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Università degli studi di Genova Facoltà di Scienze matematiche, fisiche e naturali Dipartimento di fisica Tesi di dottorato in fisica XIX ciclo 2 aprile 2007 Current fluctuations in single electron devices Federica Haupt Relatore: Prof. Maura Sassetti Correlatore: Prof. Rosario Fazio Contents Introduction iii 1 Single electron tunneling 1.1 Basic ingredients for single charge tunneling 1.2 Transport through a quantum dot . . . . . 1.3 Experimental realizations . . . . . . . . . . 1.3.1 Semiconductors 2DEG structures . . 1.3.2 Single molecule devices . . . . . . . 1.3.3 Carbon nanotubes . . . . . . . . . . 1.3.4 One-dimensional quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 6 6 9 10 12 2 Noise in mesoscopic systems 2.1 Types of noise . . . . . . . . . . . . . . . . . . . . . . 2.2 Shot noise in single electron tunneling devices . . . . 2.3 Shot-noise enhancement . . . . . . . . . . . . . . . . 2.3.1 Shot-noise in Luttinger liquid quantum dots . 2.3.2 Shot-noise in nanoelectromechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 21 22 23 25 3 The sequential tunneling regime 3.1 A general model for Coulomb blockade 3.2 The sequential tunneling regime . . . . 3.2.1 Matrix formalism . . . . . . . . 3.3 Current and shot noise . . . . . . . . . 3.3.1 The zero frequency limit . . . . 3.3.2 The two states regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 28 30 30 32 32 4 Shot noise in a 1D ring 4.1 A one dimensional quantum ring . . . . . . . . . . . . . . 4.2 The Luttinger liquid model . . . . . . . . . . . . . . . . . 4.2.1 Luttinger liquid with periodic boundary conditions 4.2.2 Luttinger liquid with open boundary conditions . . 4.3 Model and Hamiltonian . . . . . . . . . . . . . . . . . . . 4.4 The sequential tunneling regime . . . . . . . . . . . . . . . 4.4.1 Tunneling rates . . . . . . . . . . . . . . . . . . . . 4.4.2 Charge and orbital tunnel currents . . . . . . . . . 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Zero magnetic flux . . . . . . . . . . . . . . . . . . 4.5.2 Non-zero magnetic flux . . . . . . . . . . . . . . . 4.5.3 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 36 36 41 43 44 45 46 49 49 54 54 i . . . . . . . . . . . . . . systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Shot noise in NEMS 5.1 Introduction . . . . 5.2 Model . . . . . . . 5.3 Rate Equation . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 59 61 64 Conclusions 72 A Classical suppression of shot noise 74 B The current correlation function 76 C The polaron transformation 79 D The transition rates 81 ii Introduction “Noise is the signal” was a saying of Rolf Landauer [1], one of the founding fathers of mesoscopic physics. What he meant is that fluctuations in time of a measured quantity can be a valuable source of information that is not present in the time averaged value. Even if from the engineering point of view noise is essentially a nuisance which has to be as much as possible minimized, a physicist may actually delight in noise. The unique informative role of noise has been known since long time in the quantum optics comunity. It was back in 1909 when Albert Einstein first realized that the electromagnetic fluctuations are different if the energy is carried by waves or by particles [2]. The magnitude of energy fluctuations scales linearly with the averaged energy for classical waves, but it scales with the square root for classical particles. Since the photon is neither a classical wave nor a classical particle, the two contributions coexist. Fifty years later, with the Hambury Brown and Twiss experiment, it was discovered that intensity fluctuations provide information on the statistics of a photon’s source. Originally intended as a novel interferometric mean to determine the diameter of stars, the Hambury Brown and Twiss experiment is best-known today as the first experimental demonstration of photon bunching [3]. Since electrons share the particle-wave duality with photons, one might expect fluctuations in the electrical current to play a similar diagnostic role. Current fluctuations due to discreteness of the electrical charge are known as “shot noise”. This term was introduced in 1918 by Walter Schottky, drawing an analogy between the electrons moving in a vacuum tube and the small pellets of lead that hunters use for a single charge of a gun [4]. Although the first experimental observation of shot noise dates back to the 1920’s, the interest on the electronic shot noise has progressed more slowly than for photons. Much of the information it contains has been appreciated only recently from experiments on mesoscopic devices. Mesoscopic physics is a branch of condensed matter physics which studies the electrical properties of system with typical dimensions at the borderline between the macroscopic scale of the world we live in and the microscopic scale in which each atom is separately considered [5]. In mesoscopic systems, the electrons maintain their phase coherence over distances larger than the sample size, leading to interference effects which cannot be described classically. Moreover, in these devices the confining potential changes over length scales which are comparable with the electron wavelength, so that quantized states are formed. But at the same time, these devices contain from several thousand to millions of atoms so that statistical properties -such as temperature and distribution functions- still have a meaning. Moreover, mesoscopic devices can be connected and probed through macroscopic wires and amplifiers, leading to classical measurements of classical currents. Even if some mesoscopic effects are observable on the macroscopic scale - as for example the quantum Hall effect- most studies are carried out on devices of submicron dimensions. For this reason, the experimental research in mesoscopic physics has always been closely related with the development of sophisticate lithographic and crystal growth techniques. Fundamental research has also benefited much from massive iii Introduction iv industrial research and development efforts towards miniaturization of integrated electronic circuit based on semiconducting materials. Moreover, in recent years mesoscopic physics has developed into a interdisciplinary field including physics, chemistry and biology. In fact, in the relentless search for smaller electronic devices, the idea of using single organic molecules, nanotubes or DNA as active components in electrical circuits has become an important topic. Although an experimental break-through at one Kelvin may not have a direct impact on the commercial technology, the cumulative knowledge and experience gained through work at the mesoscopic scale can provide insight for conceptually new electronic devices. A notable example is the quantum dot, or single electron device, which exploits the quantum phenomenon of tunneling to control and measure the flow of single electrons inside a circuit [6]. A quantum dot is an extremely precise electrometer, thousand times more sensible to charge variations than the state-ofthe-art conventional transistors. For this reason, dots have important applications in metrology and in ultra-sensitive analog applications. As an example, coupling a single electron device to a nanoscale mechanical resonator, a displacement sensitivity of the order of 10−15 m has been recently achieved [7]. To put this number into perspective, these devices are able to detect displacement as small as ten-millionth of the size of the hydrogen atom. The electronic properties of mesoscopic system are routinely investigated by measuring the current flowing through the device in the presence of an applied voltage. However, in the last decade current measurements have often be sided by noise measurements, which have shown to be an highly efficient investigation tool. Indeed, noise in the electrical current contains information on the underlying transport processes which are not contained in the average current. As an example, Schottky’s seminal idea of extracting the charge of the current carriers from a shot noise measurement [4], has been the lever to measure the fractional charge of Laughling’s quasiparticles in the fractional quantum Hall effects [9]. Moreover, shot noise measurements are the most promising candidates to show experimental signatures of fractional statistics. Electrical noise has also become an alternative and very accurate method to determine the temperature of electrons in a solid. In the field of quantum computing, noise has been proposed as a tool to investigate the correlations caused by the entanglement. The shot noise has also proved to be very sensitive to the presence of interactions. As an example, the correlations induced by strong Coulomb interaction generally lead to a detectable suppression of the shot noise in a quantum dot. The effects of the electron-electron interaction become prominent in one dimension. Truly one-dimensional electronic system can be created either by using sophisticated lithographic techniques or by growing carbon nanotubes. Unlike in a usual three dimensional conductor, in one dimension electronic interactions affect substantially the properties of the ground state and of the low-energy excitations. For this reason, one dimensional devices are expected to show unique transport features. The Luttinger liquid model serves as a theory for interacting electrons in one dimension [11]. It is characterized by power-law correlation functions with an interaction-dependent exponent. This exponent show up in all the experimental accessible quantities, which manifest a power-law behavior. Beside electronic interaction, the properties of a mesoscopic device are strongly influenced also by the electron-phonon interaction. In solid state devices electrons generally interact with a continuous phonon-bath which is a major source of decoherence and relaxation. The situation is radically different for a molecular device. In this case the phonon spectrum is discrete and, to a good approximation, at low energies the electrons can be considered as coupled to a single phonon mode. In this thesis we study how the electron-electron interaction and the electron-phonon v Introduction interaction affects the current noise in a single electron device. We employ the Luttinger liquid description to treat electronic interaction in one dimension and an Anderson-Holstein-type Hamiltonian to harness the electron phonon interaction. We focus on the sequential tunneling regime and we calculate the current and the shot noise employing a rate equations approach. The layout of this thesis is as follow: Chapter 1. In the first chapter we discuss the most prominent features which characterize a single electron device. We first analyze the circumstances under which charge quantization effects are important and then we introduce the fundamental concepts of Coulomb blockade and Coulomb oscillations within the framework of the costant interaction model. Moreover, we discuss how the exceptional charge sensitivity of a single electron transistor can be transmuted into ultra-sensitive displacement detection. Subsequently we shortly describe a number of different realizations of single electron devices, ranging from semiconductor based quantum dots, to carbon nanotubes and molecular devices. Particular attention is devoted to carbon nanotubes and cleaved-edge overgrowth heterostructures since they represent two of the most important realizations of one-dimensional electronic system. Chapter 2. A short introduction to current fluctuations will be given in this chapter. We first summarize the generic characteristics of noise in a mesoscopic device, discussing the differences between shot noise, thermal noise and quantum noise, their distinct physical origin and the regimes where each of them become dominant. Then we focus on the zero frequency shot noise of a single electron device. We discuss the suppression of the shot noise induced by the charging energy, and then we describe mechanism known as dynamical channel blockade, which is instead responsible for a strong enhancement of the noise. Chapter 3. In this chapter we develop the theoretical tools required to evaluate the current and the current noise in the sequential tunneling regime. We introduce a general model for a Coulomb blockade mesoscopic system, discussing the characteristic energy scales and the conditions under which transport through the system can be considered as a sequential tunneling process. In the sequential tunneling regime, the dynamics of the system can be described in terms of Markovian rate equations. We show how the rate equations formalism allows to derive simple expression for the average current and for the shot noise in a compact matrix notation. Chapter 4. In order to introduce the peculiarities of a one dimensional quantum ring, at the beginning of this chapter we give a brief introduction on the main features of the Luttinger liquid (LL) model, considering both the case of periodic and open boundary conditions. Afterwards, we study current fluctuations in a 1D quantum ring, connected via tunnel contacts to external leads, in the presence of interactions. This geometry permits to analyze the interplay between interaction, Coulomb Blockade and Aharonov-Bohm interference. Signatures of this interplay reflect in a wealth of the interaction-induced noise features. We characterize the noise by considering both the charge current noise (related to the fluctuations of the tunneling current) and the magnetization current noise (related to the fluctuations of the persistent current). We study the noise both analytically and numerically and we support our observations with a Monte Carlo simulation. Chapter 5. In this chapter we discuss how phonon relaxation influence the transport properties of a single electron transistor coupled to a mechanical oscillator. We find that finite relaxation rates affects the dynamics in a highly non trivial way. Both current and noise can be either enhanced or suppressed by relaxation, depending on the electron-phonon coupling and on the considered voltage range. In Introduction vi particular, for voltages higher than a certain critical value, the Fano factor can be even suppressed below 1/2. This suppression is observed in a completely incoherent regime as a consequence of the interplay between vibration assisted tunneling and direct relaxation of different vibrational states. Finally, we give a summary of the conclusions which can be drawn from this thesis. Chapter 1 Single electron tunneling In the last two decades, an impressive number of theoretical and experimental works have been devoted to the study of single electron quantization effects in solid state devices [6, 12, 13]. Charge quantization effects are really nothing new: already in 1911 Millikan observed the effects of single electrons on the falling of oil drops [14]. Nevertheless, quantization effects play little role in conductors, where charge can be safely regarded as a continuous quantity. Because the wavefunctions of electrons in conductor are extended over macroscopic distances, the charge in any small volume is not quantized. Therefore, to observe charge quantization effects it is first necessary to localize the electrons to tiny regions of space. Progress in nanolithography and thin-film processing have opened up the possibility to confine electrons in sub-micron regions, allowing for the experimental observation of single electron quantization effects in solids. In the next section we examine the circumstances under which charge quantization effects are important. In section 1.2 we discuss the characteristics of transport through a Coulomb blockaded island at the light of the constant interaction model. Finally, in section 1.3 we will discuss a number of experimental realizations of quantum dots, ranging from semiconductor heterostructure, to carbon nanotubes and molecular devices. 1.1 Basic ingredients for single charge tunneling Let us consider a charge transport experiment in which a voltage difference is applied to two electrodes separated by an insulating gap. In the middle of the gap lies a third electrode which we call “island”, since it is surrounded by the insulator or also “quantum dot”, meaning that the electrons are confined in all three spatial directions. To travel from the source to the drain the electrons must go through the island. We assume that particle exchange occurs by quantum tunneling. This process is so fast that we can consider that the electrons are traversing the insulating gap one at a time. The key point is that during the journey from the source to the drain, the electron necessarily makes the charge of the dot vary by e. This is a tiny amount of charge if we consider an ordinary electronic device, 1 however, if the island is sufficiently small, the variation of the electrostatic potential due to the presence of an excess electron can be large enough to react back on the tunneling probabilities [15]. The change of the electrostatic potential is conveniently expressed in terms of the capacitance C of the island. An extra charge e changes the potential of the dot by the charging energy EC = e2 /C. When this energy scale exceeds the thermal 1 As a example, each charge packet in a charge coupled device (CCD) is composed of about 10 6 electrons. 1 Single electron tunneling 2 energy kB T , the quantum dot acquires a sensitivity to charge variations at the level of a single electron. A first condition to observe charge quantization effects is then e2 /C kB T. (1.1) The second requirement is that the resistance of the tunneling barriers have to be much bigger than the resistance quantum h/e2 ≈ 25.813 kΩ RT h/e2 . (1.2) This condition is obtained by requiring the RC-time constant for an electron to tunnel off of the island to be great enough that the energy uncertainty is much smaller than the charging energy, i.e. h/(RT C) EC . This condition essentially ensures that the wave function of an excess electron in the island is localized there. If RT is small on the scale provided by h/e2 , charging effects are suppressed because of quantum fluctuations. To summarize, the two conditions to observe effects due to the discrete nature of charge are RT h/e2 , 2 e /C kB T. (1.3) (1.4) When both of them are satisfied, transport through the island is dominated by charging effects and one speaks about a Coulomb blockaded mesoscopic system. The first criterion Eq.(1.3) can be met by weakly coupling the island to the source and drain, the second Eq.(1.4) by making the dot small. Modern experimental techniques permits to create electronic islands whose size is comparable with the Fermi wavelength of the host material. In this case, electrons occupy discrete “orbital like” quantum levels and have a discrete excitation spectrum. For this reason quantum dots are sometimes also called artificial atoms [16] 1.2 Transport through a quantum dot In a typical transport experiment, a quantum dot is operated in a three terminal geometry, see Fig.(1.1). The dot is cupled by tunnel barriers to a source and a drain Figure 1.1: Schematic set up for a quantum dot connected to source, drain and gate electrodes. 3 1.2 Transport through a quantum dot leads, which in turn are connected to macroscopic current and voltage meters. A third “gate” electrode is capacitively coupled to the island, and can be used to tune the electrostatic potential of the dot with respect to the reservoirs. As we will discuss in this section, in this configuration a quantum dot acts as a single electron transistor (SET). A convenient way to study the transport properties of a SET is the costant interaction model [13,17,18]. This relays on two important assumptions. (i) The Coulomb interaction among the electrons in the dot and between the electrons in the dot and those in the environment is parametrized by a single, costant capacitance C. This capacitance can be thought of as the sum of the capacitances between the dot and the source, Cs , the drain, Cd , and the gate Cg , i.e C = Cs + Cd + Cg (see Fig. 1.1). (ii) The discrete energy spectrum does not depend on the number of electrons in the dot. Under these assumptions the total ground state energy for N electron in the dot at zero temperature is given by U (N ) = N i2 X X 1 h e(N − N0 ) − n C i Vi + 2C n=1 (1.5) i=s,d,g where e is the charge of the electron, and N0 is the number of electrons in the dot in an electrically neutral situation. The terms Ci Vi (i = s, d, g), represent the charge induced on the dot by the applied external voltages. The last term of Eq. 1.5 is a sum over the occupied single particle energy levels n . The energy levels depend on the characteristics of the confinement potential. To describe transport experiments, it often is more convenient to refer to the electrochemical potential µ(N ) which is, by definition, the minimal energy required for adding the N th electron to the dot X C i Vi 1 µ(N ) ≡ U (N ) − U (N − 1) = EC N − N0 − − EC + N 2 e (1.6) i=s,d,g where EC = e2 /C is the charging energy. The electrochemical potentials for the transition between ground states with different electron number N are shown in Fig. 1.2(a). The discrete levels are spaced by the so-called addition energy Eadd (N ) ≡ µ(N + 1) − µ(N ) = EC + ∆. (1.7) The addition energy consit of a purely electrostatic part, the charging energy E C , plus the energy spacing between two discrete quantum levels ∆ = N +1 − N . The addition energy is large for small capacitance and/or large energy splitting ∆. In linear response regime µs ≈ µd , a non-zero addition energy can lead to a blockade for tunneling of electrons on and off of the dot. Infact, at zero temperature the N -th electron can tunnel into the dot from the ith-electrode only if energy is conserved, i.e. µ(N ) ≤ µi (i = s, d). Analogously, it can tunnel off only if µ(N ) ≥ µi . Therefore, if the electrochemical potentials of the dot and the leads are arranged as in Fig.1.2a, tunneling is interdicted and the number of electrons into the dot is fixed to N − 1. So, for µ(N − 1) < µs , µd < µ(N ), electron transport is blocked and this situation is known as Coulomb blockade. Coulomb blockade can be removed by changing the gate voltage to align µ(N ) into the bias window (see Fig.1.2(b)) µd ≤ µ(N ) ≤ µs . (1.8) Single electron tunneling 4 Figure 1.2: Schematic diagrams of the electrochemical potential of a quantum dot for different electron numbers. (a) The chemical potentials of the dot are misaligned with respect to µS and µD . The number of electrons in the dot is fixed to N − 1 due to Coulomb blockade. (b) As µ(N ) falls into the bias window, the number of electron can oscillate between N and N − 1 resulting in a single-electron current. (c) Both the transition between the N − 1 and the N ground states (black line) and the transition to an N -electron excited state (grey line) are energetically allowed. (d) The electrochemical potentials for N and N + 1 electrons are both comprised into the bias window. In this case two electrons can tunnel into and out of the dot at the same time. After Ref. [35]. In this case, an electron can tunnel from the source electrode into the dot and then it can tunnel off to the drain. Now, a new electron can tunnel-in and repeat the cycle N − 1 → N → N − 1. This cycle is known as single electron tunneling. By measuring the current while sweeping the gate voltage a trace as shown in Fig. 1.3a is obtained. Each peak occurs when an electrochemical potential of the dot fall into the bias window and the dot can oscillate between two adjacent charge states. Viceversa, in the valleys between the peaks the number of electrons on the dot is fixed because of Coulomb blockade. The distance in gate voltage ∆Vg between two peaks can be readily obtained from Eq.(1.6) by imposing the condition µ(N )|Vg = µ(N + 1)|Vg +∆Vg . This leads to e∆Vg = C (∆ + EC ). Cg (1.9) Therefore ∆Vg gives information about the energy spectrum of the dot. A second way to lift Coulomb blockade is to increase the source-drain voltage V (see Fig. 1.2c). Again, current can flow only when an electrochemical potential level falls within the bias window. By increasing V until both the ground state as well as an excited state transition fall into the bias window, an electron can choose to tunnel not only through the ground state, but also through an excited state of the N -th electron dot. This is visible as a change in the total current. In this way, excited-state spectroscopy can be performed. How exactly the current changes depends on the tunnel rates of the two different paths. In a typical experiment, the current I or the differential conductance ∂I/∂V , is measured while sweeping the bias voltage for a series of different values of the gate voltage. Such a measurement is shown schematically in Fig. 1.3b. Inside the diamond-shaped region, the number of electron is fixed because of Coulomb blockade and no current can flow. Outside the diamonds, Coulomb blockade is lifted and single-electron tunneling is possible. For larger bias voltage even doubleelectron tunneling is possible (see Fig. 1.2d). Excited states are revealed as changes in the current, i.e. peaks or dips in the differential conductance. From such a “Coulomb diamond”, the charging energy as well as the energy of the excited states can be read off immediately. 5 1.2 Transport through a quantum dot Figure 1.3: Transport through a quantum dot. (a) Coulomb peaks in current versus gate voltage in the linear response regime (µs ≈ µd ). (b) Coulomb diamonds in differential conductance, ∂I/∂V . versus V and Vg . The edges of the diamond shaped region (black) correspond to the onset of the current. Diagonal lines emanating from the diamonds (gray) mark the onset of transport through excited states.After Ref. [35]. SET as a displacement detector Up to now we have discussed how a quantum dot is operated as single electron transistor. This is the most sensitive electrometer, with a sensitivity which been shown to be as low as few 10−5 e Hz−1/2 , i.e. a charge variation of 10−5 e can be detected in a measurement time of 1s [19, 20]. This exceptional sensitivity can be exploited to design ultra-sensitive displacement detectors. The basic idea is frailly simple [21]: a mechanical resonator coated with a thin metallic layer and placed next to a SET acts as a mechanically-compliant gate electrode. When a fixed voltage is applied to this electrode, the motion of the resonator modulates the electrochemical potential of the SET and hence the tunneling current. Today nanolithografic techniques allow to create quantum dots which are closely integrated with a sub-micron size mechanical resonators [22–24]. Because of their small size, these resonators have small inertial masses (10−15 −10−17 kg) and vibrate at frequencies ranging from few MHz up to around a GHz [25]. By coupling a SET to 116-MHz doubly coupled suspended beam (see Fig1.4), a record displacement sensitivity of 10−15 m Hz−1/2 has been achieved at the temperature of 30 mK. To put this number into perspective, this device is able to detect displacements as small as ten-millionth of the size of an hydrogen atom. This sensitivity is roughly a factor 100 larger than the zero-point displacement uncertainty of the mechanical beam. The SET displacement detector belongs to the class of devices known as nanoelectromechanical systems (NEMS). Besides having important applications as ultrasensitive measuring devices [7, 26, 27], NEMS are interesting dynamical systems in their own right. Infact, not only the motion of the resonator affects the tunneling current but also, because of the small mass of the oscillator and the strong electrostatic coupling to the SET, individual electrons travelling through the SET can give significant displacement “kicks” to the mechanical resonator. This back-action leads to fluctuations in the oscillator position and to damping [28]. Single electron tunneling 6 Figure 1.4: (a) Scanning electron micrograph of the device used in Ref. [7]. The substrate and the suspended beam are fashioned from GaAs (blue regions). The SET and beam electrode are thin aluminum layers (yellow regions), with aluminum oxide forming the tunnel barriers. The beam is located 0.25 µm away from the island electrode (scale bar, 1 µm). The measured fundamental flexural frequency for in-plane motion is 116 MHz. (b) A schematic of the mechanical and electrical operation of the device. 