That he would not be willing to surrender his beliefs took lots of courage. I am really happy it turned out nicely in the end.
Antonio, despite being a postdoc was really well integrated among us, al-most like a Ph.D. student, though he had rather the role of the wise guy among young and na¨ıve students, knowing everything about different rumours circu-lating around. Of course, to counter this impression, he would sometimes wear hipper clothes, such as pink ”Mamma Mia” shirts.
Xuhui’s dry humour is incredible and I appreciated it very much. And I must admit I am very jealous of your blog. I hope you will keep updating it with your great entries.
With Moosa, my taste for various discussions was extremely pleased. It was really interesting to interact with someone with a truly different mindset. I hope it made me more open to other ways of thinking.
Jeroen’s ability to follow two paths, physics and music, so deeply is truly amazing. His cooking abilities were also great. I wish him the best of luck with his new roommate.
Thanks also to Babak for his great stories, to Wataru for his extremely friendly attitude , to Hayk for enduring my jokes about Armenia, and to Vitaly for his passion about politics .
The first two people that I saw graduate were Siggi and Dani. I was really impressed by the scale of the ceremony and the parties. I could not imagine how I could make it trough them. I still can’t. They were really great and friendly guys. I wish I could have spent more time with them.
One of the great things about studying in Delft is the great interaction a theoretician can have with experimentalists. They really taught me a lot, es-pecially to keep some contact with reality. In particular, I would like to thank Pablo and Sami for showing me how contrast is everything. Other people I had numerous meetings with are the NEMS people (Herre, Benoˆıt, Menno and oth-ers) thanks for making me understand NEMS through their actual realisations and for the great moments at various conferences.
There are also some experimentalists with whom I had lots of contacts, dis-turbing them daily in their busy schedules, forcing them to come to activities not related to physics and sometimes shocking them by crazy statements: Saverio, Monica, Anna and Paul, thank you and sorry for all the distractions.
There were lots of non-physicists that I met here and that made life great. It was always a pleasure to meet so many people from different places as I did in Sebastiaansbrug, for example. It is somehow a pity I did not manage to keep much contact with them, but they really helped me start my time here in a great way.
I would like to thank Vika not only from saving me from a Russian jail, but also for all the great parties she hosted, and more generally for being an incredible friend. I wish her a life full of happiness.
Among the great moments I had in The Netherlands, many are related to HSK. The relief of having a physical activity on the side helped with the stress a theoretician endures (though all these goals they scored did not always make me happy). Playing some competition was also great, but the best part was related to the numerous and great parties both in The Hague and in the tournaments we had in Belgium, France and The Netherlands. It was a fantastic experience to meet people form such various horizons, both geographically and from the point of view of their activities. It was also impressive to see how modest so important persons could be, both on the field and outside of it. Thanks guys ! Of my Geneva friends I would like to thank Micha¨el for the incredible amount of time spent on the phone talking about very deep and philosophical things and Patrick for hosting memorable parties.
I would finally like to thank my family for their support from abroad. I really wish my father was there to witness this moment. If I somehow get my diploma, my mother will have something to put on her wall. I am glad that my brothers Alim and Imam have found study paths they really seem to appreciate.
Quotes
I would like to conclude by showing some quotes people have said about me during these years:
He dared to call our favourite activities gay. We are not gay ! Dima F. and Vitaly B., Ukrainian (martial) artists
I don’t see him very often at the TU. He is such a slacker. Babak H., Iranian Blasphemator
He made me defend my country ! He turned me into a nationalist ! Sijmen G., Dutch Seismologist
Omar will become a president of something. Oleg J., Russian Energy Source
Unfortunately Omar and I have the correct orientation. Alex K., Russian Philosopher
Then, there is Omar, able to comment on anything for as long as one wishes to listen.
Jens T., German Japanophile
His statements and behaviour do not make him a good Green. Wouter W., Dutch Tree Hugger
He is so na¨ıve: I can make him believe anything ! Victoria Z., Russian Mouse Killer
I. Introduction . . . 1
I.1 What nanoelectromechanical systems are. . . 1
I.1.1 Definition . . . 1
I.1.2 Examples . . . 1
I.1.3 Shuttles . . . 2
I.2 Ultra-sensitive measurements . . . 3
I.2.1 Mass measurements . . . 3
I.2.2 Spin . . . 4
I.2.3 Charge . . . 5
I.3 The measurement of the quantum of thermal conductance. 6 I.3.1 What the quantum of thermal conductance is . . . 6
I.3.2 The experiment measuring it . . . 6
I.4 The quantum limit on the position of a harmonic oscillator. 7 I.4.1 What the quantum limit is . . . 7
I.4.2 The standard limit . . . 7
I.4.3 The limit on a linear amplifier . . . 8
I.4.4 How close observations are . . . 9
I.5 Superpositions in macroscopic objects . . . 9
I.5.1 About superposition of macroscopic objects . . . 9
I.5.2 The Cooper-Pair Box cantilever scheme . . . 10
I.5.3 The Aharonov-Bohm carbon nanotube scheme . . . 11
I.6 Single-electron Transistors . . . 11
I.6.1 What single-electron transistors are . . . 12
I.6.2 Coulomb Blockade . . . 12
I.6.3 Tunnelling rates . . . 13
I.7 NEMS and SETs . . . 15
I.7.1 Exciting the vibrationnal modes of NEMS . . . 15
I.7.2 Excitation spectra and Coulomb diamonds . . . 15
I.7.3 Observation of vibrationnal spectra . . . 16
Contents
II. Strong Feedback and Current Noise in Nanoelectromechanical
Systems. . . 27
II.1 Introduction . . . 27
II.1.1 Model . . . 27
II.2 Calculations . . . 30
II.3 Probability Distribution . . . 33
II.3.1 Discussion . . . 33
II.3.2 Expansion . . . 36
II.3.3 Two peaks . . . 37
II.4 Current and noise . . . 41
II.4.1 Current . . . 41
II.4.2 One Peak Noise . . . 41
II.4.3 Noise for Two Peaks . . . 48
II.5 tunnelling rates for which instabilities are created. . . 49
II.6 Numerical Results . . . 50
II.6.1 Probability Distribution . . . 50
II.6.2 Current . . . 51
II.6.3 Noise . . . 53
II.7 Influence of the mechanical friction . . . 54
II.7.1 Effects on the motion . . . 54
II.7.2 Effects on current and noise . . . 54
III. High-frequency limit . . . 61
III.1Introduction . . . 61 III.2Calculations . . . 61 III.3Probability Distribution . . . 65 III.3.1 Discussion . . . 65 III.3.2 Expansion . . . 66 III.4 Noise . . . 66 III.4.1 Noise . . . 66 III.5Numerical Results . . . 69
III.5.1 Probability Distribution . . . 69
III.5.2 Current . . . 70
III.5.3 Noise . . . 70
IV.Negative Capacitance Induced Current Enhancement in a Ra-diofrequency Single-Electron Transistor. . . 75
IV.1 Introduction . . . 75
IV.1.1 Negative Capacitances . . . 75
IV.1.2 Model . . . 75
IV.2 Derivation of the equations . . . 76
IV.2.1 Propagation . . . 76
IV.3 Analysis of the equations . . . 79
IV.3.1 Possible divergences at extrema of the capacitance. . . 79
IV.3.2 Cut signal . . . 80
IV.4 Delay and geometry . . . 81
IV.4.1 Cut Signal . . . 81
I.1
What nanoelectromechanical systems are.
