4 6 8 10
ω/ eV S ( ω )[ e
2eV / h ]
� ���
� � ���
Figure 2.4: Frequency-dependent noiseS(ω) at zero temperature and a single chan-nel with Fano factor F.
transmissions is shown in Fig.2.4. The frequency-dependent noise can be understood in terms of photon-assisted tunneling in which the current noise at frequencyωis due to emission and absorption of photons at the same frequency. Negative frequencies corresponds to the absorption of photons by the nanostructure whereas at positive frequency, the nanostructure emits photons. Since at zero temperature, the system is in the ground-state, the threshold for emission is set by the applied voltage (see Fig.2.4). In this sense, the frequency-dependent noise gives a measure for excitations in the nanostructure.
2.5 Nanoelectromechanical systems with carbon nanotubes
NEMS describe a wide variety of suspended nano resonators. In particular, suspended carbon nanotubes have extraordinary properties combining the stiffness and low mass of graphene [19]. The low mass of the resonator makes them ideal for mass [20] and force [21] sensing. Reducing the size as well the frequency of the nanomechanical oscil-lators have the distinct advantages to increase the sensitivity as nanoscale detectors, the quality factorQ and the amplitude of the zero-point fluctuations.
In Sec. 2.5.1, we briefly study the Euler-Bernoulli theory of beams applied to these resonators [22, 23]. Ground-state cooling of mechanical motion is discussed in Sec. 2.5.2.
2.5.1 Euler-Bernoulli theory of beams
The vibrational modes of a suspended beam can be determined by the Euler-Bernoulli equation. This equation can be applied for microscopic and macroscopic beams with a lengthlmuch larger than the width and height. The solution of the Euler-Bernoulli equation with the proper boundary conditions gives a set of eigenmodes and
corre-2.5 Nanoelectromechanical systems with carbon nanotubes sponding eigenfrequencies ωn. We can model the suspended carbon nanotube by cylindrical beam and assume that the nanotube is clamped at both of its ends. Since the mode spectrum is sufficiently sparse [24], we are concerned only with the flexural mode with frequencies in the'MHz range.
We consider now a beam with length l and apply a force to it [23,25]. The force pulls on the beam and causes a deformation (strain) of the beam. Because of the strain, internal forces induce a stress acting against the strain. The ability of a beam to be deformed when a force is applied, is expressed in terms of Young’s modules E which is the ratio between the stress and strain. The suspended beam can additionally be under tension due to the clamping at the contacts.
The Hamiltonian for the mechanical flexural motion in one plane reads [22]
Hvib=Z l
with the operator ˆu(z) corresponding to the local transversal displacement and ˆπ(z) to the conjugate operator such that [ˆu(z), ˆπ(z)] =i~δ(z−z0). For a cylindrical beam, the polar moment of inertia is I = πr4/4 with the radius r of the nanotube and T is the uniform tension on the beam. The second form of Hamiltonian Eq. (2.45) is obtained via a canonical transformation in which the local displacement reads ˆ
u(z) =Pnfn(z)un(ˆbn+ˆb†n) and ˆπ(z) =iPnfn(z)mωnun(ˆb†n−ˆbn), with the waveform fn(z), un = p~/(2mωn) and the nanotube’s mass m. The waveform satisfy the normalization condition (1/l)R0ldzfn(z)fn0(z) =δnn0.
For sufficiently strong tension T Er4/l2, one can neglect the second term in the first line of Eq. (2.45) corresponding to the elastic energy for the length-variation of an infinitesimal element caused by the local bending. Then, Eq. (2.45) reduces to a wave equation and the wave-form functions {fn(z)} are the orthonormal set solutions of the standard wave equation for an elastic string with eigenfrequencies ωn= (n+ 1)πpT /(ρl2) and fn(z) =√
2 sin[π(n+ 1)z/l] for integers n≥0.
2.5.2 Ground-state cooling of mechanical motion
Cooling a nanomechanical resonator close to the ground-state is a major challenge due to the low temperatures required to reach low occupations of the vibrational modes.
If we succeed to refrigerate the mechanical motion to temperatures much smaller than the frequency of the resonator, kBT ~ω, the resonator is in the quantum ground-state. Ground-state cooling of mechanical motion has been demonstrated in optomechanical systems, in which the nanomechanical motion is coupled to photons in optical cavities [26–28]. The coupling allows to put the resonator to the ground-state and observe superposition of quantum-mechanical states [29, 30]. Alternatively to the interaction of photons and the mechanical motion, the electron-phonon coupling can be exploited to cool nanomechanical resonators. This coupling mechanism has the advantage that the vibrational states of the resonator can be manipulated by applying a current.
In the ground-state, the resonator exhibits zero-point fluctuations with amplitude xzpm =
s
~
2mω, (2.47)
with massmand frequencyωof the resonator. An advantage of carbon nanomechancial resonators is the relatively large zero-point fluctuation ('10 pm). Additionally, car-bon nanomechanical resonators offer the additional advantage of high quality factors which can be larger than 106 [31]. However, due to the frequencies in the ' MHz range, it is very challenging to cool the mechanical motion to the ground state by conventional cooling techniques. Hence, on top of the conventional cooling, active cooling must be exploited to cool the resonator to the ground state if one aims to reach the quantum regime.
3 Josephson current through a quantum dot coupled to a molecular magnet
3.1 Abstract
We theoretically study the electronic transport of a magnetically tunable nano-scale junction consisting of a quantum dot connected to two superconducting leads and coupled to the spin of a molecular magnet. The exchange interaction between the molecular magnet and the quantum dot modifies the Andreev states due to a spin-dependent renormalization of the quantum dot’s energy level and the induction of spin flips. A magnetic field applied to the central region of the quantum dot and the molecular magnet further tunes the Josephson current and starts a precession of the molecular magnet’s spin.
We use the nonequilibrium Green’s function approach to evaluate the transport properties of the junction. Our calculations reveal that the energy level of the dot, the magnetic field, and the exchange interaction between the molecular magnet and the electrons occupying the energy level of the quantum dot can trigger transitions from a 0 to a π state of the Josephson junction. The redistribution of the occupied states induced by the magnetic field strongly modifies the current-phase relation. The critical current exhibits sharp features as a function of either the energy level of the dot, the magnetic field, or the exchange interaction.