1.3 Experimental realizations A quantum dot is a general kind of system and many different sizes and materials can be used to realize it: single molecules trapped between electrodes [29,30], metallic or superconducting nanoparticles, self-assembled quantum dots, semiconductor lateral or vertical dot [12, 16] or carbon nanotubes between closely spaced electrodes [31, 32]. In the remaining of this chapter, we will shortly discuss of a few of these experimental realizations. This is by no means intended to be an extended review, we are merely interested in introducing the typical length and energy scales of these devices and the main physical issues that characterize them. 1.3.1 Semiconductors 2DEG structures Fabrications of semiconductor based quantum dots starts with a semiconductor heterostructure, a sandwich of different semiconductor material (see Fig. 1.5). These layers, most commonly GaAs and AlGaAs, are grown one on top of each other using molecular beam epitaxy (MBE). Because these two semiconductor have nearly the same crystal structure, MBE growth results in very clean crystals. By doping the n − AlGaAs layer with Si, free electrons are introduced. As a consequence, the shape of the conduction band is modified and an almost triangular potential well develops on the GaAs side of the interface (inversion layers). Electrons accumulate in this potential well, forming a thin (∼ 10nm) sheets of electrons that can move only along the interface. Due to the confinement, the single particle energy spectrum splits in discrete subbands ~2 k 2 + n , (1.10) ~k,n = 2m∗ where m∗ = 0.067 me is the effective mass of electrons in GaAs and ~k ≡ (kx , ky ), being z the heterostructure growth direction. If the Fermi energy is such that only the lowest subband is populated, ε0 < EF < ε1 , then the system behaves effectively as two-dimensional. Assuming typical parameters for the GaAs/AlGaAs structures, the Fermi energy and the average level spacing can be estimated as EF ∼ 10meV and ∆ ∼ 20meV respectively [34]. Therefore, at temperatures kB T ∆−EF ∼ 100K 7 1.3 Experimental realizations Figure 1.5: Different layers of semiconductor in a GaAs/Alx Ga1−x As heterostructure. Due to the different band gap of GaAs and Alx Ga1−x As, a triangular potential well is formed on the GaAs side of the interface. The conduction band diagram as well as the carrier concentration have to be calculated solving selfconsistently the Poisson and the Schrödinger equation. At low temperatures kB T ∆ − EF only the energy ε0 of the first subband lies below the Fermi energy, so that the system behave effectively as two-dimensional. After [33]. the system is “frozen” into the lower subband n = 0 and is dynamically a 2D system. The so created two-dimensional electron gas (2DEG) has several properties desirable for studying mesoscopic effects. Infact they can have a high mobility and a relatively low electron density (105 −106 cm2 /Vs and ∼ 3×1015 m−2 , respectively). The low electron density results in a large Fermi wavelength (∼ 40 nm) and a large screening length, which allows us to locally deplete the 2DEG with an electric field. Two-dimensional electron gases can be easily given an arbitrary shape using lithographic techniques. This is achieved either by permanently etching a portion of the 2DEG, or by using metallic gates deposited on the top of the nanostructure. To fabricate these gates, the surface of the heterostructure is first covered by a layer of organic resist. Then the gate pattern is defined by writing in the electron- Single electron tunneling 8 Figure 1.6: Fabrication of metal electrodes by electron beam lithography. (a) A specific patter is written into the resist-layer with an electron beam. (b) A resistmask is formed over the heterostructure surface after the resist which was impressed by the electron beam has been removed. (c) Metal evaporation. (d) After lift-off, only the metal electrodes remain on the surface. After Ref. [35] sensitive resist with a focused electron beam. This locally breaks up the polymer chains, so that the exposed parts can be removed by a developer. In the next step, metal is evaporated and, because of the resist mask, it can make contacts with the heterostructure only where the resist has been exposed and removed. In the last step, the remaining resist is removed with acetone and metal electrodes are left at the places that were exposed to the electron beam. These fabrication steps are summarized in Fig. 1.6. The electron beam can accurately write very small patterns with a resolution of about 20 nm, allowing to make very complicated structures. By applying negative voltages to the gates, the 2DEG is locally depleted, creating one or more islands that are isolated from the large 2DEG reservoirs (see Fig. 1.7). These islands are quantum dots. To create freely suspended nanostructures, as the one showed in Fig. 1.4 electron beam lithography is combined to a sequence of reactive ion etching and dilute HF wet etching to remove selectively portions of GaAs [36, 37]. Another promising technique is atomic force microscope (AFM) nanolithography, where the tip of an AFM is used to define neat in-plane gates by local oxidation of the GaAs cap layer [38]. Combining this technique with additional metallic top gates an an exceptional high tunability can be reached. a b Figure 1.7: Planar quantum dot. a Quantum dot defined by 5 metallic gates fabricated on the surface of a GaAs based heterostructure (after Schönenberger’s group homepage http://pages.unibas.ch/phys-meso/). b By applying negative voltages to metal electrodes on the surface of the heterostructure, the underlying 2DEG can be locally depleted. By choosing a suitable geometry for the gates, the electrons can be confined to one or even to zero dimensions. 9 1.3.2 1.3 Experimental realizations Single molecule devices The idea of using single molecules as functional devices in electrical circuits trace back in 1974 to the pioneering work by Aviram and Ratner [39]. As active electronic units, single molecule offer several unique properties. First of all, the size of most of the simple molecules is within several nanometer, resulting into a typical energy scale of ∼eV for the electronic spectrum. Secondly, molecules are self-assembling, which is very useful in fabricating electronic devices at such a small scale. Finally, another great advantage is the huge variety of molecules with different functionalities and thus the possibility to design the electrical properties of a device through chemical synthesis. The main challenge in molecular electronics is to “wire up” Figure 1.8: Experimental data by H. Park [29] for electronic transport through a C60 transistor. The device is formed by a single-C60 molecule placed between two closely spaced Au-electrode. Upper panel: Current-voltage characteristic for different values of the gate voltage Vg . The current is strongly suppressed near zero bias followed by step-like current jumps at higher voltages. Lower panel: Differential conductance ∂I/∂V for four different devices as a function of gate and bias voltage. Dark regions correspond to the conductance gap and bright lines to peaks in the differential conductance. White arrows mark the position of vibrational sidebands. Measurements were taken at T = 1.5K. a single molecule. In recent years, a number of ingenious experimental schemes have been developed. The tip of a scanning tunneling microscope (STM) can be Single electron tunneling 10 employed as an electrode to measure the current through a molecule which rests on a substrate [40]. Othervise, a planar configuration can be employed, situating the molecule between two closely spaced electrodes realized by means of breakjunctions [41] or electromigration [42]. The molecule may or may not be chemically bound (e.g. by thiol groups) to the electrodes. Up to today, a large variety of molecules, ranging from H2 [30] to DNA [43] have been investigated experimentally in two-terminal [30,43,44] and three-terminal configurations [29, 40, 45–47]. In many of these experiments unambiguous evidence of Coulomb blockade has been reported [29, 45, 46] as well as the observation of Kondo effects [46, 47]. An important distinction between transport through semiconductor quantum dots and single molecules lies in the coupling to phonon degrees of freedom. While charge carriers typically interact with a continuum of phonon modes in quantum dots, molecules are characterized by a discrete spectrum of vibrational modes. A prominent effect of the coupling of the electrons to discrete phonon modes, is the appearance of vibrational side bands in the current-voltage characteristics [29,30,45] (see Fig. 1.8). 1.3.3 Carbon nanotubes Carbon nanotubes (CN) are macro-molecules with a hollow cylindrical shape, made up only of carbon atoms. Since their discovery, back in 1991 [48], they have been at the center of an intense research activity because of their outstanding mechanical and electrical properties. Roughly speaking, a CN can be viewed as a sheet with an hexagonal lattice of carbon atoms (graphene) which is rolled up to form a seamless cylinder. From the electronic point of view, graphene is a gapless semiconductor [49]. When a graphene sheet is rolled up to form a nanotube, its 2D-dimensional band structure split up into one-dimensional subbands, because of the boundary conditions around the circumference. Because of the symmetry and unique electronic structure of graphene, the geometric structure of the nanotube strongly affects its electrical properties [49]. The direction along which the sheet is fold can be indicated by the “folding ~ = n~a1 + m~a2 , where ~a1 and ~a2 are the elementary translation vector on vector” C the basis of the honeycomb lattice (see Fig.1.9). If m = 0, the nanotubes are called “zigzag”, if n = m they are called “armchair”, otherwise they are called “chiral”. For a given (n, m)-nanotube, if n − m = 3j, j ∈ N the tube acts as a 1D metal, with a Fermi velocity vf = 8 × 105 m/s comparable to typical metals. Moreover it has a linear dispersion relation and a flat density of states around the Fermi energy. Otherwise, if m − n 6= 3j the CN is a semiconductor with a gap between the filled hole states and the empty electron states. The bandgap is predicted to be ∼0.9eV/d[nm], where d is the diameter of the tube. These remarkable theoretical predictions has been verified scanning tunneling microscopy experiments [50, 51]. When metallic electrodes are attached to a nanotube, the properties of the resulting device depends crucially on the quality of the contacts. With good contacts, a metallic nanotube behaves as perfect electronic wire and it becomes a ballistic conductor with a conductance almost approaching the theoretical maximum value, 4e2 /h ≈ 155µS [52, 53]. Viceversa, if the the contacts between the CN and the electrodes are poor, a tunnel barrier is formed at each contact and electron transport occur via quantum tunneling. In this case, the portion of tube between the electrodes forms a quantum dot [31, 32]. As an example, in Ref. [31] transport measurements through ropes of nanotubes bridging 200nm-spaced contacts have evidenced a gap in the I − V curves at low temperature and a series of sharp peaks in the conductance as a function of a gate voltage Vg . These observations 11 1.3 Experimental realizations Figure 1.9: The graphene honeycomb lattice and its basis ~a1 ,~a2 . The folding vector ~ = n~a1 + m~a2 . The vector T~ denote the tube axis. Folding directions for zigzag is C and armchair nanotubes are also shown. are consistent with single-electron transport through a segment of a single nanotube with a typical addition energy of ∼ 10 meV and an average level spacing of ∼ 3meV. Carbon nanotubes are also ideal systems for exploring electromechanical effects because they have a low mass, are easily scalable through their length and have a Young’s modulus of the order of TPa [54], which is almost one order of magnitude higher than that of silicon. Moreover, they can be defect free and their electronic properties are nowadays well understood [49, 55]. The idea of using a doubly clamped suspended carbon nanotube as a nanoelectromechanical system was first envisioned by Sapmaz et al. [56] and then and then implemented experimentally by Sazanova et al. [57] and Witkamp et al. [58]. The basic idea is to actuate a nanotube by applying an ac-voltage to the gate, which induces a driving force on the tube due to the displacement-dependent gate capacitance. The CN starts to oscillate when the driving frequency approaches a mechanical eigenfrequency. In a semiconducting CN this lead to a modulation of the conduction, therefore the nanotube can be employed to detect its own motion. Furthermore, other experiments on suspended carbon nanotubes have evidenced the interplay between mechanical and electrical degrees of freedom also in the absence of an actuating gate voltage [59, 60]. Electrons tunneling onto the nanotube excite discrete vibrations by spending some of their excess energy provided by the bias voltage. This leads to the appearance of a series of peaks in the conductance (phonon-sidebands) which forms an harmonic low-energy spectrum. By analyzing the energy scale and the size-dependence of this spectrum, electron coupling to the radial breathing mode (RBM) [59,61] and to the longitudinal (stretching) mode [60] have been demonstrated. Moreover, the measurements show that free-hanging tubes operate in the under-damped regime of low dissipation. For the fundamental bending mode the reported quality factor is about 300 [58]; for the RBM it is estimated as high as 10000, corresponding to a phonon lifetime of the order of 10 ns [59]. Single electron tunneling 1.3.4 12 One-dimensional quantum dots The electronic properties of one-dimensional (1D) conductors have generated much interest. The reason for this excitement lies in their very rich phase diagram and the prediction that in a 1D system the Coulomb interaction should lead to a strongly correlated electron gas, called a Luttinger (LL) liquid instead of the weakly interacting quasi-particles described as a Fermi-liquid in conventional metals [62]. The reason is that, unlike their higher dimensions counterpart, in 1D conductors the electroninc interaction is poorely screened. Consequently, in 1D the fermionic quasi particle excitations that are characteristic of Fermi liquids become unstable and collective density fluctuations constitute the stable elementary excitations [63]. The difference between a Luttinger and a Fermi liquid becomes striking already in the presence of a single impurity. According to the Landauer’s theory, at zero temperature the conductance of a single non-interacting channel with a barrier is given by G = |T |2 e2 /h, where |T | is the transmission probability through the barrier. This result holds even at finite temperature, given that |T | is approximately energy independent. Viceversa, Luttinger liquid theory predict a vanishing conductance at T = 0 even for the smallest barrier, and a power law dependence G(T ) ∝ T α at finite temperature T [64]. Carbon nanotubes are one of the most promising candidates for a detailed study of 1D physics. In fact, as we will see in Sect. 4.2, the Luttinger liquid model assumes a linear dispersion relation. In metallic CN this is an unusually good approximation as their dispersion relation is linear over a large energy range of the order of several hundreds meV [49]. Moreover, due to the enormous subband separation (of the order of eV), metallic nanotubes can be considered truly one dimensional conductors even at room temperature. Experimental evidence of Luttinger liquid behavior in metallic nanotubes comes from tunneling experiments in ropes of carbon nanotubes between metal electrodes in the Coulomb blockade regime [65]. It has been shown that the conductance and the differential conductance scale as power laws with respect to temperature and bias voltage, respectively and that the functional form and the exponents are in good agreement with theoretical predictions. In this case, the one-dimensional CN-quantum dot was coupled directly to 2D-dimensional non interacting leads (the metal electrodes). Differently, by employing AFM manipulation, it is possible to create 1D quantum dots connected to leads which are themself one-dimensional. In fact, the tip of an AFM can be used to introduce local structural defects (buckles) in the CN, which behaves very much like electronic tunnel junctions. When two buckles are introduced along the nanotube, a quantum dot embedded in a onedimensional conductor is formed [66]. An example of a similar quantum dot is presented in Fig. 1.10. This technique allows to reduce the size of the quantum dot to few tens of nm. As an example, the distance between the two buckles in Fig. 1.10 is approximately 25nm. Consequently, the dot has a very high charging energy of about 120 meV, allowing for the observation of Coulomb blockade peaks up to room temperature. By studying the temperature dependence of the conductance peaks, striking evidence of power law scaling for the peak area and for the peak maximum have been found. Signatures of Luttinger liquid behavior have also been reported in semiconductor devices created by means of the cleaved edge overgrowth technique (CEO) [67]. This is an highly sophisticated molecular beam epitaxy technique, developed at Bell Labs to create low dimensional quantum structures with atomic precision [68]. The starting point to create a CEO-wire is a thin layer of GaAs sandwiched between two thicker layer of AlGaAs (Fig.1.11a). An high quality 2DEG is created at 13 1.3 Experimental realizations ) Zero-bias conductance versus gate voltage for a single conductance peak. Data are shown 4, 20, 30, 40, 70, 80, and 90 K. The solid lines are fits to / ) )] [as deT dependence of the w G Fabrication of a room-temperature single-electron transistor within an individual metallic carbon nanotube by manipulation A) Nanotube between Au electrodes on top of a substrate with a gate-independent resistance of 50 kilohm. T , which displays a linear behavior. Figure 1.10: Top: atomic force microscopy images of a single wall carbon nanotube (blue) on top a Si/SiO2 substrate (green), contacted to metallic gates (brown). After the deposition of the nanotube (A, white bar: 200nm), whose radius is 20-50 nm, two buckles are created by using the AFM tip (B, C), thus defining a short region in between, of length ≈20 nm (white bar: 20nm). Bottom left: linear conductance peaks for different temperature values ranging from 4K to 90K – dots: experimental data, solid line: theoretical fit. Bottom right: maximum (right scale) and area (left scale) for the linear conductance peaks as a function of the temperature in double logarithmic scale, shows clearly a power-law behavior. (After [66]). the interface between the semiconductors by an appropriate doping scheme. The specimen is then cleaved in ultra high vacuum and another doped AlGaAs layer is overgrown over the smooth cleavage plane (Fig.1.11b). This introduces electrons at the edge of the GaAs (Fig.1.11d), creating one or more edge states along the cleave. Electrons in such edge states are firmly confined along two directions (y and z in Fig. 1.11): on three sides, confinement is due to atomically smooth barriers and, on the fourth side, to the strong triangular potential of the cleaved-edge modulation doping sequence. Strong overlap between the 2DEG and the edge states couples both systems intimately along the entire rim. The edge state can be decoupled by applying a negative voltage to a top gate T , which was evaporated on top of the sample before the cleave. As VT is biased increasingly negative, the 2DEG splits in two sheets and a 1D wire is formed in the depletion region (Fig.1.11c). Because of the strong confinement in the y and z, the energy spectrum of the wire presents a series of 1D sub-bands, with a mean level spacing of ∼ 20meV (∼ 2K). By tuning the top voltage to increasingly negative values, the electron density in the wire can be reduced until only a single subband is populated. Conductance measurements in this low density regime show an abrupt decrease of the conductance, followed by a series of almost equally spaced sharp resonances (see inset in Fig. 1.13). These interpreted as Coulomb blockade peaks through a 1D electronic island which is formed when the highest peaks of the impurity potential cross the Fermi energy (see Fig. 1.12). In fact, despite the care taken in all the fabrication steps, some impurities are unavoidably present along the wire and are seen by the electron as a disordered potential. When the electron density is so low that the highest peak of this potential exceeds the Fermi energy, the wire is split into two parts and the conductance drops to zero. By reducing the density even further, a second barrier crosses the Fermi energy forming a 1D island and transport can then occur through Single electron tunneling 14 Figure 1.11: The creation of a CEO sample and the experimental setup for transport measurements. Insets d, e and f show a sequence of schematic cross sections of charge distribution in the wire region for different VT . The side gate S, primarily serves to vary the electron density along the entire edge. The top gate affects the electron density only in the wire region. For strongest 1D confinement, the top gate is biased negatively and the side gate strongly positively, pushing the electrons aganist the cleaved edge of the quantum well. (After [67]) Figure 1.12: Sketch of the impurity potential in a CEO wire. In the ultra-low density regime, the highest peaks of the impurity potential can exceed the Fermi energy. As soon as two peaks cross the Fermi energy, a quantum dot is formed. 15 1.3 Experimental realizations resonant states. In this case, the system consists of a 1D quantum dot connected to interacting LL leads. Signatures of Luttinger liquid behavior were indeed found in the temperature dependence of the linewidth Γi of the resonant states. In contrast to the conventional Coulomb blockade theory [12], where Γi is temperature independent, in this case Γi decrease as a power law of temperature (see Fig 1.13), in agreement with the theoretical predictions of Ref. [69]. (b) Conductance Γi (arb. units) (2e2 /h) (a) Top gate voltage [V] Deduced temperature Figure 1.13: Coulomb oscillations in a CEO-wire. a Main panel: Linear conductance as a function of the top-gate voltage for the CEO wire studied in Ref. [70]. Inset: A zoom-in of the conductance in the subthreshold region. A series of nearly equidistant peaks is clearly observable. Those peaks have been interpreted as the Coulomb oscillations of a 1D-quantum embedded into the wire. b Intrinsic linewidth Γi vs. temperature for two resonant peaks (peak # 1 and peak # 2 of the inset). Γ i decreases as a power law of temperature, indicating a Luttinger liquid behavior [69]. The dashed line are power law fit to the data Γi ∝ T 1/g−1 . (After Ref. [70]) Chapter 2 Noise in mesoscopic systems In recent years, much attention has been devoted to the statistical analysis of the current fluctuations in mesoscopic system. Initially this study has been confined to the second moment in the current fluctuations distribution, which goes under the name of noise [71, 72]. Probably the most paradigmatic example of the effectiveness of the noise as an investigation tool is the direct observation of fractional quasiparticle charges in the fractional quantum Hall effect [8–10]. In this chapter, a quick overview of the properties of noise in electronic systems will be given. In section 2.1, we discuss the main sources of noise in a electrical conductor and the different regimes where they are dominant. Afterwords, we focus on the effects of the interactions on the shot noise of a single electron device, analyzing both the suppression due to Coulomb blockade (section 2.2), and the enhancement due to dynamical channel blockade (section 2.3) 2.1 Types of noise Noise represents the spontaneous, dynamical fluctuations of a physical quantities away from its time averaged mean value. These stem from the thermal motion of matter and the discreteness of its structure and form a stochastic random process For an electrical system, not all sources of noise are inherent to the device or to the electron transport processes. These generate extrinsic noise and can be reduced or even eliminated by employing better device designs and improved fabrication technologies [71]. In this case the study of noise can help developing higher quality devices. On the other hand, intrinsic noise, which is a characteristic of the device, cannot be reduced and sets an upper limit for the accuracy that can be achieved in measurements. For quantum systems, the presence of noise represents an important source of decoherence. From this point of view, the intrinsic noise can be regarded as a nuisance. But, at the same time, the study of electrical fluctuations is an important tool to investigate the electronic properties of mesoscopic devices [72]. If we consider the electrical current through such a device, the intrinsic fluctuations contain information about the interactions that are regulating the transport. From the theoretical point of view, electrical noise is usually described in terms of time dependent current fluctuations in a conductor with a fix, non fluctuating voltage between the contacts. In the stationary regime, a detailed description of the noise in time domain is given by the correlation function CI (t) = h∆I(t + t0 )∆I(t0 )i, (2.1) where ∆I(t) = I(t) − hIi and hIi is the mean current (see Fig. 2.1). Here and in the following, the brackets h· · ·i denote an average over an ensamble of identical 16 17 2.1 Types of noise Figure 2.1: Time dependent fluctuations of the current around its average value hIi. physical systems or, equivalently, an average over the initial time t0 (ergodicity assumed). Equivalently, noise might be represented in the frequency domain by the power spectral density (the power of noise per unit frequency) Z ∞ SI (ω) = dt eiωt CI (t). (2.2) −∞ If the current I(t) is a classical observable, the correlator is real and symmetric, i.e. C(t) = C(−t) and so is the power density S(ω) = S(−ω). It is therefore enough to consider a symmetrized spectrum Z ∞ dt eiωt CI (t), ω > 0 (2.3) S(ω) = SI (ω) + SI (−ω) = 2 −∞ defined for positive frequencies ω. This is also the quantity that is detected in standard, low frequencies noise measurement. In the quantum limit, the spectrum is no longer symmetric SI (ω) 6= SI (−ω) and this classical description is not anylonger valid. However, as in the rest of this thesis we will consider only the zero-frequency current noise, we refer to Ref. [5,73,74] for for a detailed analysis of the asymmetric quantum noise. The zero-frequency component of the symmetrized noise spectral density S ≡ S(0). (2.4) is interesting for several reasons. First, it is the easiest to access experimentally. Infact, the frequency dependence of the shot noise is governed by microscopic rates, whose frequency are typically measured in units of THz. On the other hand, current and noise measurements are most usually performed ad much lower frequencies (in the range of several kHz to several 100 kHz), so that one is justified in taking the limit ω → 0 in the calculation of the shot noise. Second, the noise versus voltage curve have a rich structure, which is particularly revealing when rescaled by the average current. This can be used either to fit the parameter in the model more accurately or to provide a consistency check for parameters already determined from the average current. However, not all types of electrical noise are informative. The fluctuation current in a conductor in thermal equilibrium merely tell us the value of the temperature of the electrons T . To get more out of noise one has to bring the electrons out of thermal equilibrium. Before getting into that, let us us say a bit more about thermal noise. Thermal noise For kB T eV , thermal agitation is the main cause of current fluctuations in a conductor. These fluctuations are called thermal noise or also Johnson-Nyquist Noise in mesoscopic systems 18 noise, after the experimentalist [75] and the theorist [76] who first investigated it. At equilibrium V ≈ 0, thermal noise is directly related to the conductance G of the system by the fluctuation dissipation theorem [71, 77] S = 4kB T G. (2.5) Therefore, the investigation of thermal noise does not provide more information than the investigation of the linear conductance. Eq.