I.1.1 Definition
Nanoelectromechanical systems (NEMS) [1, 2, 3, 4, 5] are nano-to-micrometer scale oscillators coupled to electronic devices of similar dimensions.
Their masses are very small (of the order of 10−21− 10−16 kg), and they attain very high frequencies (in the MHz-GHz range, in other words, we have objects moving at radio-to microwave frequencies). These properties not only ensure that they can be used as very sensitive sensors, but make them also tools to explore fundamental issues, as we will see in this section.
I.1.2 Examples
We begin by showing a few examples of nanoelectromechanical systems. A first possibility is to have bridges, where a one dimensional system, such as a carbon nanotube [6, 7, 8, 9] is suspended between two leads (cf. Fig. I.1). In this setup, the nanotube is at the same time the mechanical part of the NEMS and its electronic part (it acts as a quantum dot).
These devices can be fabricated in the following way: first, one grows the nanotubes on a substrate, then leads are deposited at two ends of the tube. One then proceeds to remove the substrate below the tube (by wet etching, for example). This technique also allows the presence of a gate under the sus-pended portion of the tube, which enables the possibility to test the electrical (by turning on and off the current through them by modulating the gate volt-age, for example) and mechanical properties (such as their vibration modes) of the system. Alternatively, one can scan the suspended tube with an STM. In other schemes [10], the bridge itself does not carry current, but is capacitively coupled to a transistor.
I.1. What nanoelectromechanical systems are.
Fig. I.1: SEM pictures of two examples of bridge NEMS. The left device is a suspended InP nanowire made by Samir Etaki and Sami Sapmaz, TU Delft. The left one is a suspended nanotube fabricated with a PMMA etch-mask, made by Benoˆıt Witkamp, TU Delft.
I.1.3 Shuttles
Another class of devices are the shuttles depicted in Fig. I.3. There a moving part shuttles between two leads, loading electrons in the vicinity of one lead, and unloading them in the vicinity of the other. The advantage of having a very small structure is that one can obtain transport of a single electron per cycle (that can be of the order of 100 MHz). So far, experiments have required the addition of an external ac voltage to drive the shuttle [13, 14].
There is, however, another possibility, where the shuttles are self-driven [15], which we will discuss now. The system can be modelled as a metallic grain moving between a positively charged electrode and a negatively charged one. Since the tunnelling between an electrode and the grain is an exponential function of the distance between them, one can consider that the electron only gets charge from the electrode it is closer to (the sign of this charge will be the same as the one of the electrode in question). As the grain is immersed in the electrostatic field αV created by the bias voltage V, it experiences a work W = αV R
˙
Fig. I.2: NEMS motors. The left image, reproduced form Ref. [11] shows a multi-walled carbon nanotube based NEMS actuator. The right figure, reproduced from [12], illustrates the motion of a single-wall carbon nanotube based tor-sional pendulum.
quantised and is determined by the frequency of the shuttle only (I = eN ωπ ). This regime is created if the applied bias voltage exceeds a threshold value that is a function of the mechanical damping acting on the shuttle.
An interesting variation of shuttles involves the replacement of the metallic grain by a superconducting one [17]. It can then be shown that the shuttling of Cooper pairs between initially uncoupled, remote superconducting leads leads to the establishment of a superconducting phase coherence between the two leads (i.e. the phase difference will go to a fixed value) [18].
I.2
Ultra-sensitive measurements
One of the main applications of nanoelectromechanical systems is the possibility of extremely precise measurements.
I.2.1 Mass measurements
The mass oscillating part of a nanoelectromechnical system is very low (of the order of 10−21− 10−16 kg), which makes them good candidates as very sensitive mass sensors. Another key part is to have a large quality factor. It is a well-known fact that the maximal quality factor goes down with the size of the object (cf. Fig.I.4). Nevertheless, quality factors of nanoelectromechanical systems can be found in the range from 103to 105[2].
I.2. Ultra-sensitive measurements
Fig. I.3: SEM pictures of two examples of shuttling NEMS. The device on the left is a nanobell (an InP nanowire with a gold catalyst particle at the end) made by Samir Etaki, TU Delft. The right one, reproduced from Ref. [14], is a nanopillar with a gold island on top.
deposited initially [20] and by observing the shift in the resonance frequency of the oscillator resulting from the change of mass. These schemes result in a mass sensitivity of the order of a zeptogram (10−21g) [21]. These techniques can also be used to monitor the bounding of single DNA molecules to individual sites of an array of oscillators [22]. An analysis of the principles behind mass sensing given in Ref. [23] leads to an estimation of the possible maximal sensitivities of NEMS, which is of the order of a Dalton (= 1.66 · 10−27 kg, or a twelfth of the mass of a C12 atom).
