Equation Of Motion Method Research Articles

Validity of the equation-of-motion approach to the Kondo problem in the large-Nlimit

The Anderson impurity model for Kondo problem is investigated for arbitrary orbit-spin degeneracy $N$ of the magnetic impurity by the equation of motion method (EOM). By employing a new decoupling scheme, a set self-consistent equations for the one-particle Green function are derived and numerically solved in the large-$N$ approximation. For the particle-hole symmetric Anderson model with finite Coulomb interaction $U$, we show that the Kondo resonance at the impurity site exists for all $N \geq 2$. The approach removes the pathology in the standard EOM for N=2, and has the same level of applicability as non-crossing approximation. For N=2, an exchange field splits the Kondo resonance into only two peaks, consist with the result from more rigorous numerical renormalization group (NRG) method. The temperature dependence of the Kondo resonance peak is also discussed.

Transport and current noise characteristics of a T-shape double-quantum-dotsystem

We consider the transport and the noise characteristics for the case of a T-shapedouble-quantum-dot system using the equation of motion method. Our theoretical results,obtained in an approximation equivalent to the Hartree–Fock approximation, account fornon-zero on-site Coulomb interaction in both the detector and side dots. The existence of anon-zero Coulomb interaction implies an additional two resonances in the detector’s dotdensity of states and thereafter affects the electronic transport properties of the system.The system’s conductance presents two Fano dips as a function of the energy of thelocalized electronic level in the side dot. The Fano dips in the system’s conductance can beobserved for both strong (fast detector) and weak coupling (slow detector) between thedetector dot and the external electrodes. Due to stronger electronic correlations, the noisecharacteristics in the case of a slow detector are much higher. This setup may beof interest for the practical realization of qubit states in quantum dot systems.

Theoretical Study of Interplay of Superconductivity and Ferromagnetism in Hybrid Rutheno–Cuprate Superconductors RuSr2RCu2O8

We consider the extended two-band s–f model with additional terms, describing intersite Cooper pairs’ interaction between 4f (5f) and conduction electrons. Following Green’s function technique and equation of motion method, self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (mf) are derived. The expressions for specific heat, density of states, and free energy are also derived. The theory has been applied to explain the coexistence of superconductivity and ferromagnetism in hybrid rutheno-cuprate superconductors RuSr2RECu2O8 (RE = Gd, Eu). The theory shows that it is possible to become superconducting if the system is already ferromagnetic. A study of specific heat, density of states and free energy is also presented. The agreement between theory and experimental observations is quite satisfactory.

Theoretical Study of Specific Heat, Density of States and Free Energy of Itinerant Ferromagnetic Superconductor URhGe

Following the equation of motion method and Green’s function technique, the coexistence of itinerant ferromagnetism (FM) and superconductivity (SC) is investigated in a single band homogeneous system. Self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (ΔFM) are derived. It is shown that there generally exists a coexistent (Δ≠0 and ΔFM≠0) solutions to the coupled equations of the order parameter in the temperature range 0<T<min (T C,T FM) where T C and T FM are respectively the superconducting and ferromagnetic transition temperatures. Expressions for the specific heat, density of states and free energy are derived. The specific heat has a linear temperature dependence at low temperatures as opposed to the exponential decrease in the BCS theory. The density of states for a finite ΔFM increases as opposed to that of a standard ferromagnetic metal. The free energy shows that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and therefore is realized at low enough temperature. The theory is applied to explain the observations of URhGe. The agreement between theory and experimental results is quite satisfactory.

Transport through a quantum dot subject to spin and charge bias

Spin and charge transport through a quantum dot coupled to external nonmagnetic leads is analyzed theoretically in terms of the non-equilibrium Green function formalism based on the equation of motion method. The dot is assumed to be subject to spin and charge bias, and the considerations are focused on the Kondo effect in spin and charge transport. It is shown that the differential spin conductance as a function of spin bias reveals a typical zero-bias Kondo anomaly which becomes split when either magnetic field or charge bias are applied. Significantly different behavior is found for mixed charge/spin conductance. The influence of electron–phonon coupling in the dot on tunneling current as well as on both spin and charge conductance is also analyzed.

Kondo Effect in Carbon Nanotube Quantum Dot in a Magnetic Field

The out-of-equilibrium electron transport of carbon nanotube semiconducting quantum dot placed in a magnetic field is studied in the Kondo regime by means of the non-equilibrium Green functions. The equation of motion method is used. For parallel magnetic field the Kondo peak splits in four peaks, following the simultaneous splitting of the orbital and spin states. For perpendicular field orientation the triple peak structure of density of states is observed with the central peak corresponding to orbital Kondo effect and the satellites reflecting the spin and spin-orbital fluctuations.

Phonon-Assisted Kondo Resonance in Spin-Dependent Transport through a Quantum Dot

Effects of local vibrational modes on electron transport through a quantum dot attached to ferromagnetic electrodes are studied in the Kondo regime by the non-equilibrium Green function formalism based on the equation of motion method. Differential conductance is calculated for parallel and antiparallel configurations of the leads’ magnetic moments, and well defined Kondo resonance peaks and their phonon satellites are found. The influence of a compensating magnetic field on the peak positions is also discussed.