(2.5) is valid as long as ω k B T . At very high frequencies ω ≥ kB T , vacuum fluctuations contribute to equilibrium fluctuations too. In this case, Eq.(2.5) has to be changed by replacing kB T with the expression ω/2 coth(ω/2kT T ), so that noise is no longer frequency independent (“white”) but increases linearly with ω. Shot noise At low temperature kB T eV , the main source of fluctuations in the electrical current is the randomness inherent in the transmission of discrete charge quanta. These kind of fluctuations are known as shot noise and, unlike thermal noise, they carry information on the temporal correlation of the electrons which are not contained in the conductance [72, 73, 78]. We start considering the situation in which the electrons are transmitted through the device randomly and independently of each other. In this case, which is, typical of a vacuum tube or of a tunnel junction, electron transfer can be described as P a Poisson process [71, 79]. In other words, the current is expressed as I(t) = e δ(t − tn ), where tn are uncorrelated random quantities, with the average interval τ between the arrivals of consequent electrons. In this case the average current is hIi = e/τ , while the current noise is S = 2ehIi ≡ SP oisson (2.6) and it does not depend on frequency. This formula was first derived by Walter Schottky in 1918 [4], who proposed it as an alternative method to measure the charge of the electron. The possibility, envisioned by Schottky, of measuring the charge of the current carriers from the shot noise power, has been fully appreciated only recently in systems in which the granularity of the current is not the elementary charge e (e.g. in the case of the fractional Hall effect or in superconductors). In this case, Eq.(2.6) becomes S = 2qhIi, where the electron charge e is replaced by an effective charge q. Shot noise measurement performed in the fractional quantum Hall regime have allowed the direct observation of the fractional charge corresponding to the quasiparticles [8–10]. In the case of superconductor-normal metal junctions, an enhancement of the shot noise by a factor 2 was observed [80], which is attributed to the q = 2e effective charge transfer in Andreev reflection process. The Schottky formula Eq.(2.6), relays on the hypothesis that transport is a Poissson process. However, in general the electrons in a conductor are not independent and correlations cause the shot noise to deviate from Eq. 2.6. Deviations from the Poissonian value are expressed by the Fano factor F, which is defined as F ≡ S . 2ehIi (2.7) A main source of correlations is the Pauli principle. In a vacuum tube or in a tunnel junction, the mean occupation of a state is so small that the Pauli principle is inoperative, but this is not so in the general case. In a system of non interacting electrons, an efficient way of accounting for the correlations induced by the Pauli 19 2.1 Types of noise principle is Landauer’s scattering approach [81, 82]. According to the Landauer formula, the time-averaged current hIi is given by hIi = N 2e2 X V Tn h n=1 (2.8) where, V is the applied voltage, 2e2 /h is the conductance quantum (including a factor two for spin). and Tn are the transmission probabilities (0 ≤ Tn ≤ 1) of the N independent channels which characterize the system (formally, the Tn ’s are the eigenvalues of the product t·t† , where t is the transmission matrix of the conductor). At zero temperature, the noise is related to the transmission probabilities by [83,84] S = 2e N 2e2 X V Tn (1 − Tn ). h n=1 (2.9) The factor (1 − Tn ) describes the reduction of noise due to Pauli principle. The shot noise formula Eq.(2.9) has a simple statistical interpretation. Consider first a one-dimensional conductor, i.e. a device with a single channel with transmission probability T1 . Electrons in a range eV above the Fermi level, enter the conductor at a rate eV /h. In a time τ̄ , the number of attempt transmissions is τ̄ eV /h. At zero temperature, there are no fluctuations in this number, since each occupied state contains exactly one electron (Pauli principle). Fluctuations in the transmitted charge Q arise because the transmission attempts are successful with a probability T1 different from 1 or 0. The statistics of Q in not Poissonian anylonger but rather binomial. The mean-squared fluctuations h∆Q2 i of the charge for the binomial statistics is given by h∆Q2 i = e2 (τ̄ eV /h)T1 (1 − T1 ). (2.10) The relation S = (2/τ̄ )h∆Q2 i between the mean-squared fluctuation of current and the transmitted charge gives back exactly Eq.(2.9) for a single channel. Since fluctuations in different channels are independent, the multi-channel version is simply a sum over channels (as an aside, a classical derivation of the T (1 − T ) suppression of the shot noise is given in appendix A). In a tunnel barrier, electrons have a very small probability of being transmitted. In the Landauer formalism, this fact is implemented by the condition Tn 1, for all n. In this case it is easy to see that Eq.(2.9) simply reduce to the Poissonian limit S = SP oisson . The quantum shot noise formula has been tested experimentally in a variety of systems [85, 86]. As an example, Fig.2.2 shows a comparison between theoretical predictions and experimental measurements for the current and the Fano factor in a quantum point contact, which is a narrow constriction in a 2DEG. By applying a negative voltage to a metallic gate, the 2DEG is depleted and a narrow channel can be formed by an opening in the gate. By changing the gate voltage one can adjust the width of the constriction, which in turn determine the number N of discrete channels contributing to the current. According to Eq.(2.8), each open transmission channel give a discrete contribution to the average current, which is then expected to show a staircase behavior as a function of the gate voltage. Viceversa, the shot noise shows peaks in between the conductance plateaux, and it is vanish on the plateaux since fully open (or closed) channels do not contribute to Eq.(2.9). Nowadays, shot noise measurement is routinely employed in the study of atomic size contact (e.g. break junction) as, combined with conductance measurements, it allows to identify uniquely “mesoscopic PIN code” of the junction, i.e. the number of open channels and their transmission (the ) [87]. Shot noise can also be employed to probe particle statistics. Bosons emitted by a thermal source tend to bunch (more of them are emitted at the same time) 20 S/S0 S/S0 Noise in mesoscopic systems Figure 2.2: (a) Theoretical predictions [84] for the conductance G (dashed line) and shot-noise power S (full line) of a point contact versus Fermi energy. (b) Experimentally observed G and S versus gate voltage Vg at T = 0.4K (experimental data from the Reznikov’s group). After [78]. resulting in a super-Poissonian statistics [3]. Due to Pauli principle, a fermionic thermal source emits particles separately (anti-bunching), leading to sub-Poissonian statistics [88]. When a thermal stream of particle is separated at a beam splitter, the correlations between the two outgoing streams depend on the particles being bosons or fermions. By using a Hanbury-Brown and Twiss geometry, positive correlations were measured in the case of photons [3, 88] and negative correlations for electrons [89, 90]. In a macroscopic devices, like macroscopic metallic wires, the granularity of the charge flow is smeared out by inelastic scattering of the electrons with environment. This is the reason way a macroscopic conductor shows thermal noise, but not shot noise. Before finishing this section, it is worthy to stress that thermal noise and shot noise are not two different processes, but rather two different limits of the same physical effect. The crossover from thermal noise Eq.(2.5) to shot noise Eq.(2.9) generally depends on the transmission eigenvalues. 1/f noise Beside thermal and shot noise, in a real device the are other sources of noise which are not related to the transmission properties. The most common example is 1/f noise. These type of fluctuations are caused by slow changes in the device resistance and they are found in almost any conducting material. Their spectrum is proportional to the square of the injected DC current and increase as an inverse power of the frequency. The exponent is often very close to unity, hence the name “1/f ” noise. The origin of the resistance fluctuations is not universal and, despite of number of experimental and theoretical work devoted to this subject, for most system is still unknown. In this thesis we will not deal at all on 1/f noise, even if it can be truly important in nowadays mesoscopic devices (for an extensive review on 1/f noise see [5,71]). We merely remark that, as 1/f noise can obviously not be avoided in a low frequency, any measurement in units of shot noise has to be made at frequencies above those where 1/f noise is significant (typically several kHz). This is still in the zerofrequency limit as far as the shot noise is concerned, since its dependence on the frequency is governed by microscopic rates whose frequencies are typically measured in units of THz. We will be therefore justified in taking the limit ω → 0 in the calculation of the shot noise 21 2.2 2.2 Shot noise in single electron tunneling devices Shot noise in single electron tunneling devices Beside Pauli principle, another important source of correlations among electrons is Coulomb interaction. An energy scale of Coulomb repulsion is the charging energy EC = e2 /C of a single electron inside a conductor with capacitance C. In open conductors, where C is large, charging effects are expected to be negligible. However for close conductors, such as a double barrier junction or a quantum dot, EC can be as large as kB T . In this case case, as we have seen in Chapter 1, charging have a pronounced influence on the transport properties of the device. The theory of the shot noise in single electron devices has been developed both with quantum mechanical approaches [91, 92] and by means of semiclassical derivations based on the rate equations technique [93,94]. Interestingly, both approaches predict identical shot noise results, implying that the shot noise is not sensitive to quantum coherence in single electron structures. The first measurement of the shot noise in a Coulomb blockaded system was reported by Birk et al. [95] in metal nanoparticle. In this experiment the doublebarrier junction is formed by positioning the tip of a scanning-tunneling microscope above a metal nanoparticle on an oxidized substrate. Because of the tiny size of the particle it is Ec ≥ 1000kB T at T = 4K. The relative heights of the two tunnel barriers can be modified by changing the tip-particle distance. The experimental results in the case of strongly asymmetric junctions (R2 ≈ 100R1 , where 1 and 2 stand for the tip-particle and the particle-substrate junction, respectively) are represented in Fig. 2.3. The I−V current display a stepwise increase of the current with voltage (rotating by 90 the plot yields to the usual presentation of the “Coulomb staircase”). At small voltage, the current is vanishingly small I ≈ 0 because of Coulomb blockade. At each subsequent step in I, the number of excess electrons on the particle increase or decreases by one. The corresponding measured shot noise oscillates along with the step structure in the I − V curve. The full Poissonian value S = SP oisson is reached at each plateau of constant I (arrow A in Fig. 2.3), while it is suppressed for current steps in between plateaux (arrow B in Fig. 2.3). The experimental data are in excellent agreement with the theory of [93]. A qualitative understanding for the shot noise suppression caused by charging effects can be given by the two state model. Because of the large charging energy EC kB T , at each step only the two charge states with the energetically most favored number of electrons on the particle are considered. In this case, tunneling always occurs alternately across the two junctions. After an electron has been added to the particle by tunneling across the junction 1, this junction is blocked until the electron is tunnels out through junction 2. On a current plateau in the Coulomb staircase, the number of excess electrons is constant for most of the time, only disturbed during very short instants. In other words, for most of the time electron transport is blocked by the more resistive barrier R2 , which inhibits electrons to tunnel out of the particle. Once an electrons leave the dot, another one quickly takes its place tunneling from the tip of the STM. In this case the tunneling events are solely determined by junction 2, hence the shot noise is the same as for a single tunnel junction, S = SP oisson . The situation is completely different for voltages where the Coulomb staircase show a step. Here two charge states are degenerate, which result in similar tunneling probabilities for the two junctions. As both barriers are on average alternately blocked during equal times, tunneling become correlated causing a suppression of shot noise with a maximum of 1/2. It is worth noticing that measuring the shot noise of a quantum dot using conventional noise measurement techniques is very challenging. This is because of the very low currents and the corresponding low noise levels in these systems. Indeed, Noise in mesoscopic systems 22 Figure 2.3: Left panel:Current and noise measurements in a metallic quantum dot. Top panel. I-V characteristic for system formed by the tip of an STM positioned over a metallic nanoparticle on substrate. The thick line is the experimental curve at T = 4.2K, while the thin line represent the theoretical prediction. The “Coulomb staircase”, i.e. stepwise increase of the current with increasing voltage is clearly distinguishable. Lower panel. Zero-frequency shot noise as a function of the current. The dashed lines correspond to the maximal and minimal values S = 2eI and S = eI. Right panel. Measured Fano factor F = SI /2eI as a function of the current I. In both panels, diamonds: experimental data, solid line: theory by [93]. After [95]. after the breakthrough represented by Ref. [95], in the next decade few other experimental observations of the shot noise in dots have been reported (e.g. [96, 97]). An alternative way to investigate current fluctuations was introduced by Levitov et al. [98] and it is known as full counting statistics (FCS). This method relies on the evaluation of the probability distribution function of the number of electrons through a conductor within a given time period. In addition to the current and to the shot noise, which are the first and the second moment of this distribution, this methods give access to higher order moments. The most intuitive method for measuring the FCS of electron transport is to count electrons passing one by one through the conductor. Very recently, real-time detection of single electron tunneling through a quantum dot has been achieve, allowing to measure the full counting statistics of the corresponding transport process [99]. The shot noise could be extracted from the measured FCS, and it showed a clear suppression below the Poissonian limit as a function of the asymmetry of the tunneling barriers. 2.3 Shot-noise enhancement As we have seen in the previous section, for electron transport through quantum dots, the noise is typically of sub-Poissonian nature, i.e. F < 1. This is due to the Coulomb blockade, which enhance the correlation between the electrons and thereby reduces the noise. However, when several channels with different coupling strengths contribute to the electron transport, interactions can lead to more complex processes and to an enhancement of the noise [100, 101]. To explain these results let us consider the case in which transport through a quantum dot can occur both via the ground A state and via an excited state B (e.g. Fig. 1.2c). The tunnel coupling between the dot and the leads is given by the overlap of the dot and leads electronic wave functions. Since the wave functions corresponding to A and B may have different spatial distributions, it can happen that one of the two states, say A, is coupled to the leads much weaker than the 23 2.3 Shot-noise enhancement other one, B, so that its contribution to the current and noise is negligible. Because of Coulomb blockade, the dot may only hold one or zero excess electrons. This in turn implies that transport through the two states A and B is correlated: if A is occupied, current through B is blocked, while if A is empty, current through B proceeds in an ordinary way. Thus, if tunnel rates for A and B are of the order of ΓA and ΓB , respectively, transport through the system proceeds in bunches of ΓB /ΓA 1 electrons and the Fano factor achieve large super–Poisson values. This mechanism, which takes the name of dynamical channel blockade, was first introduced by Safonov et al. [102] to explain the super-Poissonian noise observed in tunneling through localized states. We stress that in order to observe bunching of electrons in a quantum dot it is necessary that the charging energy is greater than the separation between the energy levels relevant for transport. Theoretical predictions of super–Poissonian noise exist for sequential tunneling regime in quantum dots with ferromagnetic leads [103, 104] (if both leads are partially polarized, say spin-up, then spin-up electrons tend to tunnel in bunches, and spin-down electrons block the current for a long time), in a magnetic field [105]. Moreover, general dynamical channel blockade have been predicted for sequential tunneling in single [106,107] and in double quantum dots [108,109], and in quantum dots where the level coupling is mediated by non-equilibrium plasmons [110] (see below). Finally, Ref. [111] predicted Super-Poissonian noise in the inelastic cotunneling regime. Recently super-Poissonian noise was experimentally observed by Onac et al. [112] in a carbon nanotube quantum dot and by Gustavsson et al. in a semiconductor quantum dot [113]. In Ref. [112] a Fano factor up to F = 3 was observed inside the Coulomb blockade diamonds, which is therefore noise was observed inside the diamonds and is therefore associated with inelastic cotunneling. Viceversa, in Ref. [113] the super-Poissonian Fano factor was observed at the onset of the conduction region, so that it is related to sequential tunneling through an excited state and it suggests a long relaxation time for the involved excited state. As suggested by the above list of references, the shot noise of a quantum dot has been widely studied in the last decade. In the remaining part of this section we will only shortly revised those results which are more closely related to the topics dealt with in my thesis. 2.3.1 Shot-noise in Luttinger liquid quantum dots An intriguing issue is the influence of Luttinger liquid correlations on the shot noise of a one-dimensional (1D) quantum dot, such as those presented in Sec. 1.3.4. As we will discuss in detail in Chap. 4, electronic interactions plays a unique role in one dimension. Here, the usual Landau-Fermi liquid theory breaks down and it has to be replaced by the Luttinger liquid model [11]. Luttinger liquids have a number of peculiar properties. As an example, their elementary excitations cannot be described in term of fermionic quasiparticles but rather of collective charge density waves (plasmons) which have a bosonic nature. Because of this, Luttinger liquids are characterized by power law correlations functions with interaction dependent exponent. Intriguingly, this same exponent shows up also in most of the measurable observables. Signatures of Luttinger liquid behavior were recently found in the shot noise of a one-dimensional quantum dot. As an example, studying the dependence Fano factor on the applied voltage, Braggio et al. [114] have found an oscillating behavior similar to the one observed in the non interacting case. However, both the position of the deeps and the asymptotic value of the Fano factor for large V , depends strongly on the strength of the interaction (see Fig.2.4). This dependence allows to extract the value of the Luttinger liquid parameter without resorting to any fitting Noise in mesoscopic systems 24 procedure [114]. In this work, fast relaxation of the plasmonic excitations in the Figure 2.4: Fano factor F of a one dimensional quantum dot as a function of the applied voltage, for T = 0 and strong asymmetry of the tunneling barriers. Different curves correspond to different strength of the electronic interaction: g = 1 for non a interacting system and g < 1 for repulsive interactions. Color legend: g = 1red, g = 0.8 black, g = 0.6 blue and g = 0.4 gray. After [114]. quantum dot was assumed, implying that excitations created by one tunneling electron do not influence subsequent tunneling events (see also discussion in Sec.4.4.1). The opposite case was lately addressed by Kim et al. [110], who have focused on the properties and the consequences of the non-equilibrium distribution of the plasmon in the dot. This is the case when the plasmonic excitations in the quantum dot redistribute only via the single-electron tunneling events through tunnel barriers. They found that while the average electric current is only weakly affected by the non equilibrium properties of the plasmons [115], the shot noise show distinctly different features [110]. In particular, the shot noise can turn to super-Poissonian values, and the enhancement is more severe in the strong interaction limit (see Fig. 2.5a). The origin of this enhancement is again connected to a dynamical channel blockade Figure 2.5: (a) Fano factor F of a one dimensional quantum dot with nonequilibrium plasmonic excitations, for T = 0 and strong asymmetry of the tunneling barriers. εp is excitation energy of the plasmonic excitations. Different curves correspond to different strength of the electronic interaction: g = 1 green, g = 0.7 black, g = 0.5 red and g = 0.3 blue. (b) In the presence of relaxation of the plasmonic excitations, the enhancement of the shot noise is strongly suppressed. Here γp is a phenomenological relaxation rate. Different curves correspond to different values of γp (in units of the average tunneling rate Γ0 ): γp = 0 black, γp = 0.1 red, γp = 1 green and γp = 104 blue. After [110]. effect. In fact, for sufficiently high voltages, the two transport channels N → N +1→N N → N + 1∗ → N (the asterisk stand for a transition occurring through an excited state in the dot) compete in transport. As we have seen above, if the transition rate for one of the 25 2.3 Shot-noise enhancement channels is much stronger than other one, then a super-Poissonian Fano can be achieved. This enhancement of the Fano factor is intimately related to the presence of non-equilibrium excitations. In the presence of relaxation of the plasmonic excitations, the super-Poissonian Fano factor is strongly suppressed (see Fig. 2.5b). Eventually, when the relaxation rate becomes comparable with the tunneling rate, the Fano factor is always sub-Poissonian and it recovers the characteristic behavior discussed by Braggio et al. [114]. 