Laser Field -Assisted Spin Tunnelling Through Quantum Dot System

INTRODUCTION
Quantum dot (QD) is fundamental structure of zero-dimensional system, where all electrons are confined in three directions, due to the confinement, the energy levels of quantum dot will be quantized and similar to the electronic levels of real atom which give rise to be named as artificial atoms [1,2]. Generally, the tunneling effects, quantum interference, and quantized energy levels can often have key roles in the charge and spin transport processes. The modern emerging field deals with the effective manipulation of the electron’s charge and spin, this will have led to add a spin degree of freedom to the conventional charge-based technology, which has the prospective features of multi-functionality [3,4]. Study the electron and spin tunneling through quantum dot has attracted a lot of attention due to reveals new and unique effects such as Coulomb blockade effect, quantum Hall effect, Fano effect, Kondo effect, thermoelectric effect, and Photon-assisted tunneling effect (FAT), which makes the quantum dot a potential candidate as a building block for devices based on novel physics [1,5]. The (FAT), were electromagnetic (EM) field has interacted with the quantum dot system has received increased attention both experimentally and theoretically in recent year because of the potential application in quantum computing and photoelectron devices [6-8]. The physical mechanism of (FAT) effects is detected in the induced photon-current in the quantum dot system which has been investigated theoretically in quantum dot [9]. Later, different theoretical methods were proposed, such as the time-dependent Schrodinger equation, the transfer Hamiltonian method the Master equation, nonequilibrium Green function(NEGF) approach and the Keldysh nonequilibrium Green’s function method. An important characteristic of these systems is that the electron in the system can exchange an energy with the external fields, leading to several new inelastic tunneling channels [7,10], resulting in an additional conductance peak with in the Coulomb blocked regime, this additional resonance is attributed to the (FAT) [11]. The manipulation of quantum dots embedded between two leads by a microwave and (EM) field is applied on quantum dot, a multiple transport channels are opened, which utilized for the possible to fabricate an important ingredient application of quantum dots as solid state quantum bits, single-photon transistors, spin filter, heat engine, quantum-dot thermospin, refrigerators, quantum computing and photoelectron devices [7,12-15]. 
In this work, we will present an extended theoretical study for the role of laser field in spintronic properties through quantum dot embedded between two normal leads. The nonequilibrium Green’s function is used in this treatment to model the quantum dot energy level that hybridize with the leads to enable the spintronic tunneling process. An external magnetic field, the intra-dot Coulomb correlation on quantum dot and their coupling with the leads are incorporates. The occupation number on quantum dot is formulated and solved self-consistently, then it is used to derive the spin current tunneling through the system. The spin current shows three classified multiple spin currents channels due to split energy level of quantum dot to three multiple energy levels. These classified multiple spin currents channels are employed to investigate the spin tunneling properties and give us a chance to study and investigate many spin depending physical feature concerned to future spintronic applications especially in quantum computing.

THE MATHEMATICAL MODEL
The single impurity Anderson model is utilized to model the role of laser filed on system consist of single quantum (QD) embedded between normal leads, and assume that there is only two spin-split level of quantum dot duo to magnetic flux (Zeeman splitting), experience the photon mode of laser field incident directly on quantum dot.  The Hamiltonian of the system considered in this can be written as the sum of three terms Eq.1 [7, 16,17].

The first term in Eq. 1 is the Hamiltonian for noninteracting electrons in the leads. Where Ckασ+ (Ckασ) is the creation (annihilation) operators of an electron with spin σ and energy Ekασ in the lead α(≡L,R). The second term in Eq. 1, is the Hamiltonian that describes the isolated quantum dot (QD), where Edσ=Ed-σH, Ed, is the effective energy levels of the quantum dot (QD). where H denoted the Zeeman splitting energy. U represent the intra-quantum dot Coulomb correlation  between spin up and spin down electrons on the (QD)  and ndσ is the occupation number on the (QD) with spin σ where ndσ=Cσ+ (Cσ)=(nLσ+nRα)/2 and Cσ+ (Cσ) is the creation (annihilation) operator of a localized electron on the level Edσ. Where ndσ is the occupation number on the QD with spin σ due to the lead α and is given by Eq. 2 [18].