I.2.2 Spin
Another metrology application of NEMS is magnetic resonance force microscopy (MRFM) [24], which can be used to detect single spins [25, 26, 27]. This has several potential promising applications: the resolution of magnetic resonance imaging could be dramatically improved. One could also manipulate individual spins.
Fig. I.4: A figure (reproduced from Ref. [2]) representing quality factors of different devices as a function of the logarithm of their volume. It can be seen that the quality factor diminishes with the dimensions of the system.
spins well below the surface of the sample (up to 100 nm), and two- and three-dimensional imaging applications are within reach.
Fig. I.5: A schematic of the MRFM experiment of Ref. [25], reproduced from [27].
I.2.3 Charge
I.3. The measurement of the quantum of thermal conductance.
is close to a gate electrode [28]. As in the case of the mass measurements, the critical parameters for a precise measurement are high quality factors and high oscillation frequencies.
Another interesting experiment is to use a shuttle [13]. The current can be so low that we have transport of individual electrons by the shuttle, at its resonance frequency (100 Mhz in this experiment).
I.3
The measurement of the quantum of thermal
conductance.
I.3.1 What the quantum of thermal conductance is
A remarkable property of one dimensional structure is that the electric conduc-tance in ballistic systems is quantized in units of e2
h [29]. This is independent of the actual structure of the setup.
Since some nanoelectromechanical systems can be considered as one dimen-sional systems, one could ask if there is such a law regarding the phonon thermal conductance. The answer, first derived in Ref. [30], is positive. Using the Lan-dauer formalism, they show that the low temperature limit phonon thermal conductance of a one dimensional system is, for the lowest, massless modes (i.e. for which ω(k = 0) = 0):
g0= π2kB2T
3h . (I.1)
This quantum of thermal conductance represents the maximum possible value of energy transported per phonon mode.
It should also be noted that the thermal conductance of ballistic electrons is also given by g0 [31]. This makes the quantum of thermal conductance not only independent of the structure of the setup, as it is the case for the quantum of electrical conductance, but also independent of the statistics of the particles we look at (i.e. bosonic for phonons, fermionic for electrons).
I.3.2 The experiment measuring it
In order to measure the quantum of thermal conductance, the setup depicted on the left part of Fig. I.6 was designed [32]. It consists of a four micrometer size square suspended phonon cavity (a thermal reservoir), attached by four phonon waveguides to other thermal reservoirs.
will give the thermal conductance. In order to know the behaviour of the con-ductance of this system for vanishing temperatures, one must know how many massless modes are available (that is how many modes have a conductance quan-tised in units of g0). There are sixteen of them, as there are four beams, and each carrying four modes [33]. This means that by diminishing the temperature the conductance should reach a plateau at sixteen conductance quanta. This is indeed what is observed in Ref. [32], as can be seen on the right part of Fig. I.6.
Fig. I.6: The measurement of thermal conductance [32]. On the left, there is an image of the structure used for the measurements. It consists of a four micrometer size square suspended phonon cavity connected to four reservoirs via one dimensional waveguides. The right plot shows the thermal conductance (in units of the thermal conductance quantum g0. The factor 16 represents the
16 possible fundamental modes (four per connecting beam)) as function of the temperature. Both figures are reproduced from [32].
I.4
The quantum limit on the position of a harmonic
oscillator.
I.4.1 What the quantum limit is
The fact that nanoelectromechanical systems reach such small dimensions raises the question of whether there is a fundamental limit to the precision of position measurements. The combination of the Heisenberg uncertainty principle and of the fact that an amplifier is needed to perform a measurement results in a positive answer to this question.
I.4.2 The standard limit
I.4. The quantum limit on the position of a harmonic oscillator.
oscillator are related by:
δxδp ≥ ¯h
2. (I.2)
This does not put a limit on the precision of a single measurement of the position: δx can indeed be arbitrarily small. The first uncertainty will have an effect on a second one, as there will be an error on the prediction due to the uncertainty on the initial conditions (δx for the initial position and δp for the initial momentum). This error will oscillate between δx and mωδp [34]. If we want the maximum of the error to be as small as possible, we have to set δp = mωδx, which can be introduced in Eq. I.2 to obtain the standard quantum limit for a harmonic oscillator : δxSQL≥ r ¯ h 2mω0 . (I.3)
I.4.3 The limit on a linear amplifier
The measurement of the position of a nanoscale oscillator has to be performed via an amplifier. This has consequences on the minimal uncertainty that can be reached, as the amplifier will have some back action on the oscillator. A general limit for linear amplifiers has been derived in Ref. [35]. Here we will look at a simplified case to illustrate how this limit can be obtained. A linear amplifier can be modelled as a sum of harmonic oscillators. We will consider a single mode of input, given by ˆa, a single mode of output ˆb and a phase preserving amplifier. In this case, we can write :
ˆb = αˆa + ˆk, (I.4)
where |α|2 = G, and where G is the gain of the amplifier. The operator ˆ
k describes the internal degrees of freedom of the amplifier not related to the input mode. The output mode has to be unitary, i.e.: hˆb;ˆb†i= 1. This leads to the following condition for the internal degrees of freedom of the amplifier:
1 = G +hˆk; ˆk†i. (I.5)
The next step is to look at the mean-square fluctuation of the output mode: ∆ˆb 2 = G |∆ˆa|2+ ∆ˆk 2 . (I.6)
A convenient tool to characterise the noise added by the amplifier is the added noise number A = |∆ˆk|
2
mean-square fluctuation of the internal degrees of freedom of the amplifier, one gets: ∆ˆk 2 ≥ 1 2 Dhˆk; ˆk †iE . (I.7)
By combining this with the unitarity condition on the output mode given by Eq. I.5 and the definition of A, we can get the following theorem:
∆ˆk 2 ≥1 2|1 − G| , (I.8a) A ≥ 1 2 1 − 1 G . (I.8b)
This means that A is larger or equal to one half for an infinite gain. One can use the fact that phonons are distributed according to a Bose-Einstein distribution to transform this limit on A on a minimal temperature:
Tmin= ¯ hω kBln 3
. (I.9)
This temperature can be combined with the equipartition theorem to get a new, larger limit for the uncertainty of the position of a harmonic oscillator coupled to a linear amplifier:
δxQL≥ r ¯ h mω0ln 3 ≈ 1.35 · δxSQL. (I.10)
I.4.4 How close observations are
One of the most promising schemes to approach the quantum limit is to use a single-electron transistor (cf. Sec. I.6 and Ref. [36]) coupled to the resonator one wishes to observe [37, 10]. The observed positions uncertainties approach the quantum limit quite closely, down to 4.3 times the quantum limit, for a 19.7 Mhz SiN resonator coupled to a superconducting single-electron transistor [10]. In this case, the quantum limit was 26 femtometers.