2.3.2 Shot-noise in nanoelectromechanical systems Another class of devices in which non-equilibrium effects are expected to play an important role are nanoelectromechanical systems (NEMS). NEMS are single electron devices in which the electronic degrees of freedom are strongly coupled to a mechanical degree of freedom (see Sect. 5.1). Electrons tunneling onto a NEMS can excite discrete vibrational modes (phonons) by spending some of the excess energy provided by the bias voltage. Thereby, electron transport through will tend to drive the vibrational mode out of equilibrium, if the quality factor of the latter is sufficiently high. It is then a physically relevant question whether the vibrational energy is reduced by relaxation processes induced by coupling to an external environment [116] or rather because tunneling itself [117–119]. Up to now, theoretical works have focused mostly on the case of negligible relaxation, taking the opposite case of strong relaxation as a reference term. It has been predicted that the shot noise would show a radically different behavior in this two cases, both for weak and strong electro-mechanical coupling [119, 120]. Of particular interest is the case of strong coupling. In fact, a giant enhancement of the Fano factor has been predicted in the case of strong electro-mechanical coupling and negligible relaxation of the vibrational excitations, as opposed to sub-Poissonian noise for equilibrated (i.e. thermally distributed) excitations, see Fig.2.6 [120]. The origin of the giant Fano Figure 2.6: Fano factor F as a function of bias voltage (in units of the frequency ω 0 of the vibrational mode) for a NEMS with strong electro-mechanical coupling. Results are shown for both non-equilibrated and equilibrated vibrational excitations. The combined effect of the strong coupling and weak phonon relaxation leads to a giant enhancement of the Fano factor. After [120] factor stem from the combined effects of the suppression of the tunneling rates due to the strong electro-mechanical coupling and the weak relaxation of the vibrational excitations. This interplay is responsible for an avalanche like transport of electrons Noise in mesoscopic systems 26 through the device [120–122]. However, as we will discuss in Chapter 5, this peculiar transport regime is strongly inhibited even by weak relaxation, leading to a suppression of F . Chapter 3 The sequential tunneling regime In this chapter we develop the theoretical tools required for the study of current and noise in the sequential tunneling regime. In section 3.1 we introduce a general model for a Coulomb blockade mesoscopic system, discussing the characteristic energy scales and the conditions under which transport through the system can be considered as a sequential tunneling process in section 3.2. In the sequential tunneling regime, the dynamics of the system can be studied in terms of a Markovian rate equation for the occupation probability of the island. In section 3.3 we derive the general expressions for the stationary current and for the current noise within the rate equation formalism. Throughout this chapter we keep wording and notation as general as possible to stress the wide applicability of this formalism, which forms the basis for most of the following chapters. 3.1 A general model for Coulomb blockade systems In general terms, a Coulomb blockade system can be described as a small electronic island (“dot”) coupled to leads by tunneling barrier. The important point is that to travel from one lead to the other the electrons must go through the island, as there is no direct coupling between the two external electrodes. The corresponding Hamiltonian can be written as: H = Hd + Hleads + HT . (3.1) Here, Hd is the Hamiltonian of isolated the electronic island and it must contain all degrees of freedom which are suspected to be relevant for transport. Quite in general, it can be written as the sum of a single particle part and of an interaction part Hd = Hd0 + Hdint . The non-interacting part is simply given by P the single particle spectrum of the island Hd0 = νd ενd c†νd cνd , where c†νd is the creation operator for state νd , while the interaction part Hdint is specific of the considered model. Importantly, since we are considering the isolatedP island, Hd must be electron-number conserving i.e. [Hd , Nd ] = 0, where Nd = νd c†νd cνd . Hence, each many-particle eigenstate of Hd has a well defined electron number, i.e Hd |αi = Eα |αi ⇒ Nd |αi = Nα |αi. The second term in Eq.(3.1) is the Hamiltonian for the leads, which we treat as electronic reservoirs. As the left and right leads are uncoupled we can write Hleads = H1 + H2 . We denote with |Li i the eigenstates of Hi and with ELi the corresponding energy. We assume that the leads are at any time in thermal equilibrium with the local chemical potential µ1,2 . In other words, we assume that the probability to 27 The sequential tunneling regime 28 Figure 3.1: Schematic picture of a electronic island connected to source and drain leads by tunneling barriers. find the state |Li i is given by the Boltzmann distribution function. WLβi = 1 −β(EL −µi NL ) i i e Z (3.2) where β = 1/kB T , NLi is the number of electrons into the state |Li i and Z is the grancanonical partition function for the considered lead. The bias window between the leads is fixed by external voltage µ1 − µ2 = eV . Furthermore, we assume the density of the states of the leads to be flat around the Fermi energy. The reservoirs Hamiltonian can additionally incorporate a bosonic heath bath to take into account a dissipative environment [123]. Finally, the last term in Eq.(3.1) is the tunneling Hamiltonian HT which couples the electronic island to the leads: X (i) X (i) (i) (i) ∗ (tνi,νd c†νi cνd + tνi,νd c†νd cνi ) (3.3) HT , HT = HT = i=1,2 νd,νi where c†νi creates an electron in the state νi into the ith-lead. Because of dot-leads coupling, the eigenstates of the electronic island acquire a finite lifetime τ = 1/Γ, where Γ is a characteristic rate for tunneling. Equivalently, one can says that there is a broadening of eigenstates of the dot proportional to Γ. Regarding H T as a perturbation, this rate can be evaluated by means of Fermi’s golden rule. In particular, the rate for the transition between two states of the dot |αi → |βi is given by X X β (i) (3.4) WLi |hβ, L0i |HT |α, Li i|2 δ(Eα − Eβ + ELi − EL0i ), Γα→β = 2π i=1,2 L ,L0 i i where the sum is taken over all the possible initial and final states for the reservoirs and each contribution is weighted with the probability WLβi to find the initial state |Li i. Importantly, to the lowest order in HT , only transitions between states with neighboring charge are allowed, i.e. it must be Nβ = Nα ± 1. Moreover, tunneling at the different barriers gives two independent contributions to the total transition (1) (2) rate which can then be written as Γα→β ≡ Γα→β + Γα→β . 3.2 The sequential tunneling regime The general model we have introduced in the previous section, is characterized by several energy and associated time scales Their relation one to each other determines the underlaying transport regime. There are basically three energy scales one has to consider: the level broadening Γ induced by tunneling, the typical energy separation ∆ between the many-body eigenstates of the electronic island and finally the temperature T , which determines the equilibrium distribution in the reservoirs. 29 3.2 The sequential tunneling regime We are interested in the sequential tunneling regime, which means that the electrons travel through the dot via subsequent incoherent tunneling events. For this to be true, the average time between two tunneling events 1/Γ must be much longer than the coherence time in the reservoir. As the time scale over which excitations in the reservoirs are thermalized is τeq = 1/kB T , we require Γ kB T . This condition guarantees that the excitations in the reservoirs created by a tunneling event are thermalized before the next electron can tunnel. In this case, it is justified to consider transport in the lowest order in HT and to disregard any higher order contribution. Moreover, since we are not interested into the interference effects given by mixing of different states of the dot due to tunneling, we demand Γ ∆. In this case the states of the island remains well separates. When tunneling defines the lowest energy scale into the problem, i.e. Γ kB T, ∆, the dot can be described it in terms of a distribution function Pα that gives the probability of finding the island in a particular state |αi. In equilibrium this is nothing but the Boltzmann distribution function. However, when a voltage bias is applied across the system, the distribution function is not thermal anylonger and it has to be determined self-consistently from the solution of the rate equations [13,18] X d Pα (t) = {Pβ (t)Γβ→α − Pα (t)Γα→β } . dt (3.5) β where the transition rates Γα→β are given in Eq.(3.4). These equations are rather self-explanatory: the second term on the right-hand side gives the rate at which the state |αi decays. It is proportional to the probability that the system is in state |αi in the first place, multiplied by the rates for transitions from |αi to any other state |βi. The first term gives the rate for the opposite process, i.e. the rate at which the system ends up in state |αi given that it started in some other state |βi. The rate equations must include all the states relevant to the dynamics of the system. The phase space Ω one has to consider depends on external parameters such as the voltage and temperature. In general, the rate equations have to be solved numerically and the set Ω have to be determined self-consistently by means of an iterative procedure which determines also the stability of the solution itself. To obtain an analytical solution of Eq.(3.5) one has to restrict to situations in which there are only few states involved into transport. As a final remark, we stress that the rate equations are to be understood as semiclassical equations because we assumed that in between tunneling events the state if the system can be described by classical probabilities Pα . A rigorous derivation of Eq.(3.5) can be achieved through the density-matrix formalism [124, 125]. Its starting point is the formulation of the von-Neumann equation for the reduced density matrix of the island. In general, this is a complicated integro-differential equation. The crucial simplification arises from the assumption that the electronic relaxation in the leads is fast compared to the tunneling dynamics, i.e kB T . In this case, the Born-Markov approximation can be employed and the equation of motion becomes an ordinary differential equation, the master equation for the reduced density matrix. Assuming Γ ∆, coherences between different states |αi are negligible, and the master equation reduces to the rate equation. This represents the evolution of diagonal part of the reduced density matrix. 1 1 In literature, the distinction between the terms “master equation” and “rate equation” is slightly inconsistent. The rate equation it is often called Pauli master equation [125, 126], or even simply master equation [127]. Following Ref. [124], we refer to the equation of motion of the reduced density matrix as master equation and to its to diagonal part as rate equation. The sequential tunneling regime 3.2.1 30 Matrix formalism It is convenient to re-arrange the occupation probabilities Pα (t) into a vector P(t) and to express the rate equations Eq.(3.5) in terms of a matrix equation of the form d P(t) = M P(t), dt (3.6) where M is coefficient matrix with elements Mαβ = Γβ→α − δα,β Γout β , Γout β = X Γβ→α . (3.7) α From Eq.(3.6) one can immediately see that the rate equations are formally solved by P(t) = eMt P(0). (3.8) where the vector P(0) is determined by the boundary conditions at t = 0. Interpreting this expression in terms of Bayes theorem, one sees that the “evolution” operator eMt determines the conditional probability to find the system in some state at time t given a certain initial condition [ eMt ]αβ = P (α, t|β, 0). (3.9) A direct consequence of the definition Eq.(3.7) is X Mαβ = 0 ∀ β. (3.10) α This sum rule, which ensures the conservation of probability (i.e. the sum of Pα (t) is independent of time), in turn implies that the matrix M is singular and therefore it has a zero eigenvalue. In the following we will always assume the zero eigenvalue to be non degenerate. This implies that there are no metastable states in which the system can get stuck, which is a reasonable hypothesis for many physical systems. In this case, the corresponding eigenvector P(st) gives the steady probability distribution. M P(st) = 0, (3.11) 3.3 Current and shot noise Although only the net transition rate Γα→β enter in the rate equation, the current depends on whether an electron moves to the right or to the left. As an example, if we fix the sign of the current to be positive for electron transfer from left to right, then a tunnel event in which an electron enter into the dot through the first junction gives a positive contribution to the current while it gives a negative contribution if it comes across the second. To take into account the difference in sign for the two process, we introduce two new matrices, I (i) (i = 1, 2), which contains the rates for tunneling across the first and second barrier with the appropriate sign (i) if Nα = Nβ + 1 Γβ→α (i) (i) Iαβ = (−1)i+1 (3.12) −Γ if Nα = N β − 1 β→α 0 otherwise With this definition, the tunnel current across each junction can be written as X (i) hI (i) (t)i = eTr I (i) P(t) ≡ e Iαβ Pβ (t), (3.13) αβ 31 3.3 Current and shot noise where the brackets h·i denotes the statistical average over the states of the system and the “trace” of a vector is defined as the sum over its components. In particular, the stationary current is given by hI (i) i = eTr I (i) P(st) . (3.14) Due to charge conservation, the average current across the two junctions must be equal in the stationary case, hI (1) i = hI (2) i = hIi. The same formalism can be applied to calculate the current noise ij S (ω) = 2 Z +∞ −∞ dt eiωt h∆I (i) (t)∆I (j) (0)i. (3.15) Following Ref. [93], we obtain the following general expression for the correlation function of current fluctuation h∆I (i) (t)∆I (j) (0)i = e2 Cxij (t) + Cxji (−t) + δij Cai , (3.16) where the first two terms contains the correlation between two different tunneling events Cxij (t) = θ(t)Tr δI (i) eMt δI (j) P(st) with δI (i) = I (i) − hI (i) i/e (3.17) and the third is the self-correlation term Cai = δ(t)Tr |I (i) |P(st) , (3.18) σ(t) = θ(t) eMt (3.19) (a detailed derivation of the previous expression is given in Appendix B). To evaluate the Fourier transform of Eq.(3.16), it is convenient to introduce the operator which is a sort of retarded Green’s function for the rate equation. In Fourier space, the equation of motion for the operator σ(ω) becomes a simple algebraic equation −iω σ(ω) = M σ(ω) + 1, which has the solution σ(ω) = − 1 . M + iω1 (3.20) The current noise can now be readily evaluated. By substituting Eq.(3.16) into Eq.(3.15) and taking Eq.(3.20) into account we come to the following result: with S ij (ω) = 2 e2 sijx (ω) + sjix (−ω) + δij Cai , (3.21) sijx (ω) = Tr δI (i) σ(ω)δI (j) P(st) (3.22) Eq.(3.21) represents the general expression for the current noise in the rate-equation formalism. At finite frequency, this expression is very convenient for numerical calculations as the inverse of the matrix M + iω1 can be easily evaluated with standard numerical tools [128].. However, some care has to be taken in the zero frequency limit. The sequential tunneling regime 3.3.1 32 The zero frequency limit As the matrix M is singular, the operator σ(ω) does not exist for ω = 0. Nevertheless, it can be shown that the quantity sijx (ω = 0) = −Tr δI (i) M−1 δI (j) P(st) , (3.23) is well defined and that it can be evaluated by means of standard numerical techniques. To show this, let us first introduce a little bit of technical notation. Let R(M) be the range of M and N (M) its kernel, i.e. R(M) = {v ∈ Rn : v = M u for some u ∈ Rn }, N (M) = {v ∈ Rn : M v = 0}, (3.24) (3.25) where n is the dimension of the phase space of the island. Since we assumed the existence of a unique stationary solution P(st) , M is a matrix of rank n − 1 and N (M) = {αP(st) , α ∈ R}. Even if the operator M is not invertible, the quantity M−1 v exist if v is a vector in the range of M, and correspond to the preimage of v under M, v ∈ R(M) ⇒ M−1 v = {u ∈ Rn : M u = v}. (3.26) This set can always be written as M−1 v = y + N (M), where y is an arbitrary solution of the linear system My = v. It is easy to see that the range of M is given by the subset of traceless vectors. Infact, because of sum rule Eq.(3.10), each vector in R(M) is traceless, hence R(M) ⊆ Ω ≡ {v ∈ Rn : Tr[v] = 0}. The subset Ω has dimension n − 1, being defined by a single condition. But, since rank(M) = n−1, this is also the dimension of R(M) and therefore it must be R(M) = Ω. Let’s now go back to Eq.(3.23). As δI (j) P(st) is by construction a traceless vector, the quantity M−1 δI (j) P(st) is well defined and is given by M−1 δI (j) P(st) = y + αP(st) , where α ∈ R and y is a solution of the linear system My = δI (j) P(st) . This system can be easily solved by means of standard techniques (e.g. singular value decomposition). Therefore we can write Eq.(3.23) as sijx (ω = 0) = −Tr δI (i) (y + αP(st) ) = −Tr δI (i) y , (3.27) from which we see that sijx (ω = 0) is also uniquely defined. To summarize, the zero-frequency current noise in the rate equation formalism is given by the following expression S i,j = 2e2 δi,j Tr |I (j) |P(st) (3.28) −2e2 Tr δI (i) M−1 δI (j) P(st) + Tr δI (j) M−1 δI (i) P(st) , given that the symbol M−1 is not be misinterpreted as the inverse of the (singular) M. 3.3.2 The two states regime The electron transport involving only the lowest energy states of a quantum dot have been deeply studied by several authors (see, for instance [93,94]). Nevertheless, for later reference we introduce the discussion of the shot noise with the two-state process, since it can be easily solved analytically and since, it provides a reasonable 33 3.3 Current and shot noise approximation for many physical systems at low energies. For definiteness, let us call |0i and |1i the two states involved into transport and ε their energy difference, i.e ε = E1 − E0 . The rate matrix M is then simply given by −Γ0→1 Γ1→0 M= (3.29) Γ0→1 −Γ1→0 (1) (2) (1) (2) where Γ0→1 = Γ0→1 + Γ0→1 and Γ1→0 = Γ1→0 + Γ1→0 are the total rates for the transitions |0i → |1i and |1i → |0i respectively. According to Eq.(3.12), “current matrices” I (1) and I (2) are defined as # # " " (2) (1) 0 Γ 0 −Γ 1→0 1→0 . (3.30) , I (2) = I (1) = (2) (1) −Γ0→1 0 Γ0→1 0 The stationary current can be straightforwardly evaluated by employing Eq.(3.14), yielding (1) (2) e Γ0→1 Γ1→0 (1 − e−βeV ) hIi = (1) . (3.31) (2) Γ0→1 (1 + eβ∆U (1) ) + Γ1→0 (1 + e−β∆U (2) ) Here we have made use of the detailed balance relation (i) (i) (i) Γ0→1 = e−β∆U Γ1→0 (3.32) where ∆U (i) = ε − µi is the change in the free energy associated with the tunneling into the dot across the ith-junction. At zero temperature, transitions can occur only in “one direction”, which is fixed by the external bias voltage. For definiteness, in the following we assume eV > 0 so that the only non vanishing rates at T = 0 are (2) (1) Γ0→1 and Γ1→0 . In this case Eq.(3.31) reduces to (1) hIiT =0 = (2) e Γ0→1 Γ1→0 (1) (2) Γ0→1 + Γ1→0 . (3.33) Analogously, taking into account Eq.(3.28), one can evaluate the zero frequency current noise and the Fano factor, obtaining (1) F =1− (2) 2Γ0→1 Γ1→0 (1) (2) [Γ0→1 (1 + eβ∆U (1) ) + Γ1→0 (1 + e−β∆U (2) )]2 . (3.34) From this expression one can easily convince oneself that in the two state regime the noise is always sub-Poissonian and bigger than SP oisson /2, i.e. 1/2 ≤ F ≤ 1. (3.35) At zero temperature, the expression for the Fano factor becomes particularly plain (1) FT =0 = (2) (Γ0→1 )2 + (Γ1→0 )2 (1) (2) [Γ0→1 + Γ1→0 ]2 . (3.36) We wish to emphasize that although Eq.(3.34) and Eq.(3.36) look simple, they (1) (2) contain a great deal of structure because the rates Γ0→1 and Γ1→0 depend on voltage (see e.g.. Ref. [93, 114]). It is convenient to re-express Eq.(3.36) as FT =0 = 1 (1 + α2 ) 2 (3.37) The sequential tunneling regime 34 where α is a function which takes into account the asymmetry between the tunneling rates (1) (2) Γ0→1 − Γ1→0 α ≡ (1) . (3.38) (2) Γ0→1 + Γ1→0 Equation (3.37) has a clear physical interpretation: in the case of strongly asymmetric barriers α = ±1, the Fano factor reaches the full Poissonian limit F = 1 as transport is essentially dominated by a single barrier (the more opaque). Viceversa, when the two tunneling rates become comparable (α ≈ 0), tunneling events at different barriers become correlated, leading to a suppression of the shot noise. (1) (2) The maximal suppression occur for perfectly symmetric junctions Γ0→1 = Γ1→0 . It is important to stress that α is a function both of the intrinsic transparency of the barriers and of the applied voltages, so that it is generally possible to tune experimentally α between -1 and 1. Indeed, very recently the shot noise of a semiconductor quantum dot have been measured as a function of α, showing an excellent agreement between experimental data and the prediction of Eq.(3.37), see Fig. 3.2 Figure 3.2: Fano factor of a semiconductor quantum dot as a function of the asymmetry of the tunneling rates α. The experimental data show an excellent agreement with the theoretical prediction (solid line). After [99] Chapter 4 Shot noise in a 1D ring In this chapter we want to study the shot noise of a one-dimensional quantum ring [129]. Possible experimental realizations of this system include [130, 131]. The 1D ring geometry allows to study the interplay between Fermi statistics, interactions, and Coulomb Blockade. Signatures of this interplay , which were for instance found in the behavior of the linear conductance [115], will reflect in the richness of the interaction-induced noise features analyzed below. In section 4.2 we give a brief introduction to the Luttinger liquid model. Subsequently, we describe the system under investigation in section 4.3 and in 4.4 we generalize the technique developed in Chap.3 to evaluate the current noise to the case of the magnetization current. In section 4.5.1 we present results in the absence of external magnetic flux, while in section 4.5.2 the effects of the latter is briefly investigated. In section 4.5.3 we present an interpretation of the results in terms of a Monte Carlo simulation of the transport dynamics in the system. 4.1 A one dimensional quantum ring The system we want to study is schematically depicted in Fig. 4.1. It is composed + g g` g` Φ R − Figure 4.1: Schematic description of the one-dimensional ring under investigation. The ring has radius R and the one-dimensional leads are characterized by the interaction parameter g` , while the ring by the parameter g. of a one-dimensional (1D) quantum ring of radius R, connected by tunneling barriers to two external leads, which we assume to be also 1D. As we have briefly discussed in section 1.3.4, one-dimensional electronic systems are expected to show 35 Shot noise in a 1D ring 36 unique transport features [64]. In fact, unlike in two or three dimensions, where the Coulomb interaction affects the transport properties only perturbatively, in 1D it completely modifies the ground state from its well-known Fermi liquid form, and the Fermi surface is qualitatively altered even for weak interactions. Today, it is well established theoretically that the low-temperature transport properties of interacting 1D-systems are described in terms of a Luttinger liquid (LL) rather than a Fermi liquid [62, 132]. Fermi vs Luttinger liquids In three dimensions interacting Fermions are well described by the Landau’s theory for the Fermi liquid [133]. The key observation is that macroscopic properties involve only excitations of the system on energy scales small compared to the Fermi energy so that the state of the system can be specified in terms of its ground state, i.e. its Fermi surface, and its low-lying elementary excitations, a rarefied gas of quasiparticles. These quasiparticles are a superpositions of single particle excitations and particle-hole pairs which evolve continuously out of the states of the free Fermi gas when the interactions are switched on. They are in one-toone correspondence with the bare particles (adiabatic continuity), possess the same quantum numbers and obey the same statistics, but their dynamical properties are renormalized by the interaction. Moreover, beacuse of the interaction they have a finite lifetime τ . However at T = 0, as the Fermi surface is approached τ diverges according to τ ∼ (E − EF )−2 , so that the quasiparticles are robust aganist small displacements away from EF . This scenario emerges because at low temperature the phase space for scattering particles is severly restricted by Fermi statistics and only a fraction T /TF 1 participate in the scattering process. The concept of quasiparticles breaks down in one dimension. In fact, in 1D the lifetime of the quasiparticles becomes strictly zero τ = 0, undermining the very same idea of adiabatic inclusion of the interaction. Starting with the early work of Mattis and Lieb [132], it has become clear that in one dimension the elementary excitations are collective density fluctuations. The correlations between these excitations is anomalous and show up as interaction-dependent non universal power-laws in many physical quantities, where those of ordinary metals are characterized by universal (interaction independent) powers. These properties are generic for one-dimensional fermionic systems, but are particularly prominent in the Luttinger liquid model. This is a theory based on the exact diagonalization of the interacting Hamiltonian and on the bosonization of the fermionic operators. All the correlation functions of the LL can be computed exactly, so that one has direct access to all the physical properties of interest. 4.2 The Luttinger liquid model In this section we follow closely Ref. [11, 134] in giving a brief introduction to the Luttinger liquid model. For sake of simplicity, we consider only a spinless model, referring to Ref. [11] for a exhaustive analysis of the spin degrees of freedom. 4.2.1 Luttinger liquid with periodic boundary conditions In the absence of interactions, the Hamiltonian of a system of spinless fermions confined on a ring of radius R is simply H0 = X k εk c†k ck , εk = k 2 /2m (4.1) 37 4.2 The Luttinger liquid model where k = n/R (n ∈ Z) because of the periodic boundary conditions. If we include the electronic interaction, the total Hamiltonian becomes H = H0 + Hint , where Hint = 1 X V (q)c†k1 +q c†k2 −q ck2 ck1 2L (4.2) k1 ,k2 ,q represents the most generic translationally invariant interaction among the electrons. Under certain assumptions (see below), the interaction term can be treated exactly, and the total Hamiltonian H can be diagonalized in terms of density operators which obey a boson statistics. This approach, which form the basis of the Luttinger liquid description, relays on three fundamental approximations: (i) The spectrum of the free electrons is linearized around the Fermi energy, forming two distinct linear branches which we call “right-movers” and “leftmovers”. (ii) The two branches are both extended to infinity, i.e. −∞ < k < ∞. This introduce an infinite number of states with negative energy which, in order to obtain physically meaningful results, needs to be all occupied. The presence of these unphysical states is not expected to affect the low-energy physics of the model (|ω| EF , |q| kF ). (iii) Solely forward scattering is considered, i.e. electrons cannot exchange branch as a consequence of the interaction. With these assumptions, the kinetic energy and the interaction term become: X H0 = vF (rk − kF ) : c†r,k cr,k : (4.3) k,r=± and Hint = 1 X V (q){: ρr (q)ρr (−q) : + : ρr (q)ρ−r (−q) :} 4πR q,r=± (4.4) respectively. Here, c+,k (c†+,k ) and c−,k (c†+,k ) are the annihilation (creation) operators for the right- and left-moving branch, respectively (ck ≈ c+,k + c−,k ), and X † ρr (q) = cr,k+q cr,k (4.5) k is the density fluctuation operator for the rth-branch (r = ±). The “comma” denote normal ordering, i.e. : A :≡ A − hAi0 , being hAi0 the expectation value with respect to the ground state of H0 . The normal ordering convention is necessary to avoid P reference to the total number of particles k hc†r,k cr,k i which is ill-defined. Acting on the ground state, the ρr (q) behave either as creation or annihilation operator, depending on the sign of q, i.e. ρr (−rq)|0i = 0, for q > 0. Moreover, the states ρr (rq)|0i and H0 obey a simple commutation relation [H0 , ρr (q)] = vF rq ρr (q). (4.6) which means that the states ρs (q)|0i are eigenstates of the linearized Hamiltonian Eq.