Where u0α is the band bottom of the leads α, μασ is the electrochemical potential on the lead αin the presence of spin bias. fασ (E, Tασ) is the Fermi distribution function for the lead α and ρdσ is the local density of states on the quantum dot (QD) which is coupled to the lead αand is given by  Eq. 3 [7,19].

Where Γασ is the broadening in the energy levels of the QD due to the coupling with the lead α. ΓL broadening due to laser field coupling interaction.  The third term in Eq. 1 is the Hamiltonian that describes the quantum dot -leads interaction, where Vkα is the hybridization matrix elements between the quantum dot and the leadα.By using the Taylor expansion for Fermi distribution function around E=μα in the interval (-kBTα≤μα≤kBTα) we can solve Eq. 2 to calculate the occupation number on the QD with spinσ due to the lead α as Eq. 4.

The electrical current Iσ, flowing through the active region (QD) in the spin channel σ in the case of non-equilibrium due to applying spin bias, is given by the Eq. 5 [20,21 ].

By using Eqs. 2 and 3 in Eq. 4, one can write Eq. 4 in term of Eq. 2 as the following relation:

The spin current and differential conductance can be given by the following relations [21,22-24]:

RESULTS AND DISCUSSION
The role of laser field which is electromagnetic (EM) field on spin tunneling through the system of quantum dot between two normal leads well be investigated in case of weak coupling regime and the system is in thermal equilibrium at low temperature. Two parameters of the laser field are incorporate in this study, which are frequency and intensity of the laser beam applied to quantum dot, the harmonics frequencyωwith (n=-1,0,1) will be added to the energy level of quantum dot which splits each energy level into three levels (i.e.E0σ and E±1σ), while the intensity of the laser field incorporates as a tunneling rate (broadening ),due to laser field coupling interaction with the leads that affected the density of state of quantum dot energy levels. In order to investigate the spin tunneling properties through the system considered in this study, spin bias will be applied to the leads ,while the quantum dot energy level position will be controlled by gate voltage.The spin current and it channels as well as the differential conductance are calculated to investigate the spin tunneling properties, The red, green and blue solid line represent the spin current and the differential conductance through the channels Enσ,while the black dashed line represents the taotal spin current and differential conductance passing through all channels. In Fig. 1, the spin current as a function spin bias spin for different values of Zeeman splitting at positive value of quantum dot energy level is presented. In the absence of Laser field, the spin current shows step-like behavior due to the Coulomb blockade effect around spin bias equal zero, the width of step increases as Zeeman splitting  increases, see Fig. 1(a and b) thick black line. When laser field presented, the quantum dot energy level Edσ will be splitting to three energy channels (i.e.E0σ and E±1σ). At the absence of Zeeman splitting the spin current of the energy channels E0σ and E-1σ show step-like behaviors,see Fig. 1a red and green solid line, while its linear in the region around  spin bias equal zero through the energy channel E±1σ and the device behaves as resistor, see Fig. 1a blue solid line. When the value of Zeeman splitting increases to 0.2 eV, the spin current of the energy channels E0σ and E+1σ  show step-like behaviors as shown in Fig. 1b red and blue solid line, while its linear in the region around spin bias equal zero through the energy channel E-1σ and the device behaves as resistor see Fig. 1b green solid line. So that the total spin current behavior is enhanced due to tunneling through the energy channels E±1σ , see Figs. 1a and b black dashed line. But no obvious rule of the channel Edσ can be observed. Our study results show that the total spin current is increases as the laser field presented and the Photon-assisted tunneling appear. Fig. 1c and d, clarify the differential conductance as a function of spin bias. Fig. 1c black solid line, shows the differential conductance in the case of absence the  laser field and Zeeman splitting equal zero, which exhibits two splitting peaks for certain spin bias polarity. The peaks are located on spin bias polarity equal to Ed+U, one con utilize in such experiments to determine the Coulomb correlation on quantum dot. When the Zeeman splitting increases to 0.2 eV, the differential conductance becomes