Analysis of the Kondo Resonance in a Single Quantum Dot Asymmetrically Coupled to Ferromagnetic Electrodes with Non-Collinear Magnetizations

The Kondo effect is studied theoretically in the framework of the non-equilibrium Green function formalism. The system under consideration consists of a single quantum dot asymmetrically coupled to ferromagnetic electrodes, whose magnetic moments are non-collinear. The spin-dependent density of states, as well as the transport characteristics such as differential conductance and tunneling magnetoresistance through the system are obtained using the equation of motion method. Numerical illustration of the mentioned quantities for several magnetic configurations and coupling strengths is presented and discussed.

Electron-spin dephasing via hyperfine interaction in a quantum dot: An equation-of-motion calculation of electron-spin correlation functions

In this paper we develop an equation-of-motion (EOM) approach to study the non-Markovian single-electron-spin dynamics due to its inhomogeneous hyperfine coupling to the surrounding nuclei in a quantum dot. In particular, we identify an electron-spin correlation function that fully represents the electron-spin quantum coherence. Using the EOM method, we recover the exact solution of electron-spin decoherence for the case of a fully polarized nuclear reservoir. By considering nuclear-spin flip-flops mediated by virtual electron flips, which generate fluctuations in the Overhauser field (the nuclear field) for the electron spin, we find that the free-induction decay of the electron-spin correlation function for partially polarized and unpolarized nuclear-spin configurations is of the order unity instead of $O(1/N)$ ($N$ being the number of nuclei in the dot) obtained in previous studies. We show that the complete amplitude decay corresponds to the spectral broadening of the correlation function near the electron-spin Rabi frequency induced by nuclear-spin flip-flops. Our results show that a 90% nuclear-spin polarization can enhance the electron-spin coherence time by more than 1 order of magnitude. In the long-time limit, the envelope of the transverse electron-spin correlation function has a nonexponential $1/{t}^{2}$ decay in the presence of both polarized and unpolarized nuclei.

Voltage-controllable spin-polarized transport through parallel-coupled double quantum dots with Rashba spin–orbit interaction

A spin device, consisting of parallel-coupled double quantum dots and three normal metal leads, is proposed to realize spin-polarized current without the help of magnetic field and magnetic material. Based on the Keldysh nonequilibrium Green function technique and equation of motion method, the spin-dependent current formula in each lead is derived. It is shown that not only a fully polarized current but also a tunable pure spin current can be obtained by modulating the structure parameters, strength of Rashba spin–orbit interaction and bias voltages properly. It further demonstrates the dependence of the spin-polarized current on the strength of the Rashba spin–orbit interaction.

DYNAMIC STRUCTURE FUNCTION OF QUANTUM BOSE SYSTEMS: CONDENSATE FRACTION AND MOMENTUM DISTRIBUTION

We present results on the behavior of the dynamic structure function in the short wave length limit using the equation of motion method. Within this framework we study the linear response of a quantum system to an infinitesimal external perturbation by direct minimization of the action integral. As a result we get a set of coupled continuity equations which define the self-energy. We evaluate the self-energy and the dynamic structure function in the short wavelength limit and show that sum rules up to the third moment are fulfilled. This implies, for instance, that the self-energy at short wavelengths and zero frequency is proportional to the kinetic energy per particle. An essential feature in this derivation is that the short range behavior of the two-particle distribution and the long wavelength phonon induced scattering are exactly satisfied. We calculate the condensate fraction and show that our results agree very well with the Monte Carlo simulations.

Interplay of Superconductivity and Ferromagnetism in UGe2

The emergence of pressure induced superconductivity (SC) under the background of ferromagnetic state in 5f-electron based itinerant ferromagnetic superconductor UGe2 is studied in the single band model by using a mean-field approximation. The solutions to the coupled equations of superconducting gap (Δ) and magnetization (m) are obtained using Green’s function technique and equation of motion method. It is shown that there generally exists a coexistent (Δ≠0, m≠0) solution to the coupled equations of the order parameters in the temperature range 0<T<min (TC,TFM), where TC and TFM are respectively the superconducting and ferromagnetic transition temperatures. The study of electronic specific heat (C/T), density of states, free energy, etc. are also presented. The specific heat capacity at low temperature shows linear temperature dependence as opposed to the activated behavior. Density of states increases as opposed to the case of a standard ferromagnetic metal. Free energy study reveals that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and, therefore, realized at low enough temperature. The agreement between theory and experimental results for UGe2 is quite satisfactory.

Electron dynamics in the coupled double quantum dots system

We have studied the electron dynamics in different geometrical arrangements of the two coupled double quantum dot structures. Applying the equation of motion method for appropriate correlation functions the occupation probabilities of different quantum dots of the considered system has been theoretically investigated. The numerical calculations were performed for different forms of the time-dependent tunneling amplitudes and quantum dot energy levels. We found, among others, that under some conditions for the tunneling amplitudes changed in the form of Gaussian pulses it is possible to localize the electron in a controlled manner on the given dot of the considered system.