(4.3) with energy vF q. Bosonization There are three important steps in achieving a complete solution of the model: (i) the realization that due to the infinite dispersion, the ρr (q) obey exact boson commutation relation [132]; Shot noise in a 1D ring 38 (ii) a representation of the free Hamiltonian Eq(4.3) as a bilinear in these boson operator [132]; (iii) the explicit√construction of a boson representation for the fermion operators P ψr (x) = 1/ 2πR k cr,k eikx [62]. The fact that the density operators obey Bose-like commutation relations can be easily shown by writing the commutator in terms of normal ordered operators X † [cr,k+q cr,k+q0 − c†r,k+q−q0 cr,k ] [ρr (q), ρr0 (−q)] = δr,r0 k = δr,r0 X k : c†r,k+q cr,k+q0 − c†r,k+q−q0 cr,k : + δr,r0 δq,q0 X k [hc†r,k+q cr,k+q i0 − (4.7) hc†r,k cr,k i0 ] = −δr,r0 δq,q0 rqR One can safely change the summation variable in the second line of Eq.(4.7) because the operators are normal ordered; the two terms add up to zero, leaving only the contribution of the third line. This result, combined with Eq.(4.6), allows to represent the free fermion Hamiltonian H0 , as a bilinear operator of the bosons ρr (q). In fact, it can be shown that Eq.(4.3) is completely equivalent to a the following form [132], H0 = vF X vF : ρr (q)ρr (−q) : + [N 2 + J 2 ] 2R 4R (4.8) r,q6=0 where N ≡ N+ + N− and J = N+ − N −, and Nr ≡ ρr (q = 0) is the number of particles added to the ground state in the rth-branch. In other words, N measures the total number of excess electrons and J the left-right asymmetry of the fermions added to the reference state (N = N0 and J = 0). Obviously, N and J obey the selection rule (−1)N = (−1J ). The first term of Eq.(4.8) represents the energy of density fluctuations at fixed electron number, while the second term stands for the energy of N particles added to the system and put into the lowest available states above the Fermi energy. One can easily convince oneself that Eq.(4.3) and Eq.(4.8) have the same spectrum. That the multiplicities of the levels are also equal can be proved by calculating the gran partition function both in the fermion Eq.(4.3) and in the boson Eq.(4.8) representation. Thus the fermionic and the bosonic spaces are identical. A completely satisfactory boson solution of the non interacting problem also require an explicit representation of the fermion operators ψr (x) in terms of the bosons ρr (q). A detailed derivation of this procedure is given in literature [11, 62, 135]. while here we will merely summarize its main ingredients. To this end, we introduce the field operators φ(x) Π(x) = − = i X 1 −α|q|/2−iqx x e ρ+ (q) + ρ− (q) − N 2R q 2R (4.9) q6=0 J 1 X −α|q|/2−iqx e ρ+ (q) − ρ− (q) + 2πR 2πR (4.10) q6=0 where α is a cutoff parameter which has to be set to zero at the end of any calculation. The fields φ and Π obey canonical boson commutation relations φ(x), Π(x0 ) = iδ(x − x0 ) (4.11) 39 4.2 The Luttinger liquid model The single electron operator can be expressed in terms of the fields φ and Π as [62] eir(kF −1/2R)x † √ Ur exp{−i[rφ(x) − Θ(x)]} α→0 2πα ψr (x) = lim (4.12) where Ur† is an operator which decrease the particle number in the rth-branch by one unit and Z x i X 1 −α|q|/2−iqx x Θ(x) = π Π(x0 )dx0 = e (4.13) [ρ+ (q) − ρ− (q)] + J 2R q 2R −∞ q6=0 The structure of Eq.(4.12) may be intuitively justified by noting that the field operator φ is related to the local particle density via ∂φ/∂x = −πρ(x) (4.14) where ρ0 = kF /π is the average particle density in the ground state. Therefore, introducing a particle at site x creates a kink of amplitude π in the field φ, i.e. at points on the left of x, φ has to be shifted by π. This shift is operated by the Rx exponential of the conjugated momentum operator Π, i.e. by exp[i −∞ dx0 Π(x0 )]. This operator commutes with itself hence, to construct a fermionic field, one has to restore anticommutation by multiplying it with exp[irφ(x)]. Finally, the unitary ladder operators Ur , which increase by unity the number of electrons, are necessary because the bosons fields ρr (q 6= 0) conserve the total number of particles. Moreover these operators, which take the name of Klein factors, assure the anticommutation of fermionic fields in different branches [135]. Diagonalization While the fermionic and bosonic representations are true alternatives for the non interacting problem, the success of bosonization is related to the fact that the bosonic one become more “natural” once interaction are introduced. Under the assumption of considering only forward scattering, the interaction Hamiltonian can be also written as a bilinear operator of the ρr (q)’s, see Eq.(4.4), so that the total Hamiltonian reads 1 X V (q) π X : ρr (q)ρr (q) : + vF + V (q) : ρr (q)ρ−r (−q) : H = 2R 2πvF 4πR r,q6=0 r,q6=0 i vF h V (0) 2 N + J2 . + 1+ (4.15) 4R πvF In this bosonic form, the problem of a system of interacting electrons can be solved exactly by means of a canonical transformation. In fact, Eq.(4.15) can be diagonalized by a Bogoliubov transformation [132] H̃ = eiS He−iS S= 1 X ξ(q) [ρ+ (q)ρ− (−q) − ρ− (q)ρ+ (−q)] R p>0 q (4.16) where the angles ξ(q) are defined by the condition −1/2 V (q) . g(q) ≡ e2ξ(q) = 1 + πvF (4.17) The density operators transforms as ρ̃r (q) = eiS ρr (q)e−iS = ρr (q) cosh ξ(q) + ρ−r (q) sinh ξ(q) (4.18) Shot noise in a 1D ring 40 and the diagonal form of the Hamiltonian is H̃ = 1 1 X vN N 2 + v J J 2 v(q) : ρr (q)ρr (−q) : + 2R 4R (4.19) q6=0 with v(q) = vF g(q) vN = vF = 0) g 2 (q vJ = v F (4.20) For repulsive interactions, it is g(q) < 1 while for attractive interactions g(q) > 1. Importantly, there are three different velocities in the problem: v(q) is the (renormalized) sound velocity and governs the bosonic excitations, vN is related to the charge excitations and measures the shift in the chemical potential upon varying the Fermi wave vector δµ = vN δkF and, finally, vJ measures the energy required to create a persistent current into the ring. Because of the interaction they are all different. Under the Bogoliubov transformation, the fermionic field transform as, ψ̃r (x) ∝ exp[i(Θ̃(x) − rφ̃(x)], where p i X g(q) −α|q|/2−iqx x φ̃(x) = − e ρ+ (q) + ρ− (q) − N (4.21) 2R q 2R q6=0 Θ̃(x) = x 1 i X p e−α|q|/2−iqx [ρ+ (q) − ρ− (q)] + J 2R 2R q g(q) (4.22) q6=0 (the zero modes Nr are not changed by the canonical transformation). It is convenient to introduce bosonic operators s 1 X θ(rq)ρr (q) a†q = R|q| r (4.23) which satisfy canonical commutation relations aq , a†q0 = δq,q0 . With this definitions the Hamiltonian of the Luttinger model Eq.(4.19) becomes H̃ = X q6=0 ω(q)a†q aq + 1 vN N 2 + v J J 2 4R (4.24) where we have introduced ω(q) = vF |q|/g(q). As long as one is interested only in the low energy properties of the model, q ≈ 0, a reasonable assumption is to linearize the excitation spectrum ω(q) → vF |q|/g, where g ≡ g(q = 0) is the effective interaction parameter. From now on we will consider only this long wavelength limit and we will treat the interaction as q-independent. In terms of the operators aq , a†q the electronic field operator become eirkF x † irNr x/R i$r (x) ψr (x) = lim √ Ur e e α→0 2πα (4.25) where $r (x) = 1 √ iX 1 p e−iqx a†q − eiqx aq √ + r sgn(q) g . 2 g R|q| (4.26) q6=0 Persisten currents The Luttinger liquid model description can be used straightforwardly to obtain the current induced in a strictly one-dimensional ring threaded by a magnetic flux 41 4.2 The Luttinger liquid model Φ [136]. In the one-dimensional geometry, the vector field can be removed entirely from the Hamiltonian via a gauge transformation which leads to the boundary condition ψ(2πR) = e−i2πΦ/Φ0 ψ(0) (4.27) for the fermion field operator, where Φ0 = hc/e is the flux quantum. This is achieved by replacing Φ (4.28) Π(x) → Π(x) − πRΦ0 in the bosonization formula Eq.(4.12). The J-dependent part of the Hamiltonian then becomes Φ 2 vJ J −2 (4.29) HJ = 4R Φ0 giving rise to a number current j= vJ Φ Φ0 ∂E = J −2 2π ∂Φ 2πR Φ0 (4.30) At equilibrium J is chosen so as to minimize the energy. Given that at constant particle number J can only change by two units, one can easily see that the equilibrium (persistent) current has a periodicity Φ0 and reach the maximum value vJ /2πR at Φ = Φ0 /2, giving rise to the familiar Aharanov-Bohm oscillations for the current as a function of the flux. 4.2.2 Luttinger liquid with open boundary conditions Up to now we have considered the case of a Luttinger liquid with periodic boundary conditions. Even if this prescription fits perfectly with the idea of a quantum ring, it is not suited to describe a “segment” of Luttinger liquid, for which open boundary conditions have to be employed [137–140]. Non interacting electrons Consider a system of electrons confined in the finite region [0, L] by infinite walls. This condition imposes the electronic field to vanish at the boundary, i.e ψ(0) = ψ(L) = 0. (4.31) p P In this case, the Fourier transform of ψ(x) takes the form ψ(x) = 2/L k sin(kx)ck , with k, = πn/L (n ∈ N) and the Fermi surface consist of a single point k = kF . Close to this point the fermion field can be expanded as ψ(x) ≈ eikF x ψ+ (x) + e−ikF x ψ− (x) where −i r X irqx e ckF +q . ψr (x) = √ 2L q>0 (4.32) (4.33) Unlike the case of periodic boundary conditions, the left and right moving fields are not independent but satisfy the condition ψ− (x) = −ψ+ (−x) (4.34) so that one can actually work with one of the two fields only, say ψ+ (x). The annihilment condition ψ(L) = 0 implies that the right moving field ψ+ (x) have to satisfy the relation ψ+ (L) = ψ+ (−L). Therefore we can regard ψ+ as defined on the whole x axes but periodic with period 2L, i.e. ψ+ (x) = ψ+ (x + 2L). Shot noise in a 1D ring 42 The linearized Hamiltonian for the free electrons can then be re-written as Z L Z L † † † H0 = dxvF (ψ+ i∂x ψ+ − ψ− i∂x ψ− ) = dxvF ψ+ i∂x ψ+ . (4.35) 0 −L The fermion field ψR (x) can be bosonized in the same spirit as in the case of periodic boundary conditions. However, different boundary conditions yield different relations between the electron field operator and the boson operators [135,137–139]. In the present case, the appropriate expression for the field ψ+ (x) is ψ+ (x) = lim √ α→0 1 † iπN x/L iΛ(x) U+ e e . 2πα (4.36) Here, N is the number of extra electrons with respect to the ground state and the phase fiel Λ(x) is given by the expression r X π −αq/2 iqx e e aq + e−iqx a†q (4.37) Λ(x) = qL q>0 where aq are canonical Bose operators, q = πn/L and α−1 is a momentum cut-off The phase field Λ(x) satisfies periodic boundary conditions Λ(x) = Λ(x + 2L) and is related to the density of right moving electrons by ρ+ (x) = 1 N + ∂x Λ(x). 2L 2π (4.38) The corresponding expressions for ψ− (x) and ρ− (x) can be readily obtained by employing ψ− (x) = −ψ+ (−x) and ρ− (x) = ρ+ (−x). The bosonized form of the kinetic energy is Z L X πvF 2 q a†q aq + dxρ+ (x)ρ+ (x) = vF H0 = πvF N (4.39) 2L −L q>0 Interaction effects Now we turn to consider the electronic interaction Z 1 dxdx0 ψ † (x)ψ † (x0 )V (x − x0 )ψ(x0 )ψ(x). Hint = 2 (4.40) As in the case of periodic boundary conditions, we restrict only to forward scattering and, additionally, we assume short range interactions V (x − x0 ) = V0 δ(x − x0 ). In this case the interaction term simply becomes Z L Z X V0 L dxρ+ (x)ρ− (x). (4.41) Hint = dx ρr (x)ρr (x) + V0 2 0 0 r=± In order to make use of the above bosonization procedure, one has to express also Hint in terms of the right moving field ψ+ alone, at the cost of introducing a non local interaction term Z L Z L dx ρ+ (x)ρ+ (−x). (4.42) dx ρ+ (x)ρ+ (x) + V0 Hint = V0 −L −L As Hint is quadratic in the electron densities, it takes a simple form in terms of bosonic operators. By employing Eq.(4.38) we come to the following expression i V 1 V0 X h † 0 q aq aq − (aq aq + a†q a†q ) + N2 (4.43) Hint = 2π q>0 2 4L 43 4.3 Model and Hamiltonian The total Hamiltonian H = H0 + Hint can be diagonalized in a standard way by means of the Bogoliubov rotation aq → ãq = cosh(ξ) aq − sinh(ξ) a†q , with −1/2 V0 e2ξ = 1 + ≡g πvF (4.44) Note that whereas the rotation angle ξ is defined as in the case of periodic boundary conditions, there is an important difference in sign. The transformed Hamiltonian reads X πvN 2 N (4.45) H̃ = ω(q)a†q aq + 2L q>0 with ω(q) = vF q/g and vN = vF /g 2 . Comparing this expression with Eq.(4.24) one sees that there are two important differences: first, the sum over momenta is restricted to positive values only, second, there is a single zero-mode term, i.e. the topologicalPcurrent J is not conserved anylonger. Both are a consequence of the condition r ψr (rx) = 0. Under the Bogoliubov transformation the field ψ+ (x) Eq.(4.36) transforms as ψ+ (x) → ψ̃+ (x) = lim √ α→0 with Λ̃(x) = 4.3 1 † iπN x/L U+ e exp{iΛ̃(x)} 2πα √ i 1 1 1 h√ g + √ Λ(x) − g − √ Λ(−x) . 2 g g (4.46) (4.47) Model and Hamiltonian We can now consider in more detail the system represented in Fig. 4.1. We model the system as two semi-infinite LL leads weakly coupled to a one-dimensional quantum ring. As discussed above, we have to employ periodic boundary conditions and open boundary conditions for the ring and the leads, respectively. The wires are weakly coupled to the ring so that single electron tunneling is the dominant charge transport mechanism. The total Hamiltonian of the system is then given by the sum of the bosonized LL Hamiltonians describing the isolated ring and leads and the tunneling Hamiltonian, which takes into account single-electron hops through the two junctions H = Hring + Ht + Hleads , Hleads = H1 + H2 (4.48) Using the bosonization technique described in the previous section, the LL Hamiltonian can be expressed in terms of creation and annihilation operators for collective excitations. For the semi-infinite leads it reads Z ∞ Hi = dq Ω(q)a†i (q)ai (q), for i = 1, 2 (4.49) 0 where Ω(q) = vF /g` q is the continuous dispersion relation for the bosonic modes, the operators a†i (q), a†i (q) creates and annihilate a plasmonic excitation in the i-th lead, and g` is the Luttinger interaction parameter. The leads are adiabatically connected to reservoirs which keep them in internal equilibrium. The difference of their chemical potential is controlled by the external bias voltage µ2 − µ1 = eV . Specifically, we assume symmetric voltage drop on the tunneling barriers so that it is µi = (−1)i eV /2. Shot noise in a 1D ring 44 While the zero modes play no role for the semi-infinite leads, they need to be accounted for in the case of the ring, which yields Hring 2 X EN Φ EJ 2 ω(q) a†q aq . + = J −2 (N − Ng ) + 2 2 Φ0 (4.50) q6=0 where N = N+ + N− represents the total number of excess charges in the ring and J = N+ − N− represents the imbalance between right- and left-moving electrons. Obviously, the zero modes are related by the condition N + J = even. Furthemore, Eq.(4.50) takes into account the presence of an external magnetic flux Φ piercing the ring and the extra charge eNg induced by an external gate capacitively coupled to the ring. The charging energy EN represent the minimum energy cost to add an excess electron into the system. A theoretical estimate for EN is provided directly within the LL model (EN ∼ vF /2Rg 2), however as the charging energy is also strongly influenced by the coupling to external circuit, long range interactions and screening effects, in the following we regard EN as a free parameter which sets the highest energy energy scale into the problem. The orbital addition energy is EJ = vF /2R EN and it stand for the energy required to create a persistent current into the ring. Finally, the last term in (4.50) is the contribution due the bosonic modes, with a†q (aq ) the creation (annihilation) operator for an excitation with energy ω(q) = v|q|, where q = n/R (n ∈ Z) because of the periodic boundary conditions, and v = vF /g is the group velocity of the collective excitations in the ring. The tunneling events between the ring and the leads are described by the Hamiltonian i X h ti ψ (i)† (xi )ψ (ring) (xi ) + h.c. , (4.51) Ht = i=1,2 where ψ (ring)† (xi ), ψ (ring) (xi ) are the electron creation and annihilation operators of the ring, and ψ (i)† (xi ) and ψ (i) (xi ) are the corresponding fields near to the boundary of the ith-leads. 4.4 The sequential tunneling regime The eigenstates of the isolated ring are identified by the following set of quantum numbers |Si = |N, J, {lq }i (4.52) where N and J are respectively the total number and the left-right asymmetry of the electrons added to the ground state, and {lq } is the set the occupation numbers of the plasmonic excitations at different q (lq = 0, 1, 2, . . . ). In the following we will often refer to J as the angular momentum of the ring, as it is intimately related to the expectation value of the axial angular momentum. At fixed flux and gate voltage, the total energy of the state |Si is given by 2 EN Φ EJ 2 U(S) = J −2 +εq. (N − Ng ) + 2 2 Φ0 (4.53) Since the excitation spectra are linear in wave number, the energy P of the plasmonic modes depends only on the total number of excitation quanta q = n∈Z n ln , being ε = v/R = vF /Rg the excitation energy. As we consider only repulsive interactions in the ring, i.e. g < 1, we obtain the following “hierarchy” of energy scales 2EJ ≤ ε EN . (4.54) 45 4.4.1 4.4 The sequential tunneling regime Tunneling rates The occupation probabilities of the quantum states of the ring change via electron tunneling events across the two junctions. In the weak coupling regime, i.e. when the level broadening due to tunneling is much smaller than temperature and level spacing Γ kB T, EJ , the electron tunneling is the source of small perturbation to the three isolated Luttinger liquids. In this regime, we calculate the transition rates between two eigenstates of Hring to the lowest non-vanishing order in the tunneling amplitudes. In this golden rule approximation, we integrate out the leads degrees of freedom since they are assumed to be in thermal equilibrium, so that the transition rates are given as a function of the state variables and the energies of the ring only [69] Z ∞ (i) (i) 2 Γ|Si→|S 0 i = ti ΨS,S 0 dτ e−i∆U τ hψ (i)† (xi , τ )ψ (i) (xi , 0)ith (4.55) −∞ (i) where the coefficient ΨS,S 0 stand for the matrix element † ΨS,S 0 = δ∆N,1|hS|ψ (ring) (xi )|S 0 i|2 + δ∆N,−1|hS|ψ (ring) (xi )|S 0 i|2 (4.56) and ∆U (i) is the change in the Gibbs free energy associated with the tunneling across the ith-barrier ∆U (i) = U(S 0 ) − U(S) − µi ∆N, (4.57) being ∆N = N 0 − N the difference between the number of the electrons in the ring in the states |S 0 i and |S 0 i. The thermal averages are calculated with respect to the uncoupled Hamiltonian Hi and ψ (i)† (xi , τ ) = exp(iHi τ )ψ (i)† (xi ) exp(−iHi τ ). Working out the lead contribution, one comes to [69] Z ∞ π γ(E) (4.58) dτ e−iEτ hψ (i)† (xi , τ )ψ (i) (xi , 0)ith = αωc −∞ where ωc is an high energy cutoff and the function γ(E) is given by γ(E) = 1 e−βE/2 1 βE 2 2π 1/g` −1 +i Γ 2π Γ(1/g` ) 2g` 2π β ωc (4.59) where Γ(x) is the Euler Gamma function. The function γ(E) represent the local tunneling density of states at the boundary of the leads. In the case of non interacting leads g` = 1, γ(E) simply reduces to the Fermi distribution function (remember |Γ(1/2+iy)|2 = π/ cosh(πy)). At zero temperature, γ(E) is proportional to a power law of energy 1/g` −1 1 |E| (4.60) lim γ(E) = θ(−E) T →0 Γ(1/g` ) ωc vanishing for positive energy gain, as required by energy conservation. Let us now turn to the ring contribution. Up to now, we have regarded the bosonic modes in the ring as infinitely stable states. Even if this assumption is perfectly consistent with the plasmons being eigenstates of Hring , in realistic systems this is usually not the case. In fact, the coupling of the system to the environment, such as external circuit or background charge in the substrate, leads to relaxation towards equilibrium. In the following we will assume fast relaxation of the plasmonic excitations in the ring, implying that excitations created by one tunneling event do not influence subsequent tunneling events. In other words, we assume that the Shot noise in a 1D ring 46 tunneling rate is much smaller than the rate of relaxation of the bosonic excitations, so that the plasmon relax between each tunneling event according to a thermal distribution. Therefore the only dynamical variables are N and J. We can define an effective tunneling rate as X β X (i) (i) Γ|N,Ji→|N 0,J 0 i W{lq } Γ|N,J,{lq }i→|N 0 ,J 0 ,{l0 }i (4.61) q {lq } {l0q } β where W{l is the thermal probability distribution with respect to Hring . Working q} out this sum, one comes to the final form for the transition rates for kB T < ε (i) Γ|N,Ji→|N 0 ,J 0 i = (i) Γ0 ∞ X n=−∞ an γ(∆U + nε − µi ∆N ) (4.62) where ∆U is the energy difference between the states |N 0 , J 0 i and |N, Ji h h Φ 1i 1i + EJ ∆J J − 2 + . ∆U = EN ∆N (N − Ng ) + 2 Φ0 2 (4.63) Note that considering Ht to the lowest order allows only for ∆N = N 0 − N = ±1 and ∆J = J 0 − J = ±1. The weights an comes from the discrete spectral density of the ring and are given by [141, 142] i 1 1 −βε h (0) (0) (0) + an = a(0) g + e a + a − 2a (4.64) n n n+1 n−1 2 g (0) where aq are the ring weights at zero-temperature (g+1/g)/2 Γ ((g + 1/g)/2 + n) −ε/ωc θ(n) . a(0) n = 1−e Γ ((g + 1/g)/2) n! (4.65) (i) The costant (i.e. independent on the interaction and energy) prefactor Γ0 ∝ t2i defines the bare tunneling rate through the ith-barrier. At low temperature the tunneling rate Eq.(4.62) exhibits a step-like structure as a function of the external bias voltage: a new step open each time µi ∆N ≥ ∆U + nε. In this case, an electron entering (or leaving) the ring can spend some of its excess energy to excite n excitation quanta. The collective modes will then quickly relax before the next tunneling event takes place. The height of each step depends on n and on the interaction parameter g, see Eq.(4.65). In the absence of interaction in the ring (g = 1) the energy of the plasmon is exactly twice the orbital addition energy (0) ε = 2EJ , and the weights Eq.(4.65) become independent of n, i.e an ≈ (ε/ωc ). 4.4.2 Charge and orbital tunnel currents Given the transition rates, we can build up the rate equations for the probabilities PN J (t) of occupation of the ring states |N, Ji X d PN 0 J 0 (t)Γ|N 0 ,J 0 i→|N,Ji − PN J (t)Γ|N,Ji→|N 0 ,J 0 i PN J (t) = dt 0 0 (4.66) N ,J P (i) where Γ|N,Ji→|N 0,J 0 i = i=1,2 Γ|N,Ji→|N 0 ,J 0 i is the total rate for the transition |N, Ji → |N 0 , J 0 i. Introducing a compact labeling for the states |N, Ji ≡ α, we can re-write the rate equation Eq.( 4.66) as ∂t P(t) = MP(t) (4.67) 47 4.4 The sequential tunneling regime where P(t) a vector representing the occupation probabilities of the various states of the ring and the transition matrix M is if Nα = Nβ ± 1 and Jα = Jβ ± 1 Γβ→α P (4.68) Mαβ = − α6=β Γβ→α if Nα = Nβ and Jα = Jβ 0 elsewhere where we have explicitly taken into account the selection rule ∆N = ±1 and ∆J = ±1, which comes from treating Ht to the lowest order. Let us look more closely at the different tunneling events which can occur in our system. Assume that the ring is initially in the state |N, Ji and consider a transition in which an electron tunnels into the ring, N → N + 1 . This electron can either join the right- or the left-moving branch of the linearized spectrum of the ring, giving a positive or a negative contribution to the angular momentum J, respectively. Analogously, an electron leaving the ring, N → N − 1, can either increase or decrease J depending on the branch where it comes from. Therefore, one is left with four classes of events, as summarized in Table 4.1. ∆N +1 +1 -1 -1 ∆J +1 -1 +1 -1 Type of event In + In − Out + Out − Table 4.1: Table of the four classes of events, determined by the possible tunneling processes. The notation “In +” stands for a electron entering into the ring and joining the right moving branch. Analogously, “Out −” represent an electron leaving the ring from the left-moving branch, and similarly for all the other types of events. Each tunneling event can take place at both barriers. Importantly, each tunneling event can take place at both barriers. We can then generalize the definitions of current matrices given in Chapter 3 to take into account the variations of J. To this end we introduce the matrix I (i),N , with elements (i) if Nα = Nβ + 1 Γβ→α (i),N (i) i+1 Iαβ = (−1) (4.69) −Γβ→α if Nα = Nβ − 1 . 0 otherwise which is related to the charge tunneling current through the ith-barrier, and the matrix I (i),N : (i) if Jα = Jβ + 1 Γβ→α (i),J (i) i+1 Iαβ = (−1) (4.70) −Γβ→α if Jα = Jβ − 1 0 otherwise which is connected to the angular tunneling current instead. The stationary currents are then simply given by hIν(i) i = eTr[I (i),ν P(st) ], ν = N, J (4.71) where P(st) is the stationary solution of the rate equations Eq.(4.68). We now define the zero-modes matrices N and J , which are expressed in components by Nαβ = Nα δαβ , Jαβ = Jα δαβ (4.72) Shot noise in a 1D ring 48 so that the expectation value for the number of electrons into the ring and for the angular momentum at time t are given by hN (t)i = Tr[N P(t)], hJ(t)i = Tr[J P(t)]. (4.73) deriving these expressions with respect to t, and observing that N and J satisfying the commutation relations [N , M] = I (1),N − I (2),N [J , M] = I (1),J − I (2),J (4.74) we come to the following couple of continuity equations d hN (t)i dt d e hJ(t)i dt (1) (2) (4.75) (1) (2) (4.76) = hIN (t)i − hIN (t)i, e = hIJ (t)i − hIJ (t)i. From these expressions one sees immediately that the stationary currents are independent of the barriers hIν(1) i = hIν(2) i . (4.77) Moreover, Eq.