I.5
Superpositions in macroscopic objects
I.5.1 About superposition of macroscopic objects
I.5. Superpositions in macroscopic objects
This makes observing the effects of the superposition of quantum states for large objects very difficult. There is, however, no known fundamental reason that would prevent it. One direction that is used to observe increasingly large objects with quantum properties is interferometry, where superposition states have been observed for C60 molecules [39] as well as biomolecules (tetraphenylporphyrin) and fluofullerenes [40].
Nanoelectromechanical systems can be considered as macroscopic objects, as they are typically made of billions of atoms. On the other hand, they are extremely accurate sensors, as we have seen above. This makes them interesting candidates for the observation of superposition effects. Reference [5] points out that, provided the quantum uncertainty limit discussed above can be approached closely, decoherence times for a Silicon cantilever (1.6 µm long, and 0.1 µm wide and thick) would be in the range of a microsecond, which makes it possible for techniques involving single-electron transistors (cf. Sec. I.6 and Ref. [36]) to generate and detect superposition states. We will now look at some of the schemes that have been proposed.
I.5.2 The Cooper-Pair Box cantilever scheme
The idea of this scheme [5, 41] is to couple a cantilever to a two-level system , and perform a read-out through a single-electron transistor. A circuit diagram of this scheme can be found in Fig. I.7 (left).
The two-level systems has to be strongly coupled to the oscillator, so that its two basis states separate out the position states of the resonator. It also must be possible to coherently control it and produce any superposition of its basis states. Finally, its decoherence time (removing the part caused by the coupling to the oscillator) has to be comparable or larger than the period of the cantilever’s fundamental mode. An example of such a system is a Cooper-Pair box [42], which is a superconducting island connected to a reservoir via a Josephson junction. If the box is prepared in a superposition state, it becomes entangled with the cantilever and the cantilever is in a superposition state of two spatially separated states.
would have been projected onto its orthogonal by the coupling. The effect of the environment on the system is that the probability will return to a value lower than one. By analysing how the decrease of the probability grows with increasing coupling, one can infer the decoherence experienced by the cantilever.
Fig. I.7: The Cooper-Pair Box cantilever scheme from Ref. [41]. On the left, one has the circuit diagram for the coupled cantilever-Cooper Pair Box system that is read out by an SET. In the middle, one has the pulse sequence that manipulates the state of the box. On the right, the probability for the Cooper-pair box to be in its initial state is plotted as function as the wait time between the two pulses (in units of the period of the cantilever), for different values of the coupling (the larger the value, the larger the coupling). (All three figures are reproduced from Ref. [41]).
I.5.3 The Aharonov-Bohm carbon nanotube scheme
In Ref. [43], a single-wall carbon nanotube in the presence of a transverse mag-netic field is analysed. It is shown that the tunneling amplitudes get a main contribution from quasiclassical trajectories passing through the zones where the probability to find the nanotube is maximal, whereas trajectories passing through the nodes of the wave functions of the nanotube do not contribute. This situation is similar to the case where would send electrons through a screen with one (or more) hole with a quantum motion. The interference between the dif-ferent possible paths will be affected by the enclosed magnetic flux.
This creates a magnetoresistance that would indicate the presence of a quan-tized motion. For a 1 µm long single-wall carbon nanotube, the authors predict a 1-3 % relative conductance change, if the temperature is 30 mK and the magnetic field is 20-40 Tesla.
I.6
Single-electron Transistors
I.6. Single-electron Transistors
first focus on one possibility to describe their electronic part: the single-electron transistor. We will relate that to the mechanical motion in the next section.
I.6.1 What single-electron transistors are
Since their invention in 1947 by William Shockley, John Bardeen and Walter Brattain at Bell Labs, transistors have become ubiquitous. Among other uses, they are used as switches (a voltage gate can turn on and off the current between two electrodes) in digital circuits. An interesting empirical fact is Moore’s law [44] that states that the number of transistors on integrated circuits doubles every eighteen months.
As they become smaller, they will approach quantum limits, if the signal energy approaches the energy of a photon, signal charge approaches the charge of an electron, some of the device dimensions approach the electron wavelength, or the device size approaches the size of an atom [36].
An example of such a device is the single-electron transistor (SET) [36, 45, 46, 47], schematically depicted in Fig. I.8. It is made of an island connected to two leads via tunnel junctions (that is, electrons can tunnel through these junction from and unto the island). It is also capacitevely coupled to a gate (i.e. there is no tunnelling between the gate and the island). If the resistance of each junction is larger than the resistance quantum eh2 and if the charging energy associated to the capacitances e2
2C is much larger than the thermal energy kBT , then one can have electrons tunnelling one by one.