Fano effect on shot noise through a Kondo-correlated quantum dot

Transport in a semiopen Kondo-correlated quantum dot is mediated through more than one quantum state. Using the Keldysh technique and the equation of motion method, we study the shot noise $S$ for a wide range of source-drain voltages ${V}_{sd}$ within a model incorporating the additional states as a background continuum, demonstrating the importance of the Fano interference. In the absence of the interference, the noise is revealed to be a probe of the second moment of the local density of states, and our theory reproduces the well-known peak structure around the Kondo temperature in the $S\text{\ensuremath{-}}{V}_{sd}$ curve. More significantly, it is found that taking account of the background transmission, the voltage dependence of the noise exhibits rich peak-dip line shapes, indicating the presence of the Fano effect. We further demonstrate that due to its two-particle nature, the noise is more sensitive to the quantum interference effect than the simple current.

Coexistence of Itinerant Ferromagnetism and Superconductivity in ZrZn2

A microscopic coexistence of itinerant ferromagnetism and superconductivity is studied in a single- band homogeneous system, using an equation of motion method and Green’s function technique. Self-consistent equations for the superconducting order parameter Δ and magnetization m are derived. It is shown that there generally exists coexistent (Δ≠0 andm≠0) solutions to the coupled equations of the order parameters in the temperature range 0<T<min (Tc,TFM), where TcandTFM are respectively the superconducting and ferromagnetic transition temperatures. The expressions for specific heat, density of states, free energy, and critical field are also derived. The theory is applied to explain the observations in ZrZn2. The agreement between the theory and experimental observations is quite satisfactory.

Dicke-like effect in spin-polarized transport through coupled quantum dots

Spin-dependent electronic transport through a quantum dot side-coupled to two quantumdots and attached to ferromagnetic leads with collinear (parallel and antiparallel)magnetizations is analyzed theoretically. The intra-dot Coulomb correlations are takeninto account, whereas the inter-dot ones are neglected. Transport characteristics,i.e. conductance and tunnel magnetoresistance associated with the magnetization rotationfrom parallel to antiparallel configurations, are calculated by the non-equilibriumGreen’s function technique. The Green’s functions are derived by the equation ofmotion method in the Hartree–Fock approximation. The conductance spectraare shown to reveal features similar to the Dicke resonance in atomic physics.

Theoretical Study of Specific Heat, Density of States, Free Energy, and Critical Field of High Temperature Cuprate Superconductors Based on Intra and Interlayer Interactions

The role of attractive interlayer and intralayer interactions in layered high Tc cuprate superconductors have been investigated using a one-band two layer tight binding Hamiltonian. Self-consistent equations for the superconducting order parameter (Δ) and critical temperature (Tc) are derived using double time Green’s functions and equation of motion method. The expression for excitonic type correlation (γc), specific heat, density of states, free energy, and critical field are obtained. The interlayer interactions play an important role in the enhancement of Tc in layered high Tc cuprates. The oxygen isotope effect is also analyzed. The agreement between theoretical and experimental results for the system YBa2−xLaxCu3O7 (0≤x≤0.5) is quite satisfactory.

Electron dynamics in quantum qubits coupled with the single electron transistor

The system of one and two qubits coupled electrostatically with the quantum dot placed between the two electron reservoirs (single electron transistor, SET) has been studied. The qubit charge oscillations and the current flowing through the SET were calculated using the equation of motion method for appropriate correlation functions. The calculations were done for constant and harmonically oscillating qubit energy levels. We also show that the SET current flowing in response to the pulse of the tunneling amplitude between the qubit quantum dots versus the pulse duration exhibits the damped oscillatory time-dependence similar to the qubit charge oscillations.

Tunnel Magnetoresistance in Carbon Nanotube Quantum Dot

The out-of-equilibrium transport properties of carbon nanotube quantum dot in the Kondo regime are studied by means of the non-equilibrium Green function. The equation of motion method is used. The in∞uence of the polarization of electrodes and orbital level splitting, as well as leftright asymmetry, on the spin polarizations of difierential conductance are discussed. For zero bias voltage and orbitally degenerate states the SU(4) symmetry of Kondo state is preserved for antiparallel conflguration of polarizations of electrodes, whereas it is broken for parallel. In the former case a suppression of linear conductance with increasing polarization is observed. In the latter the behaviour is nonmonotonic due to splitting of the Kondo peak and bringing closer one of the peaks to the Fermi level with increasing polarization. This gives rise to giant tunnel linear magnetoresistance for large polarization.

Poor Man's Scaling and Green Function Analysis of the Kondo Anomaly in Single-Level Quantum Dots

Kondo effect in a single-level quantum dot attached to magnetic leads is studied theoretically by the “poor man’s scaling” and non-equilibrium Green function methods. From the scaling equations we derive the Kondo temperature as a function of the model parameters — in particular as a function of the angle between magnetic moments. Transport characteristics, i.e. differential conductance and tunnel magnetoresistance associated with magnetization rotation, were calculated within the non-equilibrium Green function formalism based on the equation of motion method.