(4.76) allows for physical interpretation of hIJ i. As hIN i is related to ˙ i.e. the variation of the number of electrons into the dot, hIJ i is connected to hJi, to variations of the persistent current of the ring, due to tunneling. Following the same steps described in Appendix B, we can use the elements defined above to evaluate the correlation function of current fluctuations (ν = N, J) ji i (4.78) h∆Iν(i) (t)∆Iν(j) (0)i = e2 Cx ij ν (t) + Cx ν (−t) + δij Ca ν where the cross correlation term is (i) Mt δIν(j) P(st) ] Cx ij ν (t) = θ(t)Tr[δIν e (i) (i) (4.79) (i) with δIν = Iν − hIν i/e and Ca iν = Tr{|Iν(i) |Pst } (4.80) is the autocorrelation part. The noise spectral density Sνij (ω) is then simply given by the Fourier transform of Eq. (4.78). In particular, the zero frequency noise is given by Sνij (ω = 0) = 2e2 δi,j Tr |Iν (j) |P(st) (4.81) (i) (j) (st) (j) (i) (st) 2 −1 −1 −2e Tr δIν M δIν P + Tr δIν M δIν P , where the symbol M−1 has to be dealt with as explained Chapter 3. Because of the continuity equations, at zero frequency the noise spectral density Sνij (ω = 0) is independent of the barrier index Sνij (ω = 0) = Sνkl (ω = 0) ∀i, j, k, l; therefore we adopt the simplified notation Sνij (ω = 0) ≡ Sν . For the charge tunneling current, it is customary to refer to the Fano factor FN ≡ SN . 2ehIN i (4.82) Analogously, we define the Fano factor for the angular current as FJ ≡ SJ . 2ehIN i (4.83) which is related to the fluctuations of IJ . Note that we have to define FJ with respect to hIN i because, in the absence of an external magnetic field it is hIJ i = 0 because of time reversal symmetry. 49 4.5 4.5 Results Results Assuming the charge addition energy EN to be the largest energy scale into the problem kB T, eV EN , the electron transport across the ring is governed by just two charge states N and N + 1. This condition, however, does not exclude the possibility of exciting states with an high angular momentum, by means of a sequence of transitions such as In,± Out,∓ In,± |N, 0i −→ |N + 1, ±1i −→ |N, J ± 2i −→ |N + 1, J ± 3i · · · (4.84) where we have assumed N to be even. It is useful to notice that, in this two chargestate (2CS) regime, any state of the ring is fully specified by the angular momentum J only. Infact, because of the constrain (−1)N = (−1)J , it is sufficient to give J in order to identify the corresponding N . For definiteness, from now on we will assume N to be even and we will adopt the simplified notation |Ji ≡ |N, Ji, (4.85) Importantly, in the 2CS regime the electron transport transport forms a one-step Markov process, which means that transitions are only possible between neighboring states |Ji ↔ |J + 1i. A remarkable property of one-step process, is that the stationary occupation probability P(st) obeys the balance condition (st) PJ (st) ΓJ→J±1 = PJ±1 ΓJ±1→J (4.86) which allows us to write down immediately the stationary occupation probability (st) PJ (st) = P0 J−1 Y j=0 (st) where the constant P0 4.5.1 Γj→j+1 , Γj+1→j (st) (st) P−J = P0 J−1 Y j=0 Γ−j→−(j+1) Γ−(j+1)→−j (4.87) is fixed by the normalization condition Tr[P(st) ] = 1. Zero magnetic flux Before discussing the transport properties of the ring, it is worthy to look at the phase space of our system. As we have discussed in Chapter 3, the number of states involved into transport depends on the temperature, on the applied voltages and, in the present case, on the magnetic flux Φ. At low temperature kB T EJ , to a good approximation there is only a finite number of states involved in the transport J ∈ {Jmin , . . . , Jmax }, where Jmin(max) (V, Ng , Φ) can be determined solely on the base of energetic consideration. We start considering the case where there is no external magnetic field, Φ = 0. In this case, it is ΓJ→J 0 = Γ−J→−J 0 (J > 0) because of the degeneracy of the states |Ji and | − Ji. Therefore it is also Jmin = −Jmax . From Eq.(4.86), we see that in the stationary regime a certain state |J + 1i participate to the dynamic of the system only if: (i) the previous state |Ji is occupied; (ii) the transition rate ΓJ→J+1 is non vanishing. Taking into account dependence of the tunneling rates Eq.(4.62) on the external voltages, we can subdivide the (V, Ng )-plane according to the different possible transitions (remember, µi = (−1)i eV /2). For example, the transition |0i → |1i through the right barrier is possible only in the half plane defined by the condition (2) Γ|0i→|1i 6= 0 ⇔ Ng > −eV /2EN + N̄ (4.88) Shot noise in a 1D ring 50 where N̄ = N + (EN + EJ )/2EN . Analogously it is (1) Γ1→0 6= 0 ⇔ Ng < +eV /2EN + N̄ (4.89) 6= 0 ⇔ Ng < +eV /2EN + N̄ − 2EJ /EN (4.90) 6= 0 ⇔ Ng > −eV /2EN + N̄ − 2EJ /EN (4.91) 6= 0 ⇔ .. . Ng > −eV /2EN + N̄ + 2EJ /EN (4.92) (1) Γ1→2 (2) Γ2→1 (2) Γ2→3 and so on. For definiteness, from now on we assume V > 0. A little care have to be taken while considering transitions between states with high angular momentum. Consider for instance the transition |2i → |3i across the right barrier. Such an event can take place only if the state |2i is occupied. In other words, the transition |2i → |3i is relevant (i.e. it contributes to the tunneling current) when both Eq.(4.92) and Eq.(4.88 – 4.91) are satisfied. This give rise to the characteristic fish-bone structure of Fig. 4.2. Here, we have denoted with different colors regions characterized by different Jmax . As an example, in the yellow region it is Jmax = 1, which means that there are only three states accessible to the system {|0i, | ± 1i}, while in the cyan region it is Jmax = 2, and the relevant states are {|0i, |±1i|±2i}. In the following we will refer to the yellow and cyan regions as three- and five-states region, respectively. The boundaries of zones with different Jmax corresponds to the conductance peaks ∂I/∂V of an ideal current measurement. Similar considerations holds for transitions a 0 → 3 → → 2→ 0→ 1 1 → 2 1 →2 → b Ng → 1→ → Ng V → V Figure 4.2: (a) Scheme of the relevant transport regions in the (V, Ng ) plane for Φ = 0 and N even. Solid lines denote detectable conductance peaks, the involved |J| values are denoted. The black lines represent transitions which do not involve plasmonic excitations. The green line represents the transition 2 → 1 with the excitation of one plasmon. Yellow region: three-states regime Jmax = 1. Cyan region: 5 states regime Jmax = 2. (b) Same scheme as in (a), for 0 < Φ < Φ0 /2. Each transition line is now split in two (for the sake of simplicity, the plasmon-related green lines have been omitted). Arrows indicate the lines evolution increasing Φ. involving an excited state. As an example, the transition |2i → |1i involving a plasmon excitation into the final state, which we indicate as |2i → |1∗ i, can occur only if Eq.(4.88 – 4.91) are all satisfied (i.e. if the initial state |2i is populated) and if the bias voltage provides the necessary energy (2) Γ|2i→|1∗ i 6= 0 ⇔ Ng > −eV /2EN + N̄ + (ε − 2EJ )/EN . (2) (4.93) Note that for 0.5 < g < 1 the onset of the transition Γ|2i→|1∗ i cut the five states region into two zones which we call I and II. For a non interacting ring g = 1, there is only zone II, as the plasmonic excitations becomes degenerate with the topological currents, and finally for g < 0.5 we have only zone I. 51 4.5 Results We now focus on the regions of the (V, Ng ) plane where Jmax ≤ 2. In this case, the transition matrix M is a 5 × 5 tridiagonal matrix and rate equations and Eq.(4.71) and Eq.(4.81) can be evaluated analytically. Even in this simple case the expression for current and for the noise at finite temperature are extremely cumbersome, therefore in the following we will consider only the zero-temperature limit T = 0. In this case, we obtain (2) hIN i = e (2) (1) (1) 2Γ0→1 Γ2→1 (Γ1→0 + Γ1→2 ) (1) (2) (2) (1) (4.94) (1) Γ1→0 Γ2→1 + 2Γ0→1 (Γ1→0 + Γ1→2 ) for the charge currents while, as expected, hIJ i = 0. The charge Fano factor is (1) FN = 1 + 2 (1) (2) (2) (2) (2) (1) (2) (2) (1) Γ1→0 Γ1→2 (2Γ0→1 − Γ2→1 )2 − 2Γ0→1 Γ2→1 (Γ1→0 Γ2→1 + 2Γ0→1 Γ1→2 ) (1) (2) (2) (1) (1) [Γ1→0 Γ2→1 + 2Γ0→1 (Γ1→0 + Γ1→2 )]2 (4.95) and the one related to the angular tunnel current is FJ = 1 − 2 (1) (1) (1) (1) (1) (1) Γ1→2 Γ1→0 − Γ1→2 Γ1→0 Γ1→0 + Γ1→2 . (4.96) Comparing these expression with the exact numerical result, we have found that (2) they describe very accurately the temperature regime kB T ≤ 0.1EJ . For Γ1→2 = 0 (three states regime), they give back the well known results for a degenerate electronic single level [143, 144] (2) hIN i = e (1) 2Γ0→1 Γ1→0 (2) (1) 2Γ0→1 + Γ1→0 (2) , FN = (1) (2Γ0→1 )2 + (Γ1→0 )2 (2) (1) (2Γ0→1 + Γ1→0 )2 (4.97) and FJ = 1. An important difference between the three- and the five-states regime in that while in the first case in is always 0.5 ≤ FN ≤ 1, the second addend in Eq.(4.95) may be positive, increasing the value of the Fano above the Poissonian limit. It is useful to introduce the average dwell time for the Jth-state, i.e. the average time spent by the system in the state |Ji. In our model, a reasonable estimate for the dwell times τ|J| is given by τ0 = 1 (2) 2Γ0→1 , τ1 = 1 (1) Γ1→0 + (1) Γ1→2 , τ2 = 1 (2) Γ2→1 . (4.98) Moreover, we define the adimensional parameters (1) r1 = (2) Γ1→2 r2 = (1) Γ1→0 2Γ0→1 (4.99) (2) Γ2→1 In terms of these quantities, the expressions for the charge and the angular Fano factor Eq.(4.95) and Eq.(4.96) become FN = 1 + 2 FJ r1 (1 − r2 )2 − (1 + r1 r2 )(1 + r1 )τ1 /τ0 = 1 − 2r1 [1 + r1 r2 + (1 + r1 )τ1 /τ0 ] 1 − r1 . 1 + r1 2 , (4.100) (4.101) From Eq.(4.101), one sees that the orbital Fano factor depends only on r1 , which represents the “populating ratio” between the states |J| = 2 and J = 0. In other Shot noise in a 1D ring 52 words r1 measures the transition probability per unit of time for the transition |1i → |2i with respect to the one of |1i → |0i. In order to achieve FJ > 1 it is necessary to have what we call an inversion of the populating ratio r1 > 1, i.e. the transition |1i → |2i leading to the more energetic state have to be stronger than the one going to the ground state |1i → |0i. Note that the r1 > 1 does not imply an (st) (st) invertion of populations P2 > P1 . Infact, from Eq.(4.87) one can easily see that (st) (st) (st) (st) the ratio P2 /P0 depend on both r1 and r2 . Indeed, it is 2P2 /P0 = r1 r2 . The parameter r2 is defined by the ratio between the dwell times τ0 and τ2 r2 = τ2 /τ1 , (4.102) and it plays a fundamental role in determining the properties of the charge Fano factor FN . From Eq.(4.100) we see that in order to observe FN > 1, it have to be r2 6= 1 (and r1 6= 0). As we will discuss below, having τ2 6= τ2 is a signature of the electronic interactions in the ring. Furthermore , it is necessary to have asymmetric tunneling barriers. More precisely, to have FN > 1, the asymmetry of the barriers (1) (2) A ≡ Γ0 /Γ0 , have to be bigger than a certain critical value defined by (2) Ac = (i) (i) 2Γ̄0→1 (1) Γ̄1→0 + (1) Γ̄1→2 (1 + r1 r2 )(1 + r1 ) , r1 (1 − r2 )2 (4.103) (i) (i) where Γ̄α→β ≡ Γα→β /Γ0 are the the normalized rates. Note that because of Γ̄α→β , the critical asymmetry strongly depends on the voltages and on the interaction parameters g and g` , see Fig 4.4. Note that in the A Ac , one has τ1 τ0,2 , so that the transport dynamics is dominated by the two last time scales τ0 and τ2 . Before discussing the characteristics of the critical asymmetry, let us look in more detail to the properties of the Fano factors FN and FJ . 1.6 FN a 1 0.8 2 FJ b 1 0.6 0 (eV − 2.43EJ )/ε 2.5 Figure 4.3: Fano factors as a function of (eV − 2.43EJ )/ε, where V varies according to Ng = 1/2 + 0.93EJ /EN − eV /2EN , with kB T = 0.02EJ and A = 20. (a) FN for an interacting ring, g = 0.7, and different interactions in the leads: g` = 1 (red), 0.9 (cyan), 0.5 (green), 1.2 (blue); magenta: g = g` = 1. (b) FJ , parameters and colors as in (a). In Fig. 4.3, the values of FN (a) and FJ (b), calculated along the line Ng = 1/2+0.93EJ /EN −eV /2EN (red dash-dot line in Fig. 4.2a), are shown as a function of the offset adimensionalized voltage (eV −2.43EJ )/ε. Both exhibit clear jumps as a consequence of the excitation of plasmonic modes in the transitions |J| = 1 → J = 0 and |J| = 1 → |J| = 2. In the absence of interactions (g = 1 = g` , magenta curves) the plasmonic modes are degenerate ε = 2EJ . Since E1→2 = E1→0 + 2EJ and 53 4.5 Results E2→1 = E0→1 + 2EJ , from (4.62) it is clear that r1 < 1 and r2 = 1, leading always to at most Poissonian noise. With an interacting ring, the degeneracy is lifted (ε > 2EJ ) so that it is possible to have r2 6= 1. In this case it is possible to achieve FN > 1, if A > Ac . The super-Poissonian charge noise is robust against the leads interactions as can be seen in Fig. 4.3 (a). In particular, and in striking contrast with the behavior of FJ , the charge noise is not sensitive to the presence of repulsive g` < 1 or attractive g` > 1 interactions in the leads. On the contrary, for the magnetization noise, it is crucial to have g` > 1 in order to obtain superPoissonian values. For an interacting ring, in region I it is always r2 = 1. Moving along the red dash-dot line Ng = 1/2+0.93EJ /EN −eV /2EN , one can identify two energy regions σ± (n) σ− (n) if (n − 1)ε < eV − 2.43EJ < (n − g)ε σ+ (n) if (n − g)ε < eV − 2.43EJ < nε . In these two regions, for r1 one has ( 1 in σ− (n) r1 = n/(n + 1) in σ+ (n) In σ− (n), for A Ac and g` = 1, the Fano factors reach the asymptotic limits FN = 1 + 2/9 and FJ = 1. By tuning the voltage V , one periodically enters the regions σ+ (n), where the Fano factor depends on the details of the interactions. For noninteracting leads one has FN = 1 + 2n(n + λ) , (3n + λ)2 FJ = 1 − 2nλ . (n + λ)(2n + λ) (4.104) with λ = (g +g −1 )/2. For interacting leads, one finds smoothed steps in FN and FJ , with a power law behavior as a function of V . For particularly strong interactions g . 0.5, the smoothing of the step-like structure is almost complete. It is interesting to notice the strong increase of FN , as compared to the suppression of FJ (green curves). In Fig. 4.4, the critical asymmetry Ac is plotted in the plane XY , where 0.08 a A∞ Y 0.03 b Y 0 1 0 X 0.2 Figure 4.4: Color-contour plot of Ac for g = 0.7, g` = 0.8, kB T = 0.02EJ in the XY plane (see text). (a) Region I, A∞ = 4100. (b) Region II, A∞ = 2.6 105 . X = 1/2 − Ng − 1.5EJ /EN + eV /2EN and Y = −1/2 + Ng − 0.5EJ /EN + eV /2EN . Note that Y = 0 corresponds to the transition line 0 → 1, Y = 0.03 to the line E2→1 = ε, and X = 0 to the line 1 → 2 (cf. Fig. 4.4 (a)). In the panels (a) and (b), the regions I and II are shown. In both regions, near certain lines parallel to the X axis, for g` it is possible to have r2 = 1. Here, Ac diverges and no super-Poissonian charge noise can be achieved. Away from these lines, increasing Shot noise in a 1D ring 54 the voltage, Ac decreases because of the increasing number of excited plasmons present in the output transition. The latter decrease the dwell time τ1 increasing (1) the ”output” rates Γ1→0,2 . For sufficiently high voltages, one can eventually reach the condition Ac < 1. 4.5.2 Non-zero magnetic flux When Φ 6= nΦ0 /2 (n ∈ Z), the degeneracy of the states with ±J is lifted. As a consequence, the transition lines J → J 0 split in two and move in opposite directions increasing Φ, with a separation proportional to EJ Φ/Φ0 . In Fig. 4.2 (b) an example of such a situation is shown. The plane (V, Ng ) is subdivided in many different regions, in each of which different values of Jmin and Jmax are found. This makes an analytical diagonalization of the matrix M̄ extremely difficult and we have to resort to numerical evaluations. Many interesting results are found. We restrict our attention on the 0 < Φ < Φ0 /2 regime. The results shown in Fig. 4.5 represent 0.5 1.21 Φ FN a 0 0.5 FJ Φ 0 0.86 1 0 (eV − 3.2EJ )/ε b 1 0.6 Figure 4.5: Density plot of FN (a) and FJ (b) as a function of (eV − 3.2EJ )/ε (voltage moving according to Ng = 1/2 + 1.7EJ /EN − eV /2EN ) and Φ, in units Φ0 . Parameters are: g = 0.8, g` = 1, kB T = 0.02EJ and A = 20. FN and FJ evaluated along a diagonal line which lies inside zone II at Φ = 0 and are specular with respect to the Φ = Φ0 /2 when Φ0 /2 < Φ < Φ0 . As a whole, the results are periodic, with period Φ0 . Increasing Φ, many moving transition lines cross the diagonal, fixed one. Therefore, it is possible to study the correlation effects on the noise spectrum in a wide range of transport regions. As Fig. 4.5 confirms, the noise exhibits an extremely rich dynamics by tuning the flux. At Φ = 0 one has A < Ac , so that FN < 1. However, increasing the flux, super-Poissonian charge noise is reached for Φ ≈ 0.4Φ0 , signalling the crossover to a transport regime where A > Ac . The onset of this region is at Φ∗ = 0.35Φ0, given by the intersection of the line along where V is varied, Ng = 1/2 + 1.7EJ /EN − eV /2EN , with the upper moving transition line 2 → 1 with one plasmon, Ng = 1/2 + EJ /2EN (4/g − 3 + 4Φ/Φ0 )−eV /2EN , i.e. Φ∗ = [1.7+(3−4/g)/2]Φ0/2. The possibility of crossing over between sub- and super-Poissonian behavior as a function of the flux is a signature of the presence of interactions. Indeed, in a noninteracting ring we always find subPoissonian behavior, regardless of the number of states supporting the transport. Concerning the magnetization noise, interactions in the ring and finite flux are not enough to induce FJ > 1 (Fig. 4.5 (b)). 4.5.3 Monte Carlo simulation As discussed in section 4.5.1, the interaction-induced separation of the time scales τ0 and τ2 is responsible for the super-Poissonian behavior of SN . In order to better 55 4.5 Results τ0 τ1 1 2 1 τ1 → 2 1 0 τ0 → → 0 1 → → 1 0 1 → a 0 understand this mechanism we set-up a BKL Monte Carlo simulation [145] in the T = 0 regime. An effective time evolution for the system is modeled by a sequence of discrete steps. At the k-th step, the ring makes the transition Jk → Jk+1 . The transition is supposed to be instantaneous, while the average time between the k-th and the k + 1-th steps is simply the dwell time τ|Jk | . The situation is schematically τ2 τ Figure 4.6: Schematic description of the Monte Carlo simulation. The subsequent simulation steps are denoted by red crosses. An example of possible transitions and corresponding dwell times is displayed (see text). depicted in Fig. 4.6. The k-th transition is performed according to a conditioned probability given by X ΓJk →Jk+1 ΓJk →J 0 J 0 =Jk ±1 (we omit here the barrier indexes for ease of notation) with −2 ≤ Jk , J 0 , Jk+1 ≤ 2 and |Jk+1 − Jk | = 1. It is possible to show that the average time spent in each state converges, in the asymptotic (k → ∞) limit, to the stationary occupation probability as calculated by solving the rate equation 4.66 in the t → ∞. In the a b c d τ /τ0 Figure 4.7: Output sequences of a Monte Carlo simulation for tunneling events at junction 2, with eV = 3EJ , Ng = 0.48, kB T = 0.02EJ and A = 20. Black (white) dots denote a clockwise (anticlockwise) entering electron. Colored bars are the sequence of the orbital value |J|, green: oscillations 0 → ±1 → 0; red: oscillations ±2 → ±1 → ±2; (a) g = g` = 1; (b) g = 0.7, g` = 1; (c) g = 0.7, g` = 0.5; (d) g = 0.7, g` = 1.2. following, we consider A Ac , Φ = 0 and V > 0 in the 5 states region. Figure 4.7 shows typical outcomes of the simulation, represented as sequences of tunneling events for clockwise (black dots) and anti-clockwise (white dots) electrons tunneling through the input barrier. Since τ1 τ0,2 , the ring state ”oscillates” alternatively around J = 0 and |J| = 2. The colored bars describe these oscillations: green represents the transitions 0 → ±1 → 0, red represents ±2 → ±1 → ±2. We denote these two sequences as S0 and S2 . The average time interval between tunneling is τ0 inside S0 and τ2 inside S2 and. Correspondingly, the average number of transitions in S0 is n0 = (1 + r1 )/r1 with an average duration time T0 = n0 τ0 , while in S2 we have n2 = (1 + r1 ) with an average duration time T2 = n2 τ2 . It is interesting to notice that T2 represents the average sequential-tunneling relaxation time of the Shot noise in a 1D ring 56 states |J| = 2 [146]. Without interactions, Fig. 4.7 (a), the tunneling events are uniformly distributed since (τ0 = τ2 ). In the interacting case, the removal of the degeneracy of τ0 and τ2 is reflected in a bunching tendency of the tunneling events as can be clearly seen in panels (b,c,d) of Fig. 4.7. The bunching can obviously be present either in S0 – panels (b,d) – or in S2 – panel (c). In all of these cases we have FN > 1. Quite generally, while with superpoissonian noise we always find a bunching of the tunneling events, the converse can be false: if A < Ac the bunching might still occur but the interplay of the two barriers with comparable transparency gives rise to negative correlations and a depression of the noise below the Poissonian limit, thus giving FN < 1. The interpretation of FJ is distinctly different (Fig. 4.7 (d)). Here, the time scales τ0,2 do not play any role. The important parameter is the number of events n2 inside the sequence S2 in which the ring oscillates around the excited state J = 2. The condition FJ > 1 is fulfilled only for n2 > 2, independent of the bunching mechanism, which in principle could even be absent. Chapter 5 Shot noise in NEMS In this chapter we study the effect of relaxation on the transport properties of a nanoelectromechanical system (NEMS) [147]. We describe the NEMS as a quantum dot coupled to an oscillator with incoherent dynamics and we consider the sequential tunneling regime. We show that relaxation induces qualitative changes in the transport properties of the dot, depending on the strength of the electron-phonon coupling and on the applied voltage. In particular, critical thresholds in voltage and relaxation are found such that a suppression below 1/2 of the Fano factor is possible. The Chapter is organized as follows. The model Hamiltonian is defined in Sec. 5.2, while in Sec. 5.3 we introduce the rate equation and the formal expressions for the current and the noise. In Sec. 5.4 numerical results for the current and Fano factor are presented: in particular, the suppression of the Fano factor is discussed in detail for a wide range of parameter. Finally, analytic expressions for the current and the Fano factor are derived within a toy model employing few phononic states. 5.1 Introduction Nanoelectromechanical systems (NEMS), are MEMS scaled to the submicron dimensions. Micromechanical systems (MEMS) have been studied for decades [148, 149], with an increasing interest because of growing commercial applications. Nowadays, micromechanical devices are routinely employed in a wide field ranging from ink-jet printers, accelerometers, scanning probe microscopy and optical communications. Reducing the size of mechanical devices to the nanoscale may provide a revolution in applications such as sensors, medical diagnostics, display and data storage [22, 23]. Indeed, with resonant frequencies from kilohertz to gigahertz, low dissipation and small masses [25], NEMS are the ideal candidate to ultra-sensitive detection [7,26,27,150]. Moreover, their dimension not only make them susceptible to local forces, but also make it possible to integrate and tightly couple them to a variety of electronic structures, such as quantum dots or point contacts [36]. From the fundamental point of view, the dynamics of nanoelectromechanical systems have attracted a considerable interest because of the novel transport mechanisms they can give rise to [151, 152] and their interesting non linear and chaotic properties [153, 154]. Furthermore, when cooled to ultralow temperatures, high-frequency resonators are expected to display quantum mechanical behavior so that NEMS might be used to explore the cross over from quantum to classical behavior in mechanical systems [155]. A prerequisite for attaining the ultimate potential from NEMS is displacement sensing, that is reading out the motion of the device induced by an applied stimulus. Perhaps not surprisingly, transduction schemes which are important for MEMS do 57 Shot noise in NEMS 58 not prove optimal for nanoscale devices. For example, the sensitivity of opticalinterferometry approaches is severely limited by diffractions, since the dimension of NEMS are small compared to the wavelength of light. Some of the most successful transduction efforts are focused on coupling a nanomechanical system to quantum dots or single electron transistors (SET) used in configurations where the motion of the mechanical part modulate the electron transport properties. This can be achieved by coating the mechanical resonator by a thin metallic layer, and placing it next to a dot [7, 27]. The resonator acts as a capacitor whose presence affects the current flowing through the SET. The capacitance, and therefore the current, depend on the position and hence on the motion of the oscillator. The exceptional charge sensitivity of the SET allows to motion detection with a sensitivity down to the quantum limit (see discussion on Sec. 1.2). However, because of the small inertial mass of the oscillator, the movement of the electrons through the SET acts back on the resonator affecting its dynamics in a important way. Intriguingly, in these devices the tunneling of a single electron may induce a displacement of the movable structure. Another class of systems for which the electro-mechanical coupling plays a fundamental role are molecular devices (see Sec. 1.3.2). This coupling originates from the fact that, when a molecule is employed as an active part in an electrical circuit, its nuclear configuration needs to adjust to the addition or removal of electrons. Because of this, electron transfer is generally accompanied by excitation or de-excitation of molecular vibrations (phonons). Recent experiments on singlemolecule junctions have shown that these excitations are reflected in the current voltage characteristics as vibrational sidebands [29, 40, 45]. As an example, signatures of the center of mass mode of a C60 molecule between gold electrodes have been recently reported [29], whereas in a experiment on C140 dimers, vibrational sidebands due to an internal mode have been observed [45]. Moreover, as electrons tunneling on to the molecule can excite vibrational modes by spending some of their excess energy provided by the external bias, they will tend to drive the vibrations out of equilibrium, if the quality factor of the vibrational mode is sufficiently large. Such nonequilibrium vibrations have so far been observed in at least one experiment on a suspended carbon nanotube as absorption satellites of Coulomb blockade peaks [59] Despite the obvious differences in design and characteristic energy scales, electron transport through NEMS and single molecule devices shows many common features. Therefore the acronym NEMS is customarily used for both class of devices. On the theoretical side, NEMS are often described in terms of a very simple phenomenological model, based on the following series and simplifications: (i) Transport is assumed to be dominated by tunneling through a single electronic level with single-particle energy ε. Due to coulomb interaction, double occupation of the level is associated with an additional charging energy U > 0. (ii) Only a single vibrational mode with frequency ω0 is taken into account within the harmonic approximation. (iii) The coupling between the charge on the dot and the mechanical mode is assumed to be linear The key features of this model, which is also known as the Anderson-Holstein model, are schematized on Fig. 5.1. Even within these rude simplifications, many peculiar features such as negative differential conductance [156–158], shuttling instability [151, 159] and strong mechanical feedback [160] have been predicted in the case of an underdamped oscillator. 