I.6.2 Coulomb Blockade
An analysis of the characteristic circuit on the right of Fig. I.8 can give us the change of electrostatic energy for an electron to tunnel from the left lead to the island, by taking into account the change in charging energies created by the tunnelling processes, as well as the work done by the charge transfer [48]:
V VG V VG CL,RL CR,RR CG V VG CL,RL CR,RR CG V V VG CL,RL CR,RR CG VG CL,RL CR,RR CG VG VG CL,RL CR,RR CG CL,RL CL,RL CCRR,R,RRR CG CG CG
Fig. I.8: A schematic (left) and circuit representation of a single-electron transistor.
are allowed, then there is no current and the charge of the island is fixed. This phenomenon is called Coulomb blockade.
This is illustrated in Fig. I.9. On the left part, there is a schematic view of the situation: the dashed lines represent the tunnelling conditions (each line corresponds to a zero of the transfer energy in the gate voltage–bias voltage plane). If the four possible tunnelling events from a given charge state (leaving the island, going to the island, for each lead) are forbidden, the charge cannot change. This defines the shaded Coulomb diamonds. The left picture shows the results from Ref. [51], where light-coloured Coulomb diamonds appear.
I.6.3 Tunnelling rates
I.6. Single-electron Transistors
Fig. I.9: Left: A schematic view of Coulomb diamonds. The numbers in the shaded zones indicate the charge on the dot. Right: Experimental Coulomb dia-monds on a carbon nanotube, reproduced from [51].
Γn→n+1L,R = e 2 RL,R ∆EL,Rn→n+1 1 − exp −∆E n→n+1 L,R kBT , (I.12a) Γn→n+1L,R = e 2 RL,R fF ∆En→n+1L,R , (I.12b)
where the first line corresponds to the case where there is a continuum of elec-tronic levels in the island, and where the second line represents the case where only one electronic level is available (fF is the Fermi function). The zero tem-perature limits of these functions are given by:
Γn→n+1L,R = e 2
RL,R
∆EL,Rn→n+1θ∆EL,Rn→n+1, (I.13a)
Γn→n+1L,R = e 2
RL,R
θ∆EL,Rn→n+1, (I.13b)
where θ is a Heaviside step function. One must mention that these rates are computed for potential barriers modelled by delta functions. In Sec. II.5, we will discuss different barrier models.
I.7
NEMS and SETs
In this section, we will have a closer look on the possible relations between the electronic and vibrationnal parts of NEMS. In particular, we will look at how we can either stimulate or detect different vibrationnal modes coupled to the electronic degrees of freedom of the devices.
I.7.1 Exciting the vibrationnal modes of NEMS
We begin by discussing the driving of a suspended carbon nanotube or a GaAs beam by an oscillating gate voltage [9, 37]. An oscillating gate voltage will modify the current in two ways: there will be a standard transistor gating effect, which would also happen if the device could not move, and an effect due to the oscillations of the nanotube. The main difference between these two effects is that the latter one is strongly frequency-dependent, as it peaks at the resonant frequency of the suspended nanotube. The left panel of Fig. I.10 clearly shows a resonance peak in the frequency dependence of the current for a suspended carbon nanotube [9].
The dependence of the resonance frequency on the dc component of the gate voltage allows to describe the different possible states of the carbon nanotube, such as behaving like a hanging chain or an elastic string [7]. The dc gate voltage dependence of the resonance frequency for the experiment of Ref. [9] is shown in the right panel of Fig. I.10.
I.7.2 Excitation spectra and Coulomb diamonds
Another possibility to observe oscillations in nanoelectromechanical systems is to look at excitation spectra in plots of the current in a gate voltage-bias voltage plane. The discussion of Sect. I.6.2 explained the appearance of Coulomb dia-monds. They were explained to correspond to suppressed current due to certain tunnelling conditions being satisfied or not. If one performs a differential cur-rent (with respect to the bias voltage) scan in the gate-bias voltage plane, one can also see lines running parallel to the edges of some diamonds, and ending on the edge of another diamond, as is illustrated by the dotted line on the left side of Fig. I.11. These lines correspond to excitations of the island. These can be for example electronic (caused by the finite size of the island) or vibrationnal (i.e. the tunnelling electron can produce a phonon).
I.7. NEMS and SETs
Fig. I.10: Driving a suspended carbon nanotube with an oscillating gate voltage [9]. The left panel shows the current through the nanotube, as a function of the driving frequency. A clear peak appears, indicating the resonance frequency. The black fit has a frequency of 55 MHz and a quality factor of 80 as fitting parameters. The right panel shows the current in a dc gate voltage-frequency plane. The lines, shown again in the inset correspond to the resonances. Both figures are reproduced from Ref. [9]
level of the right lead, which allows electrons to leave the island through the left lead. The excited level E1,Ewill satisfy the condition set by A at a fixed distance from the edge A (it adds a term ∆E to the tunnelling condition for the energy. Since this term is independent of the voltages, it does not affect the slope of the condition, but just shifts it.). This is depicted by a dotted line, which indicates at which voltages the energy of the excited level will be below the left Fermi energy, and therefore reachable through the left lead. The crossing point of line B and the dotted line corresponds to having both conditions satisfied: the ground state will be at the right Fermi energy and the excited state will be at the left Fermi energy, as illustrated on the right side of Fig. I.11. At this point, the energy difference between the ground and excited states will correspond to the energy associated to the bias voltage. Note that if the charge one ground state was below the Fermi energy of the right lead, it would necessarily be occupied and one could not use the excited level of charge one: only a higher bias would lift the Coulomb blockade and restore the current.