59 5.2 Model Figure 5.1: Potential surfaces for a NEMS-model featuring a single, spin-degenerate electronic orbital coupled to a single vibrational mode. The model Hamiltonian approximates the potential surfaces by a harmonic-oscillator potential. The electronphonon coupling, parametrized by the coupling strength λ, corresponds to a charge-dependent x-shift of the potential surfaces. 5.2 Model According to the Anderson-Holstein model, the Hamiltonian of a NEMS is given by Hs = Hn + Hb + Hn,b (5.1) where Hn , Hb and Hn,b are respectively given by Hn Hb = ε n + U n ↑ n↓ = ω0 (b† b + 1/2), (5.2) (5.3) Hn,b = λ ω0 (b† + b) n. (5.4) The operators nσ = d†σ dσ and n = σ nσ represent the spin-resolved and the total occupation number of the single level. The single particle energy ε is measured with respect to the zero bias Fermi energy, and it may be tuned by applying a voltage Vg to the gate electrode ε = ε(Vg ). Vibrational excitations with frequency ω0 are created b† and their ground state is defined as the zero-phonon state when n = 0. The frequency of the oscillator ω0 can range from the tens of MHz of a nanometrical cantilever [25] to a dozen of THz in the case of molecular devices or suspended nanotubes. [45, 59] The coupling betwen vibrational and electronic degrees of freedom is described by the term Hn,b . The dimensionless parameter λ represents the strength of the e–ph interaction. For example, λ ∼ 1 was reported for the C60 devices [29] and for suspended carbon nanotubes, [60] while values of λ between 0.4 and 3 have been found in different C140 samples. [45] We are interested in the case where the NEMS is coupled both to external electronic leads and to a dissipative environment. The total Hamiltonian then reads P H = Hs + Hleads + Ht + Henv + Hb,env . (5.5) Shot noise in NEMS 60 The Hamiltonian for the non interacting leads is X X Hleads = εi,k c†i,kσ ci,kσ (5.6) i=1,2 k,σ where c†i,kσ (c†i,kσ ) is the creation (annihilation) operator for an electron with momentum k and spin projection σ in lead i (i = 1, 2). For vanishing bias voltage V = 0, both leads assume the same Fermi energy, which we define as the zeropoint of our energy scale. At finite bias, the chemical potentials of the two leads are are shifted and their difference is fixed by µ1 − µ2 = eV . The exact voltage splitting betwen the left and right junction depends on the details of the junction capacitance’s [161]. In general, the voltage splitting may be described in terms of a capacitance dependent splitting parameter 0 ≤ η ≤ 1 such that µ2 = (1 − η)eV. µ1 = ηeV, (5.7) However, as the case of asymmetric voltage splitting η 6= 1/2 can always be compensated for by tuning the gate voltage, in the following we may restrict our discussion to the case of symmetric voltage drop µ1,2 = ±eV /2. The coupling between the NEMS and the leads is determined by the tunneling Hamiltonian XX Ht = ti (c†i,kσ dσ + d†σ ci,kσ ), (5.8) i=1,2 k,σ The strength of this coupling is parametrized by the tunneling matrix element ti . For simplicity, in the following we will assume symmetric barriers t1 = t2 ≡ t0 . The coupling between the oscillator and the environment is included as a linear coupling to a bath of harmonic oscillators in the spirit of the Caldeira and Legget theory [123] X (5.9) Henv = ωj (a†j aj + 1/2), j Hb,env = X χj ωj (a†j + aj )(b† + b). (5.10) j Here a†j are the creation operators of the bosonic bath modes and χj are the coupling constats for the oscillator-bath interaction. The environmental coupling is usefully characterized by its spectral function [162] X J (ω) = 2π ωj2 χ2j δ(ω − ωj ). (5.11) j Polaron transformation The coupling term Hn,b can be eliminated from Hs by means of a canonical transformation similar to the Lang-Firsov transformation [163, 164] (see Appendix 5.12 for details). This transformation corresponds to a basis change to polaron-type quasiparticles, i.e. electrons surrounded by clouds of vibrations. Importantly, because of the coupling term Hb,env , the transformation must include both the operators of the oscillator and those of the environment [116] X Ō = eAn Oe−An , A = κ(b† − b) − 2κ χj (a†j − aj ), (5.12) j where κ= 1−4 λ . 2 j χj ωj /ω0 P (5.13) 61 5.3 Rate Equation Applying this transformation to the total Hamiltonian Eq.(5.5) we obtain H̄ = H̄n + Hb + Hleads + Henv + H̄t + Hb,env where H̄t = X t0 (c†k,i e−A d + d† eA ck,i ). (5.14) (5.15) k,i=1,2 and H̄n = ε̄ n + Ū n↑ n↓ with ε̄ = ε − λκω0 , Ū = U − 2λκω0 . (5.16) Hence, the electron–phonon coupling term Hn,b is eliminated at the cost of introducing displacement operators in the tunneling matrix elements t0 → t0 e−A . Moreover, the single-particle energy and the charging energy are renormalized (polaron shift). The renormalization of the charging energy U is irrelevant since we assume this to be the largest energy in the problem, however, let us note that for weak Coulomb interaction the polaron shift can even result in a negative effective Ū [120, 165]. From now on, we will proceed with the transformed Hamiltonian. 5.3 Rate Equation There are four different energy scales relevant for the analysis of transport in the Anderson-Holstein model: the phonon energy ω0 , the bare tunneling rate Γ = 2πνt20 (with ν the density of states of the leads), the relaxation rate Γrel which characterizes the coupling to the dissipative environment and the temperature T . Ifthe phonon energy is large compared to the level broadening induced by tunneling Γ, then different vibrational states do not get mixed and the system completes many oscillation cycles between two tunneling events. On the other hand, for phonon energies small compared to the level width a strong mixing between vibrational states occurs and coherences between them must be taken into account. A scenario typical of this limit is the case of the quantum shuttling in nanoelectromechanical devices [159, 166]. Analogously, due to the presence of the environment, the discrete vibrational excitations become broadened by Γrel . The ratio Q = ω0 /Γrel , which is known as the quality factor, characterizes the energy loss of the vibrational mode per period (1/ω) due to damping. For large quality factors the broadening of the vibrational excitations can be neglected. Finally, the temperature determines the time scale τeq = 1/kB T over which excitations in the leads decay . As we have seen in Chapter 3, the sequential tunneling regime requires τeq to be the shortest energy scale into play. Therefore, in the following we will always assume Γ, Γrel ω0 kB T which is a typical experimental situation [29, 45, 59]. In this regime, the coupling Hamiltonian Hint = H̄t + Hb,env can be treated perturbatively and the calculation of transport characteristics may be carried out within the framework of the rate equation, as discussed in Chapter 3. The transition rates can be evaluated to the lowest order in Hint by means of Fermi’s golden rule (see appendix D) Γ|ii→|f i = 2π|hf |Hint |ii|2 δ(Ei − Ef ) = 2π|hf |H̄t |ii|2 δ(Ei − Ef ) + 2π|hf |Hb,env |ii|2 δ(Ei − Ef ). This leads to two different kind of contributions: the tunneling rates proportional to t20 , and relaxation rates, which depend on χ2j . Shot noise in NEMS 62 Tunneling rates. The tunneling Hamiltonian H̄t is responsible for the transfer of one electron from a lead into the electronic level or vice versa. Because of the e–ph coupling, the electron transfer is generally accompanied by excitation (or excitation) of molecular vibrations. Assuming the electrons in the leads are at equilibrium with their chemical potential, one obtains the following expressions Γi 0→σ = Γ Xl0 l f µ(l0 − l) − µi , (5.17) l→l0 σ→0 0 Γi l→l0 = Γ Xl0 l 1 − f µ(l − l ) − µi , (5.18) where f (x) is the Fermi function and µ(l 0 − l) = ε̄ + ω0 (l0 − l) is the addition energy of the NEMS, i.e. the energy required to add an electron into the system while changing the vibrational state l → l 0 . The coefficient Xll0 denotes the matrix element for the transition l → l 0 , and is given by Xll0 = |hl|e−λ(b † −b) 0 2 |l i| , (5.19) These terms are called Franck–Condon (FC) factors and they will discussed in detail later in this chapter. Importantly, even if the displacement operator in H̄T Eq. (5.15) includes also the momenta of the environment, the tunneling rates depends only on λ(b† − b). This is a consequence of treating the Hint to the lowest order in the coupling constant t0 , χi . Therefore, as the tunneling rates are already ∝ |t0 |2 , we have to be careful to disregard all terms which includes χi . Relaxation rates The relaxation rates represent transitions between vibrational excitations without change of the electronic state βω0 rel Γrel Γ(l−1)→l = l l→(l−1) = e J (ω0 ) , 1 − e−βω0 (5.20) where J (ω0 ) is the spectral density of the phonon bath Eq. (5.11), evaluated at the frequency of the oscillator. Note that treating Hb,env at second order allows only transitions between neighboring states (i.e. |l 0 − l| = 1). Transitions with |l 0 − l| ≥ 1 can be included considering different relaxation mechanisms [120, 156]. Given the tunneling and the relaxation rates, we can write down explicitly the rate equation for the occupation probability for the eigenstate of H̄s X X d 0→σ rel P0l = [Pσl0 Γ σ→0 [P0l0 Γrel l0 →l − P0l Γ l→l0 ] + l0 →l − P0 l Γl→l0 ] dt 0 0 σ,l l X X d rel σ→0 [Pσ l0 Γrel [P0 l0 Γ l0→σ Pσ l = 0 →l − Pσ l Γ l→l0 ] + l0 →l − Pσ l Γl→l0 ] dt 0 0 l (5.21) l 0 ν→ν = where Pνl (ν = 0, ±σ) is the occupation probability of the state |ν, li and Γ l→l 0 P ν→ν 0 is the total tunneling rate. Γ 0 i i l→l Since all the transition rates do not depend on the spin projection σ, we may introduce the total probability P1 l = P↑l + P↓l for having one electron into the level and sum Eq.(5.21) over σ. This result in spin averaged rate equations X X d 0→1 rel P0 l = [P1 l0 Γ l1→0 [P0 l0 Γrel 0 →l − P0 l dΓ l→l0 ] + l0 →l − P0 l Γl→l0 ] dt 0 0 l l X X d rel 0→1 1→0 [P1 l0 Γrel [P0 l0 dΓ l0 →l − P1 l Γ l→l0 ] + P1 l = l0 →l − P1 l Γl→l0 ] dt 0 0 l (5.22) l 1→0 σ→0 0→1 where we have called Γi 0→σ l→l0 ≡ Γi l→l0 and Γi l→l0 ≡ Γi l→l0 and where the factor d takes in account the spin degeneracy. We have introduced the notation d in order 63 5.3 Rate Equation to have the possibility to disregard the spin degeneracy by simply inserting d = 1. From Eq.(5.22) is evident that, in the case of strong electronic interaction, a spindegenerate electronic level is formally equivalent to a spinless single level, apart for an effective doubling of the tunneling-in rates [167]. For sake of simplicity, in the following we will consider the spinless case d = 1, however our results can be readily generalized to the case of a spin-degenerate level. In the remaining of this section we will shortly discuss two of the main features of the rate equation Eq.(5.22). One is the dependence of the Franck-Condon factors Xll0 on the strength of the e–ph coupling λ, the other is the harmonic spectrum of the oscillator, which allows for vibration assisted cascades. Cascades and accumulation of vibrational energy Due to the harmonic spectrum, the addition energy for the transition |0, li ↔ |1, l 0 i depends only on the change of the difference ∆l = l 0 − l and not on the individual vibrational numbers l, l 0 . This means that everytime that the addition energy µ(∆l) comes into the bias window, all the transitions |0, li ↔ |1, l + ∆li for all l and fixed ∆l become possible. This is an intrinsic property of the harmonic spectrum and gives rise to vibrational assisted cascades. As an example, imagine that the addition energies µ(0) and µ(1) lie within the bias window. An electron tunneling onto the NEMS can therefore excite the vibrational mode by one quanta by the transition |0, 0i → |1, 1i. When it tunnels off, it can leave the system in the excited state by the transition |1, 1i → |0, 1i. If we assume that the vibrational energy does not relax between tunneling events, this excitation is conserved until the next tunneling event which may even even increase the excitation by the transition |0, 1i → |1, 2i. The system can then reach a highly excited state via subsequent tunneling events |0, 0i → |1, 1i → |0, 1i → |1, 2i → |0, 2i → |1, 3i → |0, 3i → |1, 4i → . . . The amplitude of this cascade depends on the details of the FC factors and on all other possible transitions. Franck–Condon factors Even if the addition energy depends only on the difference ∆l, the tunneling rates acquire a strong dependence on the vibrational numbers l, l 0 because of the Franck– Condon factors Eq.(5.19). The properties of the FC factors are best elucidated by † expressing the matrix element hl 0 |e−λ(b −b) |li in coordinate representation as the overlap of two displaced harmonic oscillator wave functions φl and φl0 X ll0 Z = +∞ −∞ dx φ∗l (x 2 √ 2 2 0 l< ! |l−l0 | 0 + 2λ x0 )φl (x) = e−λ λ2|l−l | Ll< (λ2 ) l> ! (5.23) p where x0 = 1/M ω0 is the the zero-point displacement of the oscillator and l< = min{l, l0 }, l> = max{l, l0} and Lnl (x) is a generalized Laguerre polynomial. The coupling parameter λ however determines the shift between the two equilibrium positions of the oscillator. However, from Eq.(5.23) we see that the relevant scale is not λ but λ2 as one could have expected observing that λ2 is the elastic energy in units ω. 1 1 In other words, λ2 ω is the elastic energy gained by the system when it is charged because of the deformation. This can be easily seen imagining a classical spring which is elongated by an √ amount 2λ. Then the potential energy that the spring gains is exactly λ2 ω. Shot noise in NEMS 64 The FC factors are symmetric under the exchange of indices Xll0 = Xl0 l and obey the following sum rule X X Xll0 = Xll0 = 1 (5.24) l l0 which sets an upper bound for the current at large bias. A consequence is the inequality Xll0 ≤ 1 for all l, l0 . The behavior of the FC factors depends crucially on the strength of the e–ph coupling. Roughly speaking, one can distinguish three regimes: (i) λ2 1. Transitions which conserve the vibrational number have a dominant amplitude up to high excitations l, l 0 . For λ → 0 the FC factors are given by Xll0 = δll0 due to the or thonormality of the harmonic oscillator eigenfunctions. (ii) λ2 ∼ 1. Transitions slightly changing the vibrational number are favored but the global maximum of the FC factors still lies at small vibrational numbers. For λ2 = 1 it is X01 = X00 = 1/e, while for λ2 < 1 it is X01 < X00 and for λ2 > 1 it is X01 > X00 . (iii) λ2 1. Transitions which change l considerably are strongly favored and the maximum FC factors lie at l = 0, l 0 = λ2 . For small l, l0 a gap opens in which the FC factors increase exponentially with increasing l + l 0 . To summarize, we can say that the effect of the e–ph interaction on transport is two fold: on one hand it suppresses the effective tunneling rate (Xll0 ≤ 1, ∀l, l0 ) and on the other, it induces a highly non trivial dependence on the phononic indices l, l 0 . 5.4 Results As discussed in Chapter 3, within the rate equation approach the current and the current noise can be evaluated by means of standard techniques [94]. During numerical calculations, we introduce a cut-off lmax 1 for the vibrational number. In this case the occupation probability vector P is a vector of length 2lmax and the transition matrix M is a 2lmax × 2lmax matrix (see Chapter 3 for the definitions of P and M). The phonon cut-off has to be chosen quite carefully because, due to the harmonic spectrum, for weak relaxation the possibility of vibrational cascades exists, i.e. in principle there is an infinite number of molecular states. To choose lmax , we check the convergence of the probabilities. Full solution of the rate equation The dynamics of the system is characterized by two competing time scales: the average time spent by an electron in the dot τel and the phonon relaxation time τph . If τel τph , the vibrational excitations tend to relax between each tunneling event (eq) to the thermal Bose distribution Pl = e−βlω0 (1 − e−βω0 ). In this limit, charge (eq) and vibrational degrees of freedom decouple Pnl = Pn Pl and the dynamics of the system reduces to an effective two–state sequential tunneling process. [116] The analytic expressions for current and noise are well known [72] and, for kB T eV , are respectively given by I (eq) = e Γ̃1 Γ̃2 , Γ̃1 + Γ̃2 F (eq) = Γ̃21 + Γ̃22 . (Γ̃1 + Γ̃2 )2 (5.25) 65 5.4 Results λ=1 λ = 16 Xll0 λ = 0.04 l0 l0 l l0 l l Figure 5.2: Franck–Condon factors for weak (λ2 = 0.04), intermediate (λ2 = 1) and strong (λ2 = 16) e–ph coupling. Panels in the top and middle row show the FC factors as a function of the initial and final phonon state l and l 0 respectively. The graphs on the bottom visualize the displacement of the harmonic potential surfaces for the neutral and the charged system. P P Here Γ̃1 = Γ(0) l al f1 (lω0 ) and Γ̃2 = Γ(0) l al [1 − f2 (−lω0 )] are the renormalized rates for tunneling in and out of the dot and the coefficients al are Poissonian weight 2 factors 2 al = θ(l)e−λ λ2l /l!. In this case the smallest possible value of the Fano factor is F (eq) = 1/2. Vice versa, if τel τph the tunneling electrons drive the vibrations out of equilibrium and peculiar features such as negative differential conductance (NDC) [156, 158, 168] and super–Poissonian shot–noise [120] have been predicted. 2 −1 −1 In our model, a rough estimate of τel is given by τel = Γe−λ , i.e. by the effec−1 tive transparency of the barrier set by the e–ph coupling, while τph is determined −1 by environment spectral density τph = J (ω0 ). It is useful to define a dimensionless parameter for the relaxation strength w = J (ω0 )/Γ(0) . (5.26) In terms of w, the condition for equilibrated phonons τel τph reads w exp(−λ2 ). It is then evident that the e–ph coupling defines a characteristic scale for relaxation: the stronger is the coupling, the more sensitive is the system to phonon relaxation. (st) (st) (st) This is reflected by the stationary phonon distribution Pl = P0 l + P1 l . For (st) (eq) increasing relaxation strength w, Pl tends monotonically to Pl but with λ2 – depending speed (see Fig.5.3). For strong e–ph coupling, the phonon distribution is narrow already in the non–relaxed case [119] w = 0 and it reaches equilibrium for values of w which are sensibly smaller than for weak λ2 . 2 The general expression ` for the factors is: al = ´weight exp[−λ2 coth(βω0 /2)] exp[lβω0 /2]Il λ2 / sinh(βω0 /2) , where Il is the modified Bessel function of first kind. Shot noise in NEMS 66 1 w=0 w=1 w = 10 w=∞ w=0 w=1 w = 10 w=∞ (st) Pl 0 1 (st) Pl 0 0 6 0 l 6 0 l l 6 0 l 6 (st) Figure 5.3: Stationary phonon probability distribution Pl for different values of w. Upper panels: λ2 = 0.4, lower: λ2 = 7. The rightmost panels (w = ∞) represent the (eq) thermal Bose distribution Pl = e−βlω0 (1−e−βω0 ). Other parameters: eV = 3 ω0 , ε̄ = 0 and kB T = 0.02 ω0. w w w w w 100 F =0 = 0.1 =1 = 10 =∞ 10 1 0.5 0 2 4 6 8 10 12 14 eV /ω0 Figure 5.4: Fano factor as a function of voltage for λ2 = 16 and for different values of the relaxation strength w. The dotted line w = ∞ corresponds to case of thermally distributed phonons. Other parameters: ε̄ = 0 and kB T = 0.02 ω0. 67 5.4 Results (st) (eq) Since Pl converges monotonically to Pl for growing w, one expects most of the features of the non equilibrated case to be washed out by increasing relaxation. This is particularly evident in the case of the giant Fano factor observed at low voltages for strong interaction (λ2 1), which is strongly suppressed even by weak relaxation (see Fig. 5.4). This behavior can be easily understood observing that F 1 depends dramatically on the non equilibrium distribution of the vibrational excitations induced by tunneling. [120, 121] In fact, for large λ2 transitions between low lying phonon states are exponentially suppressed (see Eq. (5.23)). Therefore, the main contribution to the current comes from high excited vibrational states (states with large l) but at low voltages the occupation probability of those states is strongly suppressed. [119, 158] These conditions leads to avalanches of tunneling processes which, in turn, are responsible for the huge values of F . [120, 121] Direct phonon relaxation inhibits this mechanism reducing even further the occupation of states with large l and, consequently, both the current and the Fano factor are strongly suppressed. For very strong relaxation (w → ∞), F → 1/2 as one would expect for equilibrated phonon on resonance (ε̄ = 0). Similarly, relaxation has a destructive effect on NDC (not shown) as this is also a consequence of the peculiarity of the nonequilibrium phonon distribution induced by tunneling itself. [158, 168] One could be tempted to conclude that considering explicitly the effects of relaxation simply results in an “interpolating” behavior between the opposite limits of no relaxation and thermally distributed phonons. However, we find that finite relaxation rate can induce unexpected features. Let’s first consider the case of moderate coupling λ2 = 3. In Fig. 5.5 we plot the Fano factor as a function of voltage for different values of w and for ε̄ = 0. It 0.7 w w w w w 0.6 =0 =5 = 12 = 100 =∞ F 0.5 0.4 0 2 4 6 8 10 12 14 16 18 20 eV /ω0 Figure 5.5: Fano factor as a function of voltage at λ2 = 3 and for different values of the relaxation strength w. The dotted line w = ∞ corresponds to case of thermally distributed phonons. Other parameters: ε̄ = 0 and kB T = 0.02 ω0. appears that F has a non systematic dependence on w: it can be either enhanced or suppressed by relaxation depending on the considered voltage range. For eV < 6ω 0 it is always F ≥ 1/2. In particular, for eV < 2ω0 it is F = 1/2 as the tunneling electrons cannot excite vibrations and the system behaves as an ordinary single level. More interestingly, for eV > 6ω0 , relaxation can suppress F even below 1/2. In this voltage range, then, relaxation of the mechanical mode induces a tendency to ordered transfer of electrons through the SET. It is worthwhile to stress that such a suppression of current fluctuations is not merely an effect of the coupling of the SET to the oscillator, but it is a consequence of Shot noise in NEMS 68 a non trivial interplay between vibration–assisted tunneling and phonon relaxation which induces correlation between different current pulses via emission-absorption of phonons. 20 20 0.6 λ2 = 0.4 w 0.6 λ2 = 1 w ← ← 0.48 0 0 eV /ω0 15 20 λ2 = 3 0.46 0 0 eV /ω0 0.86 20 w 15 λ2 = 7 1.5 w ← 0.43 0 0 eV /ω0 20 0.41← 0 0 eV /ω0 20 Figure 5.6: Density plot of the Fano factor as a function of bias V and relaxation w for different values of λ. In all the panels: dark gray F < 1/2, medium gray F = 1/2 (indicated by the arrow in the color map ) and light gray F > 1/2. The white line, corresponding to F = 1/2, represents wt (V ). The black line in the 4th panel delimits the region where noise is superpoissonian, F > 1. Other parameters: ¯ = 0 and kB T = 0.02 ω0. This peculiar behavior is observable in a wide range of parameter. Infact, from a numerical analysis, it emerges that F < 1/2 can be found for any value of the e–ph coupling. In particular, we observed that it exists a voltage threshold set by the e–ph interaction eVt (λ) = 2ω0 int[λ2 ] (5.27) such that, for V > Vt (λ), relaxation larger than a certain threshold value wt (V, λ) suppresses the Fano factor below 1/2. This is shown in Fig. 5.6, which represents a grayscale plot of the Fano factor in the (V, w)-plane, for different values of λ2 . The white contour line corresponds to F = 1/2 and separates two different regions in the (V, w)-plane: the one to the right of the contour, where F < 1/2 and the other one where F > 1/2. In other words, the white line denotes wt as a function of V at given λ2 . The threshold voltage Vt (λ) corresponds to the position of the vertical asymptote of wt (V, λ). For λ2 < 2 the critical voltage coincides with the onset of vibration assisted tunneling eVt (λ) = 2ω0 ; vice versa for strong e–ph coupling (λ2 1), Vt (λ) becomes very large and this is why F is always higher than 1/2 in Fig. 5.4. The minimal value assumed by the Fano factor Fmin depends itself on the e–ph coupling (see Fig. 5.7). For weak coupling, Fmin differs only slightly from 1/2. For 69 5.4 Results stronger coupling (λ2 > 1) it decreases logarithmically and it only reaches the value Fmin ∼ 0.4 for considerably strong interactions. Note that each point in Fig. 5.7 corresponds to different values of voltage and relaxation strength, as the position of Fmin in the (V, w)-plane depends on λ2 . The inset shows the voltage Vmin where the minimum is found. 