I.7.3 Observation of vibrationnal spectra
be-Fig. I.11: A schematic representation of the situation for excited levels. On the left, the dotted shows the limit at which an excited level can participate to the current. The lines A and B show limits at which adding an electron through the left contact and removing one from the right are possible. The right figure shows the two (ground and excited) energy levels for charge one at the crossing point of line B and the dotted line. The bias voltage at this point corresponds to the energy difference between the ground and excited states.
tween gold electrodes, on top of a gate [52]. In addition to the Coulomb dia-monds, one can see excitation lines, highlighted by arrows. The energy of the excitations (5 meV), as well as the fact that excitations at multiples of this energy are observed, are compatible with centre-of-mass oscillations of the C60 molecules. It is also found that each electron excites approximately one quan-tum of vibration, which means that the system is in a strong coupling regime.
I.8. This thesis
phenomenon is attributed to a non-equilibrium phonon distribution induced by the current through the nanotube.
The left panel shows the stability diagrams of an experiment on suspended carbon nanotubes on top of a gate [6]. The excitation energies are found to be compatible with stretching oscillations of the nanotubes, both from their values and their length dependences. By analysing the current as a function of the bias voltage, it can be found that the coupling is of order, as it was the case for the C60experiment.
Fig. I.12: Excitation spectra in a gate voltage-bias voltage plane, for C60 (left) and
suspended carbon nanotubes (right). The central figure shows the peaks in conductance for a suspended carbon nanotube placed under an STM. These figures are reproduced from Ref. [52] (left), Ref. [8] (middle) and Ref. [6] (right).
I.8
This thesis
As we have seen in this introduction, the experiments are not enough refined yet to show quantum motion of the oscillatory parts of nanoelectromechanical systems. It is therefore important to understand the relations between a classical motion of the oscillator and the electrical properties of a nanoelectromechanical system.
We will focus on the following type of systems: single-electron transistors with a central island that can move. As a current goes through the island, its charge fluctuates with time. This creates a random force on the island, through the electric fields generated by the different terminals (gate, leads). The question is to find out if this random force can make the island move and what the consequences of this possible motion are on quantities such as the electrical current noise.
Fokker-Planck one. The key element is the feedback due to the tunnelling processes. If negative, it has a role similar to mechanical friction and damps oscillations. If positive, it enhances the oscillations. The sign of this feedback is proportional to the slope of the average charge on the island with respect to the gate voltage. This corresponds to a capacitance between the island and the gate. For the probability distribution as a function of the gate and bias voltages, four charac-teristic behaviours of the system can be found: in the first one the capacitance is always positive, which means that the system is always dissipative with respect to the oscillations. This in turn means that there are no significant oscillations. In the other three cases, the capacitance may be negative, which leads to hav-ing either the oscillator movhav-ing at a given amplitude or switchhav-ing between two given amplitudes (those two cases corresponding to an unstable system). The last possibility corresponds to having the system switching between a moving one and a system that does not move. The last two cases correspond to bistable systems. These four behaviours have important consequences on the electrical current and noise. In the first, stable one, the current is not modified. The noise can however be large (even superpoissonian). In the second case, the current is strongly modified, while the noise is always superpoissonian. The third and fourth case also provide a strong modification of the current. The noise, which is in these two cases a telegraphic noise resulting from the bistabilty of the sys-tem becomes even bigger than in the second case. We then provide an example of a system where these four behaviours appear by looking at tunnel barriers with a transparency that depends on the energy of the incoming electrons. This example is related to actual, realistic systems. Effects on the current and noise are shown.
In Chapter III, we look at the same system, but in a different limit: this time the tunnelling events occur only once in a large number of periods of the oscillator. Following a procedure similar to the previous case, we find that the feedback is expressed in a different manner. The main difference is that only the first two behaviours occur (i.e. an immobile system and a system moving with a fixed, finite amplitude: there is no bistability). The current behaves in a similar way to the previous case: it is not modified for the immobile case, while the current in the other case is strongly modified. The noise behaves in a different manner in that it is always smaller than the shot noise (i.e. subpoissonian). Current and noise plots are again provided.
I.8. This thesis
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II.1
Introduction
II.1.1 Model System under consideration
Fig. II.1: The setup we are interested in.
II.1. Introduction
The goal of this work is to find conditions under which this happens and how physical quantities such as the current and noise are affected.
It was recognized already [2, 3, 4, 5] that for a strong coupling between a SET and a mechanical device , mechanical degrees of freedom strongly influence transport through a SET device, leading, for instance, to polaron physics and Franck-Condon effect. However, the weak-coupling regime is characteristic for most of NEMS and will be considered below.
Naively, the effect of the oscillator on transport current in this regime must be small and proportional to the coupling parameter. However, an underdamped oscillator can be swung up to big amplitudes even by a weak random force orig-inated from stochastic electron transfers through the device [6]; this amplitude may provide a strong feedback on the current. A less obvious effect is the extra dissipation due to electron tunnelling [7] which has been erroneously disregarded in [6].
We demonstrate in this chapter that such electron-induced dissipation may become negative, resulting in the generation of mechanical oscillation and in strong mechanical feedback. This takes place if the average charge accumulated in the SET device is a non-monotonous function of gate voltage.
The strong feedback is the most manifest in the current noise. The natu-ral measure of noise in nanostructures is the Poisson value [8], SP = 2eI. We demonstrate that in the strong feedback regime the noise is always parametri-cally bigger than SP due to long-time correlations of oscillator amplitude. If the generation is bistable, we predict a telegraph noise that can be exponentially big. Even if the strong feedback is absent, the noise may still exceed SP. The experimental observation of the enhanced noise thus would provide a strong evidence for mechanical motion.
Definitions
V and VG are the bias (we set the right voltage to zero) and gate voltages, respectively. The energy differences associated to tunnelling electrons are (+ indicate that we add an electron to the dot,- that we remove one)
∆EL+ = −∆EL−= −W + eV − F x, (II.1a) ∆E−R = −∆E+
R = W + F x, (II.1b)
also drop the lead index to indicate that we the the sum over the indices and use Γt for the sum of the four rates.
Regime considered
We now describe the limits that define the regime we used in our approach. The charging energy of the device is much larger than the applied voltages, which implies that the number of electrons on the island can be either zero or one. It is also important to notice that although all the examples given below consider the case where the level spacing is much larger than the applied voltages (meaning that the device is a single level one), it is not a limit we took in our approach, which is also valid in more general cases.
Concerning the time scales, we consider that many tunnelling events occur in an oscillating period (i.e. the typical tunnelling rate is considered as much higher than the oscillation frequency). The large ratio between the typical tunnelling rate and the oscillation frequency is denoted by α.We need this limit to be allowed to compute stationary solutions. In the next chapter, we will look at the opposite limit.
The temperature of the gas surrounding the molecule will be considered as very small (as will the mechanical friction created by this gas) compared to all other energy scales. We will however make no such assumption for the temperature of the electrons.
There are two scales for the energies of the oscillations. The first one is EQ, the quantum scale (i.e. ¯hω0). The second one, EΓ, is the energy to affect the rates. This happens when the work Fx is of the same order as W. The amplitude εΓ of oscillations with energy EΓ is:
εΓ= r 2EΓ M 1 ω0 . (II.2)
This means that :
EΓ= W2M ω2 0 2F2 = W 2 W λ¯hω0 , (II.3)
where we have defined:
λ = F 2 ¯ hM ω3 0 . (II.4)
λ is a coupling parameter that tells us how much individual tunnelling events displace the oscillator. We will consider the weak coupling regime, where λ is small.
To recapitulate, the scales are:
II.2. Calculations
We note that we are in the weak coupling (λ << 1) and classical (W >> ¯hω0) regime. We also have to take another classical limit (W >> ¯hΓ), which is needed to use master equations (basically that means that the tunneling events are well separated and that we don’t have a superposition of states with different charges, i.e. no coherence).
II.2
Calculations
The master equations for the probability densities to be in state n(=0,1), with velocity v and position x at time t are (see e.g. Refs. [6, 10, 11, 12]):
L0Pn+ n F M ∂Pn ∂v = (2n − 1) Γ +P 0− Γ−P1 , (II.6a) L0f ≡ ∂f ∂t + v ∂f ∂x − ω 2 0x ∂f ∂v − ω0 Q ∂vf ∂v , (II.6b)
where Q is the quality factor of the oscillator. We write the total probability density to have position x and velocity v as P = P0+ P1.
We will transform these equations by writing:
P1= Γ+
Γt
P + δP = ¯N P + δP. (II.7)
The quantity ¯N = ΓΓ+
t is the average charge on the island (it is a stochastic average). We also note that, since the rates only depend on the position, only the position derivative term of L0acts on ¯N . This means that:
L0N P = ¯¯ N L0P + vP ∂ ¯N
∂x = ¯N L0P + vPF ∂ ¯N
∂W, (II.8)
since the rates depend on W + F x. We will make an adiabatic approximation for the term δP, i.e. it does not depend on the variables at first order.
The master equations at order zero in the adiabatic approximation are:
L0P0= L0P − L0P1= 1 − ¯N L0P − vPF ∂ ¯N ∂W = ΓtδP, (II.9a) ¯ N L0P + ¯N F M ∂P ∂v + vPF ∂ ¯N ∂W = −ΓtδP, (II.9b) for n equal to zero and one (we used (Γ+P0− Γ−P1) = −ΓtδP).
−vPF∂ ¯N ∂W + ¯ N − 1N¯ F M ∂P ∂v = ΓtδP. (II.10)
The first order term for δP in the adiabatic expansion is then:
δP = ¯ N N − 1 F¯ M Γ2 t ∂P ∂v − vPF Γt ∂ ¯N ∂W. (II.11)
We can now go to the next order in the adiabatic expansion, by restoring the derivatives of δP. Most of the derivatives can be eliminated by taking the sum of the two master equations:
L0P + ¯N F M ∂P ∂v + F M ∂δP ∂v = 0. (II.12)
Inserting Eq. II.11 in this equation and regrouping our terms by the different derivatives of P, we get: ∂P ∂t + v ∂P ∂x + F (x) M ∂P ∂v = γ(x) ∂vP ∂v + D(x) ∂2P ∂v2. (II.13) This is a Kramers equation. It is similar to the Kramers equation decribing a Brownian particle subject to a force dependent on the position [13]. The difference lies in the origin of the different coefficients. Let us have a look at them.
The applied force term is: F (x) = −M ω2 0 x + ¯N√λεQ+ v ω0Q ≈ −M ω02x +ω0v Q , (II.14) where εQ = r ¯ h M ω0 (II.15) is the zero-point quantum motion. We can therefore neglect the constant term
¯
N√λεQ because it is smaller than the quantum scale, and we consider a clas-sical motion. The remaining terms are as expected the force −M ω2
0x and the dissipation −Mω0v
Q of a harmonic oscillator.
The diffusion coefficient is due to the tunnelling events (for a Brownian particle, it is due to the collisions with the surrounding gas). It is given by:
D(x) = F 2 M2 ¯ N 1 − ¯N Γt . (II.16)
II.2. Calculations
If this is positive, it can be interpreted as a dissipation. As we will see later, it also can be negative and have very important effects. It is composed of two terms: the first is due to the tunnelling events and the second due to the different friction mechanisms (surrounding gas, clamping points of the device, for example). This term is also present for a Brownian particle.
We will use the following parameterisation:
x = ε sin ϕ (II.18a)
v = ω0ε cos ϕ, (II.18b)
where ϕ is a phase and
ε = r 2E M 1 ω0 (II.19) is the amplitude. E is the energy of the oscillator).
To proceed further, we mention that we are interested on the influence of the tunnelling on the motion. It influences mainly the energy of the oscillations (i.e. makes certain energies more probable than others), since there is an exchange of energy between the flow of electrons and the mechanical oscillations. The coupling between the phase and the tunnelling (i.e. how much the phase is influenced by the tunnelling events) is much weaker, since it can only be indirect. Since our interest is to describe the coupling between the tunnelling and the energy of the oscillator, we will take an average over the phase. We will also disregard the phase dependence of the probability distribution, since is changes very little with the phase. The other quantities will retain their phase dependence.