0.52 0.5 0.48 45 0.46 Fmin 0.44 Vmin 0.42 0.40 0 0 14 λ2 0.38 10−3 10−2 10−1 1 10 λ2 Figure 5.7: Main panel: Fmin as a function of λ2 . Each point corresponds to different values of w and V . Inset: voltage Vmin where the minimum is found as a function of λ2 . Finally, let’s observe that for λ2 > 2, the threshold voltage Vt (λ) corresponds to the onset of the transition l : 0 → int[λ2 ]. In this case Vt (λ) is a characteristic voltage also for the current which can be either suppressed or enhanced by relaxation depending on V being smaller or larger than Vt (λ) (see Fig. 5.8). Relaxation contributes to populate the low lying phonon states and then, at low voltages, it inhibits the current as the transitions between those states have exponentially suppressed rates. However, for V > Vt (λ) the transition l : 0 → int[λ2 ] is allowed and, as it correspond the greatest Franck–Condon factor, [158] it gives a substantial contribution to the current. In this case relaxation has the opposite effect and it sustains the current “feeding” the population of the vibrational ground state. For V ∼ Vt (λ), these two mechanisms coexist and, consequently, the current depends only weakly on relaxation (see inset in Fig. 5.8). This observation fits nicely what is reported in literature. [119, 120] In fact, for λ2 < 2 the critical voltage is smaller than the energy required to have phonon– assisted tunneling and then the current is enhanced by phonon relaxation at any voltage, consistently to what was observed in Ref. [119] Vice versa for very strong e–ph coupling the enhancement of the current due to relaxation can be hardly seen [120] as Vt (λ) shifts to very large voltages. All the previous results are not qualitatively modified by considering a spindegenerate electronic level. As we have discussed above,in the case of strong electronic interaction, a spin-degenerate electronic level is formally equivalent to a spinless single level, apart from an effective doubling of the tunneling-in rates. [167] In particular, at low temperature the analytic expressions for current and noise in the limit of strong relaxation, when the system behaves as a single spin de(eq) (eq) generate electronic level, become: [144] Isd = e 2Γ̃1 Γ̃2 /(2Γ̃1 + Γ̃2 ) and Fsd = (eq) (4Γ̃21 + Γ̃22 )/(2Γ̃1 + Γ̃2 )2 The lowest value for Fsd , in this case is not 1/2 but 5/9. Performing numerical analysis we have found in the case of a spin degenerate electronic level qualitatively the same behavior observed in the spinless case, i.e. the Fano factor shows a non monotonous dependence on the strength of the relaxation rate. Infact, without relaxation the coupling to an harmonic oscillator generally Shot noise in NEMS 70 0.4 w w w w w 0.3 I =0 =1 = 10 = 100 =∞ 0.2 0.22 0.1 0.21 4 eV /ω0 6 0 0 2 4 6 8 10 eV /ω0 Figure 5.8: Current as a function of voltage for λ2 = 3 and for different values of the relaxation strength w.The dotted line w = ∞ corresponds to case of thermally distributed phonons. Other parameters ε̄ = 0 and kB T = 0.02ω0 . Inset: zoom of the plateau around eV = 5ω0 . Current in units eΓ. leads to an increase of the current noise with respect to the uncoupled electronic level. However, relaxation of the phononic mode can introduce negative correlation between different current pulses via emission-absorption of phonons and lead to a suppression of the Fano factor below 5/9, which is the minimal expected value without e–ph interaction. Moreover, as for the spinless-case, we have found that such a suppression is observable only if the source drain voltage V > Vt (λ) - see Eq. (5.27) -. For certain choices of the parameters we also registered F < 1/2, but for the spin-degenerate case such a value does not represent any relevant limit. Toy model To get some insight in these results, we focus on the low voltage region eV < 4ω 0 and we consider a toy model with only four accessible state, i.e. n = 0, 1 and l = 0, 1. In this case analytical expressions for current and noise can be derived. For sake of simplicity, we report only the solutions on resonance (ε̄ = 0) and at zero temperature I = eΓ(0) hX 00 2 + θ(eV − 2ω0 ) X01 (w + 2X01 − ∆) i , 2(w + 2X01 ) (5.28) and F = 1 θ(eV − 2ω0 )X01 ∆[w2 + w(2X01 − ∆) − X01 ∆] − , 2 (w + 2X01 )2 K (5.29) where ∆ = X00 − X11 and K = [w(X01 + X11 + ∆) + X01 (2X01 + 2X11 + ∆)]. From Eq. (5.28) it is easy to show that the current is an increasing function of w only for ∆ > 0 (that is, for λ2 < 2, see Eq. (5.23)). Vice versa, for ∆ < 0 (λ2 > 2) the current decreases for increasing relaxation, in agreement with what was previously observed. Moreover Eq. (5.29) tells that ∆ > 0 (λ2 < 2) is the necessary condition to have F < 1/2 in the region 2ω0 < eV < 4ω0 . In fact only in this case, it exist a threshold value for relaxation 2wt (λ) = ∆ − 2X01 + q 2 , ∆2 + 4X01 (5.30) 71 5.4 Results such that for w > wt the Fano factor is smaller than 1/2. For stronger e–ph coupling λ2 > 2 (∆ < 0), it is always F > 1/2. This confirms the numerical estimate eVt (λ) = 2ω0 as the threshold voltage for any λ2 < 2. Despite the coarseness of the model, Eq. (5.29) accords qualitatively with the exact numerical solution for eV < 4ω0 (see Fig. 5.9). The agreement is reasonably good even in the case of weak e–ph coupling, where the phonon distribution is mostly broadened [119] and one expects the four state approximation to be more inaccurate. A better agreement can be obtained considering a six state model with n = 0, 1 and l = 0, 1, 2 but, in this case, the analytic solutions become quite cumbersome and we don’t report them here for simplicity. The agreement of the four states model with numerical result suggests that F < 1/2 rather depends on the interplay between relaxation and vibration–assisted tunneling, than on the possibility to access an high number of vibrational states. 0.68 0.6 0.62 F F F λ2 = 3 λ2 = 1 λ2 = 0.4 0.5 0.5 0.5 0.44 0.46 0 w 50 0 w 50 0 w 50 Figure 5.9: Fano factor as a function of relaxation strength w for eV = 3ω 0 and for different values of the e–ph interaction. Red (solid) line: exact numerical result; blue (dash-dotted) line: result for the four states model - Eq. (5.29); green (dashed) curve: result for a six states model with n = {0, 1} and l = {0, 1, 2}. Dotted line: F = 1/2. Other parameters ¯ = 0 and kB T = 0.02ω0 . Conclusions In this thesis we have analyzed the effects of two different kind of interactions, namely the electron-electron interaction and the electron-phonon interaction, on the shot noise of a single electron device. While studying the influence of the electronic interaction, we have considered a one-dimensional quantum ring connected to leads by tunneling barriers, as this peculiar geometry allows to analyze the interplay between Luttinger liquid correlations, Coulomb blockade and Aharanov-Bohm interference. Subsequently, we have focused our attention on the interaction between a single electronic level and a local phonon mode. This peculiar kind of coupling is typical of molecular transistors and, more in general, of a number of nanoelectromechanical systems operating in the Coulomb blockade regime. Our main results are summarized in the following. Quantum ring. We have investigated the influence of the electronic interaction on the transport properties of a one dimensional quantum ring. In addition to the charge tunnel-current, we have considered the angular tunnel-current related to variations of the persistent current into the ring induced by tunneling. The shot noise of the charge and angular currents show distinctly different properties. In the presence asymmetric tunnel barriers, the interactions can drive the shot noise of the charge current to super-Poissonian values. In this case, a Fano factor larger than 1 indicates bunching and super-Poissonian statistics. Viceversa the angular current noise is insensitive to the asymmetry of the barriers and to bunching effects, while it is closely related to the occupation dynamics of states with large angular momentum. The Fano factor of the angular current can be larger than one only in the presence of attractive interaction in the leads, when an inversion of the populating ratio is achieved. In order to test these effects experimentally, one needs to separate from the total noise the magnetization current contributions. This might be done by measuring the magnetization of a single-wall nanotube ring or semiconductor ring attached to the leads. Nanoelectromechanical system. While in the case of the quantum ring we have assumed an equilibrium distribution of the plasmonic excitations, in the case of a nanoelectromechanical system we have concentrate on non-equilibrium effects and, in particular, on how relaxation of the phononic excitations influence the current shot noise. We have found that while the occupation probability distribution of the states of the system evolves monotonically towards thermal equilibrium for increasing relaxation strength, the Fano factor exhibits an highly non–monotonous behavior. Moreover, for finite relaxation rates it can be even suppressed below 1/2, which is the minimum expected value in the case of fully relaxed excitations. This relaxation–induced tendency to order of the electronic transfer through the dot is unexpected, since we are dealing with an oscillator with incoherent dynamics coupled to a quantum dot in the sequential tunneling regime. We have analyzed in detail the onset of this behavior as a function of relaxation, electron-phonon interactions, and external voltages. We have found that for any value of the electron72 73 Conclusions phonon coupling, it exists a critical value such that for voltages larger than this threshold a suppression of the Fano factor below 1/2 is possible. At low voltages, these results are qualitatively predicted by a four states toy model. Appendix A Classical suppression of shot noise The shot noise of a single barrier can be understood from simple classical arguments treating transmission through the barrier as a stochastic process with probability T of success. Consider a stream of particles incident on a barrier. If the particles behave independently, the probability that nT charges are transmitted through the barrier given that there were n incoming ones, is expressed by a binomial distribution: n P (nT |n) = (A.1) T nT (1 − T )n−nT . nT The average number of transmitted particles hnT i is simply given by hnT i = = ∞ X n X n=0 nT =0 ∞ X nT P (nT |n)Pτ (n) Pτ (n) n=0 n X nT =0 nT n T nT (1 − T )n−nT = hniτ T, nT (A.2) (A.3) where Pτ (n)is the probability P of having exactly n incident particles in the measurement time τ and hniτ = n nPτ (n). Following the same steps, one can evaluate the variance h(∆nT )2 i obtaining: h(∆nT )2 i = hniτ T (1 − T ). (A.4) Knowing hnT i and h(∆nT )2 i it is immediate to turn to current and current fluctuations. Infact the mean current flowing through the barrier is hIi = q hnT i = qνT , τ (A.5) where ν = hniτ /τ is the mean rate of incidence. Moreover, in the limit of an infinitely long time of measurement the zero frequency shot-noise is simply related to the variance of the number of transmitted particles: 2q 2 2q 2 h(∆nT )2 i = hniτ T (1 − T ). τ →∞ τ τ S = lim (A.6) Expressing S in term of the average current Eq.(A.5) S = 2qhIi(1 − T ), 74 (A.7) 75 it emerges clearly that the shot-noise of a tunnel barrier is always smaller than to the Poisson limit SP oisson = 2qhIi. In particular, the shot-noise vanish in the limit of a truly ballistic system (T = 1). The shot-noise measure the extra randomness introduced into the flow of particles by the transmission process; if T = 1 there is no randomness and therefore there is no shot noise. In the opposite limit of very small transparency T , the binomial distribution Eq.(A.1) can be approximated by the Poisson distribution and the Schottky formula S = 2qhIi is recovered again. Finally, we remark that even if the Schottky problem [4] (i.e. electron motion in vacuum tubes) is often referred as ballistic, it is in fact a problem in which carriers have been emitted by a source into the vacuum by tunneling through a barrier with very small transparency. This simple derivation fails for small biases eV < kB T as we have not included thermal fluctuations and we assumed that all the noise is due to current flowing through the sample. Consequently, Eq.(A.7) has to be thought of as the limit of the shot-noise for very large voltages (eV kB T ), where thermal fluctuations are not important. Appendix B The current correlation function In this appendix we review Korotkov’s technique [94] for computing the current shot noise in the rate eqution formalism. As discussed in Chapter 3, in the sequential tunneling regime the dynamics of the dot can be described in terms of a distribution function Pα (t) hat gives the probability of finding the island at time t in a particular state |αi. The time evolution of Pα (t) forms a Markov process and the corresponding rate equation is given in Eq.(3.5). The conditional probability to find the system in the state |αi at time t given that it was in |βi at t = 0 is given by P (α, t|β, 0) = eMt αβ (B.1) where M is the “evolution” matrix defined in Eq.(3.7). Given P (α, t|β, 0), we can easily evaluate the correlation function of any observable relative to the dot X (i.e. of any observable which is only a function of the state of the dot Xα ≡ X(|αi) ). As an example, if we consider the number N of electrons into the dot we have X hN (t)N (0)i = Nα Nβ P (α, t|β, 0)Pβ0 , t > 0 (B.2) α,β and the solution for t < 0 is determined by the symmetry hN (t)N (0)i = hN (0)N (−t)i. However, an expression simlar to Eq.(B.2) for h∆I (i)(t) I (j)(0) i is not valid as the current is not a function of the state of the dot. Hence the current correlator has to be calculated anew, taking into account that in the current in the sequential tunneling regime is generated by discrete charge transfer process. The current across each junction may then be written as a sum of δ-like contributions X X (i) (i,α,β) I (i) (t) = e ςαβ δ(t − tk ), (B.3) k α,β (i,α,β) where tk (k = 1, 2, . . . ) represent the times at which the transition |αi → |βi (i) through the i-th junction take place and the quantity ςαβ ≡ (−1)i (Nα − Nβ ) stands for the number of electrons transfered through the barrier in that particular transition. Its sign reflects whether the transfer occurs in the positive or negative direction. As in the sequential tunneling regime transitions are possible only between (i) states with neighbouring charges, it is ςαβ ∈ {−1, 1}. If we take the average of this expression over the stationary state and we compare with Eq.(3.14), we obtain the relation X (i,α,β) (i) hδ(t − tk )i = Γα→β Pα0 . (B.4) k 76 77 Let us now turn to the current-current correlator. When substituing Eq.(B.3) into the correlator hI (i) (t0 )I (j) (t)i, we obtain a multiple sum where each summand is proportional to the average of the product of two δ-functions X X X (i) (j) (i,α,β) ςαβ ςγδ hδ(t0 − tk )δ(t − tq(j,γ,δ) )i. (B.5) hI (i) (t0 )I (j) (t)i = e2 k,q α,β γ,δ (i,α,β) As the set of all times tk forms the time lattice of the Markov process, the times (i,α,β) (j,γ,δ) are considered to be pairwise different, i.e. tk 6= tq whenever {k, i, α, β} 6= {q, j, γ, δ}. In other words, as in the sequential tunneling regime the probability of simultaneous tunneling events is equal to zero, different tunneling events occur at different times. The correlator can then be separated into auto-correlation and cross-correlation contributions hI (i) (t0 )I (j) (t)i = hI (i) (t0 )I (j) (t)ia + hI (i) (t0 )I (j) (t)ix (B.6) where XX (i) = e2 δ i,j hI (i) (t0 )I (j) (t)ix = “terms with tk k α,β (i) (i,α,β) ςαβ ςαβ hδ(t0 − tk hI (i) (t0 )I (j) (t)ia (i,α,β) (i,α,β) )δ(t − tk 6= tq(j,γ,δ) ” )i, (B.7) (B.8) We begin considering the self-correlation of a given tunnel event with itself. Using (i,α,β) (i,α,β) (i,α,β) the fact that hδ(t0 − tk )δ(t − tk )i = δ(t0 − t)hδ(t − tk )i and applying Eq.(B.4), we obtain X (i) hI (i) (t0 )I (j) (t)ia = e2 δi,j δ(t0 − t) Γα→β Pα0 . (B.9) α,β (i) since (ςαβ )2 = 1. attenzione!!!! Dire che stiamo considerando il regime stazionario e che quindi Pα (t) → Pα0 The cross-correlation terms now involve only distinct processes at different times. These contributions can be treated in the following way. Let us denote with dQ(i) (t) = I (i) (t)dt the charge passed through the i-th junction in the interval (t, t + dt). The cross-correlation term can then be written as hI (i) (t)I (j) (0)ix = hQ(i) (t)Q(j) (0)i dt0 dt (B.10) The product dQ(i) (t)dQ(j) (0) is non vanishing only if we have a tunneling event in junction i during (t, t + dt) and in junction j during (0, dt0 ). The average value hdQ(i) (t)dQ(j) (0)i is then given by X X (i) (j) (i) (j) hdQ(i) (t)dQ(j) (0)i = e2 ςαβ ςγδ P (ςαβ , t; ςγδ , 0), (B.11) α,β γ,δ (i) (j) where P (ςαβ , t; ςγδ , 0) is the joint probability for having the transition |αi → |βi at the i-th junction at time t0 and the transition |γi → |δi at the j-th junction at the initial time. As transport is a stationary Markov process, it is given by (i) (j) (i) (j) P (ςαβ , t; ςγδ , 0) = Γα→β dtP (α, t0 − t|β, 0)Γγ→δ dt0 Pγ0 , (B.12) where we have assumed t > 0. This leads to X X (i) (j) (i) (j) hI (i) (t)I (j) (0)ix = e2 ςαβ ςγδ Γα→β P (α, t|β, 0)Γγ→δ Pγ0 , (B.13) α,β γ,δ The current correlation function 78 for t > 0. The expression for t < 0 can be readily obtained by exploiting the symmetry hI (i) (t)I (j) (0)i = hI (i) (0)I (j) (−t)i. (B.14) Let us emphasize that in Eq.(B.13), the initial state for the evolution (during time t) is not γ but δ. One can say that the current at t = 0 behaves as an operator which change the state of the dot. We can write the the previous results in a compact form by adopting the notations of Chaper 3 (i) (i) (i) P (α, τ |β, 0) = eMτ α,β Iαβ = ςαβ Γα→β , (B.15) With this definitions the current-current correlator becomes hI (i) (t)I (j) (0)i = δi,j δ(t)Tr[ |I (i) | P0 ] (i) Mt (j) +θ(t)Tr[ I e I P0 ] + θ(−t)Tr[ I (B.16) (j) −Mt (i) e I P0 ] The correlation function for current fluctuations h∆I (i) (t)∆I (j) (0)i can be simply obtained by replacing I (i) with δI (i) ≡ I (i) − hI (i) i/e in the previous expression. Appendix C The polaron transformation In this appendix we give a detailed derivation of the canonical transformation Eq.(5.12), which we introduced to eliminate the e–ph coupling Hn,b . Without coupling to the environment Hb,env , the problem is formally analogous to the independent boson model [164], and the system Hamilonian Eq.(5.2) is diagonalized by the Lang-Firsov transformation H̄ = eS He−S (C.1) with S = λ(b† − b)n. In the presence of Hb,env this transformation have to be generalized to include also the momentum operators of the environment. We consider the following form for the generator of the transformation X κj (a†j − aj ), (C.2) S = An, A = κ(b† − b) + j where κ, κj ∈ R. With this definition, we calculate the transformation of the relevant annihilation operators applying the Baker-Campbell-Hausdorff formula eA Be−A = ∞ X 1 1 1 [A, B]m ≡ B +[A, B]+ [A, [A, B]]+ [A, [A[A, B]]]+. . . (C.3) m! 2 3! m=0 In this way we obtain ∞ X Am [n, dσ ]m = dσ e−A m! m=0 d¯σ = eAn dσ e−An = b̄ = eAn b e−An = āj = eAn aj e−An = c̄i,kσ = eAn ci,kσ e−An = ci,kσ ∞ X κm n m † [(b − b), b]m = b − κ n m! m=0 ∞ m X κm j n [(a†j − aj ), aj ]m = aj − κj n m! m=0 (C.4) (C.5) (C.6) (C.7) where we have used [n, dσ ]m = (−1)m dσ and [b† − b, b] = −1. The corresponding creation operators are imediately obtained by taking the Hermitian conjugate d¯†σ = d†σ eA , b̄† = b† − κ n, ā†j = a†j − κj n. (C.8) Inserting these expressions into the transformed Hamiltonian H̄ = eAn Ha−An = H̄s + H̄leads + H̄t + H̄env + H̄b,env 79 (C.9) The polaron transformation 80 one obtains H̄s H̄leads H̄t = Hd + Hb + ω0 (λ − κ)(b† + b)n + ω0 κ2 n2 = Hleads X = t0 (c†k,i e−A d + d† eA ck,i ) (C.10) (C.11) (C.12) k,i=1,2 H̄env H̄b,env = Henv − X = Hb,env − j ωj [κj (a†j + aj )n − κ2j n2 ] X j (C.13) χj ωj [2κj (b† + b)n + 2κ(a†j + aj )n − 4κκj n2 ] (C.14) Is now a matter of simple algebra to show that all the e–ph terms are eliminated by choosing λ P and κj = −2χj κ (C.15) κ= 1 − 4 j χ2j ωj /ω0 This way, one comes to the final result H̄ = H̄d + Hb + Hleads + H̄t + Henv + Hb,env (C.16) where we have included all the terms proportional to n2 = n + 2n↑ n↓ into H̄d = ε̄n + Ūn↑ n↓ ,renormalizing the the single-particle and of the charging energy, i.e. ε → ε̄ = ε − λκω0 and U → Ū = U − 2λκω0 . This is exactly Eq(5.14). Appendix D The transition rates In this appendix we evaluate the transition rates induced by Hint = H̄t + Hb,env to the lowest order in the coupling constant ti , χi . For convenience, we report here the Hamiltonian of the NEMS in the canonically transformed base H̄ = H̄s + Hleads + Henv + H̄t + Hb,env (D.1) with H̄s = Hleads = H̄d + Hb = ε̄n + Ū n↑ n↓ + ω0 (b† b + 1/2) X X εi,k c†i,kσ ci,kσ (D.2) (D.3) i=1,2 k,σ Henv X = ωj (a†j aj + 1/2), (D.4) j Ht XX = ti (c†i,kσ e−A dσ + d†σ eA ci,kσ ) (D.5) i=1,2 k,σ Hb,env X = χj ωj (a†j + aj )(b† + b). (D.6) j We treat the interaction term Hint = H̄t + Hb,env as a perturbation to H0 = H̄s + Hleads + Henv . The transition betwen two eigenstates of H0 can be evaluated to the lowest order in Hint by means of Fermi’s Golden rule Γ|ii→|f i = 2π|hf |Hint |ii|2 δ(Ei − Ef ). (D.7) In the absence of coupling, the eigenstates of H0 are simply given by the tensor product of the eigenstates of Hs , Hleads and Henv . We denote the eigenstates of H̄s as |ν, li, with ν ∈ {0, σ, 2} and l ∈ {0, 1, 2, . . . }, and we call Eν,l corresponding eigenenergies. Analogously, we indicate with |Li, |Ri and EL , ER the eigenstates and the eigenenrgies of Hleads and Henv . For definiteness, let us call |ii = |ν, li|Li|Ri and |f i = |ν 0 , l0 i|L0 i|R0 i. Writing down |hf |Hint |ii|2 explicitely |hf |Hint |ii|2 = |hf |H̄t |ii|2 + |hf |Hb,env |ii|2 + hi|H̄t |f ihf |Hb,env |ii + hi|Hb,env |f ihf |H̄t |ii (D.8) one can easily convince oneself that there are only two non vanishing contributions, namely |hf |H̄t |ii|2 and |hf |Hb,env |ii|2 . The mixed terms vanish as the number of particles cannot be conserved. i.e. hf |H̄t |ii ∝ δν 0 ,ν+σ and hi|Hb,env |f i ∝ δν 0 ,ν therefore hi|Hb,env |f ihf |H̄t |ii = 0. 81 The transition rates 82 Relaxation rates Let us start considering the transitions induced by |hf |Hb,env |ii|2 , which we call relaxation rates 2 Γrel (D.9) |ii→|f i = 2π|hf |Hb,env |ii| δ(Ei − Ef ). Actually, we are only interested in transitions between two different states of the system, while the state of the leads and of the environment remains unobserved. The total rate for the transition |ν, li → |ν 0 , l0 i is obtained from Eq.(D.9) by summing over all the initial and final states for the environment and the leads, and weighting each contribution with the probability to find the initial state occupied XX β β Γrel WL WR |hν 0 , l0 , L0 , R0 |Hb,env |ν, l, L, Ri|2δ(Ei − Ef ) |ν,li→|ν 0 ,l0 i = 2π L,L0 R,R0 (D.10) where Ei = Eν,l + EL + ER and Ef = Eν 0 ,l0 + EL0 + ER0 and where WLβ , WRβ are the equilibrium distributions for the leads and for the bosonic environment. The modulus squared of matrix element is given by |hν 0 , l0 , L0 , R0 |Hb,env |ν, l, L, Ri|2 X χ2j ωj2 |hν 0 , l0 , L0 , R0 |(a†j + aj )(b† + b)|ν, l, L, Ri|2 = j = X j χ2j ωj2 |hR0 |(a†j + aj )|Ri|2 |hl0 |(b† + b)|li|2 δν 0 ,ν δL0 ,L = δν 0 ,ν δL0 ,L [(l + 1)δl0 ,l+1 + l δl0 ,l−1 ] X j χ2j ωj2 |hR0 |(a†j + aj )|Ri|2 from which we see that considering Hb,env to second order allows only transitions between states with neighbooring vibrational number l ↔ l ± 1. Let us focus on the transition l → l − 1. Inserting this expression into Eq.(D.10) we obtain X X β χ2j ωj2 WR |hR0 |(a†j + aj )|Ri|2 δ(ω0 + ER − ER0 ). Γrel |ν,li→|ν,l−1i = 2π l j R,R0 To evaluate the contribution of the bosonic environment we exploite the fact that the eigensates |Ri can be written in terms of the occupation numbers nj of each of the vibrational mode of the environment ωj , i.e. |Ri = |n0 , n1 , . . .i (nj ∈ {0, ∞}), and that the occupation probability factorize WRβ = Πj Wnβj with Wnβj = e−βωj nj /Z. This way we obtain X β WR |hR0 |(a†j + aj )|Ri|2 δ(ω0 + ER − ER0 ) = R,R0 = ∞ X 1 −βωj nj 1 δ(ω0 − ωj ) e [nj δ(ω0 + ωj ) + (nj + 1)δ(ω0 − ωj )] = Z 1 − e−βωj n =0 j Collecting results, the rate for the transiton |ν, li → |ν, l − 1i is given by Γrel |ν,li→|ν,l−1i = 2π l ∞ X χ2j ωj2 j=0 1 δ(ω0 − ωj ). 1 − e−βωj Introducing spectral density of the the environmental coupling X J (ω) = 2π ωj2 χ2j δ(ω − ωj ) j (D.11) (D.12) 83 we come to final result Eq.(5.20) Γrel l→l−1 = l J (ω0 ) . 1 − e−βω0 (D.13) where we have eliminated the reduntant index ν for simplicity. Following the same steps one can calculate the rate for the upward transition l → l + 1, obtaining Γrel l→l+1 = (l + 1) J (ω0 ) . eβω0 − 1 (D.14) Comparing this result with Eq.(D.13) we see that the relaxation rates obey the detailed balance βω0 rel Γrel Γl−1→l . (D.15) l→l−1 = e Tunneling rates Let now turn to evaluate the tunneling rates. A little care has to be taken while considering |hf |H̄t |ii|2 , as the renormalized tunneling matrix element ti e−A contains terms of any order in the couplig constant χj (see Eq.(5.12)). Writing e−A in powers of χj one obtains ti e−A = ti e−λ(b † −b) 1 − 2λ X j χj (a†j − aj ) + . . . . (D.16) Since we are treating Hint to the lowest order in the coupling constants ti , χj , when evaluating the tunneling rates we have to discard all the terms of Eq.(D.16) which contains χj . Therefore we are left with 2 XX † † ti hf |(c†i,kσ e−λ(b −b) dσ + d†σ eλ(b −b) ci,kσ )|ii δ(Ei − Ef ) Γ|ii→|f i = 2π i=1,2 k,σ XX † † = 2π t2i |hf |c†i,kσ e−λ(b −b) dσ |ii|2+|hf |d†σ eλ(b −b) ci,kσ )|ii|2 δ(Ei − Ef ). i=1,2 k,σ (D.17) From this expression we see that there are four kind of independent contributions, which correspond to an electron tunneling into or out of the island from the two different barriers. In other words, to the lowest order in ti , there is no coherence between the two tunneling junctions. Analogously to what we have done for ralaxation rate, to obtain the total rate for the transition |ν, li → |ν 0 , l0 i we have to sum Eq.(D.17) over all initial, weighted with the probability to find these states, and over all final states. For definiteness, we consider the transition |σ, li → |0, l 0 i through the first junction. X X β βX † WL WR t21 |h0, l0 , L0 , R0 |c†1,kσ0 e−λ(b −b) dσ0 |σ, l, L, Ri|2 × Γ1 σ→0 l→l0 = 2π L,L0 R,R0 k,σ 0 × δ(ε̄ + ω0 (l − l0 ) + EL − EL0 + ER − ER0 ) X X † = 2πt21 |hl0 |e−λ(b −b) |li|2 WLβ1 |hL01 |c†1,kσ0 |L1 i|2 δ(µ(l − l0 ) +EL1−EL01 ) k L1 ,L01 (D.18) (1) 0 where we have introduced the notation Γ1 σ→0 l→l0 ≡ Γ|σ,li→|0,l0 i and µ(l − l ) ≡ ε̄ + ω0 (l −l0 ). The only possible states with non vanishing matrix element hL01 | c†1,kσ |L1 i are |L1 i = | . . . , 0kσ , . . .i and |L01 i = | . . . , 1kσ , . . .i where this notation means that The transition rates 84 the state with wavevector k and spin projection σ is unoccupied in |L1 i while it is occupied in |L01 i. The occupation of states with other quantum numbers is arbitrary. Therefore, we obtain X 2 0 −λ(b† −b) Γ1 σ→0 |li|2 [1 − f (ε1,k − µ1 )] δ(µ(l − l0 ) − ε1,k ) (D.19) l→l0 = 2π t1 |hl |e k where we have exploited the fact that the probability WLβ1 factorize, i.e. WLβ1 = Πkσ Wnβ1,kσ and that W1β1,kσ = f (ε1,k − µ1 ) and W0β1,kσ = 1 − f (ε1,k − µ1 ), since we have assumed the leads to be in equilibrium with the local chemical potential. Replacing the sum over k with an integral over energy, we obtain Z σ→0 2 0 −λ(b† −b) 2 Γ1 l→l0 = 2π t1 |hl |e |li| dερ1 (ε)[1 − f (ε − µ1 )] δ(µ(l − l0 ) − ε) (D.20) P where ρ1 (ε) = k δ(ε−ε1,k ) is the density of the states (DOS) of the lead. Assuming the DOS to be constant over the relevant energy interval ρ1 (ε ≈ ρ1 ), we come to the final simple result 0 Γ1 σ→0 l→l0 = Γ1 Xll0 [1 − f (µ(l − l ) − µ1 )] where Γ1 = 2π t21 ρ1 is the bare tunneling and Xll0 = hl0 |e−λ(b Condon matrix element. (D.21) † −b) |li|2 is the Franck- The remaining tunneling rates can be evaluated in a similar fashion. This way we obtain 0→σ Γi l→l 0 Γi σ→2 l→l0 Γi σ→0 l→l0 2→σ Γi l→l 0 = Γi Xll0 f (µ(l0 − l) − µi )], (D.22) = Γi Xll0 f (µ(l0 − l) + Ū − µi ), (D.23) = Γi Xll0 [1 − f (µ(l − l ) − µi )], (D.24) 0 = Γi Xll0 [1 − f (µ(l − l0 ) + Ū − µi )]. (D.25) Franck-Condon factors Now, we are only left with the evaluation of the Franck-Condon factors Xll0 = hl0 |e−λ(b † −b) |li|2 . (D.26) † To evaluate the matrix element hl 0 |e−λ(b −b) |li, it is convenient to “disentangle” the † displacement operator e−λb eλb by means of the relation 1 eA+B = eA eB e− 2 C , C = [A, B] (D.27) which holds if [A, C] = [B, C] = 0. This way we obtain hl0 |e−λ(b † −b) 1 2 † |li = e− 2 λ hl0 |e−λb eλb |li. (D.28) We can then easily calculate s ∞ l X X λm m l! λm e |li = b |li = |l − mi m! m! (l − m)! m=0 m=0 s l0 X l0 ! (−λ)n 0 −λb hl0 − n| hl |e = 0 n! (l − n)! n=0 λb (D.29) (D.30) 85 Eploiting the orthogonality relation hl0 − n|l − mi = δl0 −n,l−m (D.31) and assuming l0 > l one obtains 0 hl |e −λ(b† −b) |li = e −λ2 /2 = e−λ 2 /2 (−λ) l0 −l 0 (−λ)l −l r r l l0 ! l! X (−λ2 )m 0 l ! m=0 m!(l − m)!(l0 − l + m)! (D.32) l! l0 −l 2 L (λ ) l0 ! l where Lkn (x) is the associated Laguerre polynomial Lkn (x) = k X (−x)m m=0 m!(n + k)! . (n − m)!(k + m)! † The expression for l > l 0 can be obtained using hl 0 |e−λ(b the end, we get the following result for arbitrary l, l 0 s hl0 |e−λ(b † −b) |li = e−λ 2 /2 0 [sgn(l − l0 )λ]|l −l| −b) |li = hl|eλ(b l< ! |l0 −l| 2 L (λ ) l> ! l < (D.33) † −b) 0 ∗ |l i . In (D.34) where l< = min{l, l0 } and l> = max{l, l0}. 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