We now have to compute the changes in the derivatives. Using the chain rule, we have: ∂f ∂E = r 1 2EM 1 ω0 sin ϕ∂f ∂x + ω0cos ϕ ∂f ∂v , (II.20a) ∂f ∂ϕ= r 2E M 1 ω0 cos ϕ∂f ∂x − ω0sin ϕ ∂f ∂v . (II.20b)
Combining these, we get: ∂f ∂x = ω0 √ 2EM sin ϕ∂f ∂E+ cos ϕ 2E ∂f ∂ϕ (II.21a) ∂f ∂v = √ 2EM cos ϕ∂f ∂E − sin ϕ 2E ∂f ∂ϕ . (II.21b)
the left hand side of the Kramers equation (Eq. II.13) after a phase averaging, except for the time derivative.
We take the average on the remaining terms, and use the following notation: f0= hf i, f1=cos2ϕf . (We do not use this for the probability distribution, since we consider it to be independent of the phase). The averaged Kramers equation is: ∂P ∂t = γ0P + 2Eγ1 ∂P ∂E + M D0 ∂P ∂ε + 2EM D1 ∂2P ∂E2. (II.22) Using Eq. II.21b, the fact that the diffusion and the feedback are independent of the velocity and periodic in the phase, because they are just functions of the tunnelling rates, one can compute the derivative of both γ1and D1. After some manipulations, the averaged Kramers equation becomes:
∂P ∂t = LP = ∂ ∂E 2Eγ1P + 2EM D1 ∂P ∂E . (II.23)
This equation is for the probability density P. We can also look at the distribution function at point E, which is the probability that the energy is between 0 and E. It is defined by:
P (E) = E Z
0
P(E0) dE0. (II.24)
If we integrate Eq. II.23 from 0 to E, we get: ∂P (E)
∂t = 2Eγ1P + 2EM D1 ∂P
∂E. (II.25)
We can look for the stationary solution (a stationary P implies a stationary P (E)): P(E) = P(0) exp − E Z 0 γ1(E0) M D1(E0) dE0 . (II.26)
II.3
Probability Distribution
II.3.1 Discussion
II.3. Probability Distribution
W >> EQ, (II.27a)
W = 2EΓ λ¯hω0
W << EΓ. (II.27b)
One can also see that defines a maximal quality factor Qmax obtained by equating the two terms of Eq. II.17:
Qmax= Γt ω0 1 λ¯hω0∂W∂ ¯N ∼ Γt ω0 W λ¯hω0 . (II.28)
The decay scale given by the ω0
Q term is λ¯hω0 ω0
ΓtQ. This will be of the same order as EΓ for a quality factor Qr, which is the required quality factor that we would need to see effects on the current if the term with the derivative of the charge was not present in the feedback. The scale of this quality factor is:
Qr∼ Γt ω0 W λ¯hω0 2 = W λ¯hω0 Qmax. (II.29)
The required quality factor is much larger than the maximal quality factor that we get (because of the charge term in the feedback). We will consider the case where only the charge term appears (i.e. the high quality factor limit).
This means that the probability distribution decays on a classical scale that is too small to significantly affect the rates.
The only possibility to have a current modified by the motion is to have a probability distribution that has peaks at non-zero values. To achieve that, we need the argument of the exponential in the probability distribution to have positive values for a range of oscillator energies. The only term that can create such contributions is the charge . A sine qua non condition to observe effects of the motion induced by the tunnelling events is that the charge must have a negative slope at some values. We have to look for a range of (total) energies for which this is the case.
We now discuss some general properties of the case where there is only one zone where this condition is satisfied (an example of a system where this is the case will be provided later. Fig. II.3 can be used as a guide in the following discussion): the slope is positive up to a value W1, then negative, and positive again above a value W2 (it has to be positive for large values of the energy (in absolute value), since the probability distribution has to go to zero). These two values are indicated by dashed vertical lines in Fig. II.3.
P(E) E (a) P(E) E (b) P(E) E (c) P(E) E (d)
Fig. II.2: A schematic representation of the possible behaviours of the probability distribution. The width of the peaks has been widened for readability. In panel a, the oscillator does not move. In panel b, the oscillator moves at a given amplitude, in c, the oscillator can either move at a finite amplitude, or not move. In panel d, the oscillator can move at two different amplitudes.
II.3. Probability Distribution
oscillator) and to an unstable one if the peak is at a finite value (i.e. we have a moving device).
There are two other possibilities that happen if the charge density has a negative slope at zero oscillator energy. In this case, the probability density will initially grow, until we reach values that will bring us to the threshold energies W1,2. Since the values of oscillator energy at which W1 and W2 can be different, it is possible that the one that is reached first is not important enough that change the sign of the argument of the exponential. We will then need the second one to start decaying. In this case, the probability distribution has one maximum (panel b). If the first zone is important enough to switch the sign of the argument of the exponential before we reach the second zone, we can have one or two maxima (panel d). The latter case can occur if the negative terms that we include between the values of the oscillator energy for which we had the first peak and the ones where we start including the second positive zone are sufficient to compensate the positive contributions. These two possibilities correspond to bistable systems, where the oscillator can be in two different states (moving and not moving (panel c) or moving at two different amplitudes (panel d).
In summary, one can have one peak at zero oscillator energy, one peak at a non-zero oscillator energy, one peak at zero and one at a non-zero value, or two peaks at non-zero values.
II.3.2 Expansion
The fact that the probability distribution is peaked allows us to expand the argument of the exponential around a given extremum (i.e. around zeroes of the feedback), or around E=0, even if the feedback is not zero there.
Zeroes of the feedback
We first look at the case where the feedback is zero at energy En (which can also be zero).
We first have to rewrite Eq. II.26 as:
P(E) = P(En) exp − E Z En γ1(E0) M D1(E0) dE0 . (II.30)