relatively higher and the peaks become more separated, the peaks are lying at spin bias equal to Zeeman splitting, one can experimentally deduce the Zeeman splitting depending on the measured differential conductance, see the black solid line in Fig.1d. The red, green and blue lines, represent the differential conductance of the energy channels E0σ and E±1σ. In the presence of laser field and Zeeman splitting equal to zero, the laser field induce a peak emerge in the differential gab region around spin bias equal zero due to the spin tunneling current through the channel E+1σ in the Coulomb blockade regionsee see Fig. 1c blue solid line, further two peaks appear in the negative and positive spin bias regions represent the spin tunneling through the energy channel E-1σ,these two peaks are located at spin bias polarity equal to 2E+1σ, see Fig. 1c green solid line. one can experimentally deduce the energy level of quantum dot depending on the measured differential  conductance. Our study results show that the laser field enhanced the total differential conductance, the peaks that appear in the total differential conductance are belong to the enrgy channels. One can experimentally deduce the behavior of the total differential conductance from the behavior of channels of differential conductance. As the Zeeman splitting increases to 0.2 eV, the feature of differential conductance of the energy channels E+1σ becomes relatively lesser and shows two peaks for certain spin bias polarity located at spin bias polarity equal to the Zeeman splitting, see Fig. 1d blue solid line, while the differential conductance of the energy channel E-1σ exhibits three peaks, one of these peaks emerges in the differential gab region around  spin bias equal zero due to the spin tunneling current through the channel E-1σ in the Coulomb blockade region (spin bias equal zero), the other two peaks are laying at the spin bias polarity equal to the energy 2E+1σ + H, see Fig. 1d green solid line, So that the Zeeman splitting induced more peaks in the total differential conductance, these peaks reflects exactly the peaks of the channels E±1σ, see dashed black line in Fig. 1c and d. From the results above , one can deduce the behavior of total differential conductance from the differential conductance of the energy channels E±1σ, also one can control the channel induce peak in the differential gab region around  spin bias polarity , depending on the measured quantum dot energy level and Zeeman splitting.
In order to report physical nots about the position of quantum dot energy levels, all the above mentioned calculation are also achieved for negative value of quantum dot energy levels. In the case of presence laser field and absence of Zeeman splitting, the spin current relation through the energy channels E0σ and E+1σ show step-like behaviors see Fig. 2a red and blue solid lines, while its linear in the region around  spin bias equal to zero  through the channel E-1σ and the device behaves as resistor see Fig. 2a green solid line, while the laser field induces a peak emerge in the differential  gab region around  spin bias equal zero due to the spin tunneling current through the energy channe E-1σ l  in the Coulomb blockade region, see Fig. 2c green line, further two peaks appear in the negative and positive spin bias regions represent the spin tunneling through the channel E+1σ, see Fig. 2c blue solid line. By comparing Fig. 1 with Fig. 2 in the case of absence Zeeman splitting, one can control experimentally which channel induces the current and the peak of differential conductance spectrum in the Coulomb blockade region, also one can deduce the behavior of quantum dots energy level with the spin bias variation depending on the measured differential conductance. When the Zeeman splitting increases to 0.2 eV, the spin current tunneling through all channels show step-like behaviors, no spin current induced in the region around  spin bias equal to zero and the Coulomb blockade appear see Fig. 2b red , blue and green lines, the most important physical features are that the spin current through the two channels E±1σ have same value and behavior which mean that the spin current through these channels are coherent current. The differential conductance spectrum shows corresponding splitting peaks for each spin polarity belong to the energy channels E±1σ see Fig. 2d blue and green lines, these peaks reflected the splitting peaks in the total differential conductance spectrum see Fig. 2d black dashed line. 
In order to report physical nots about the position of quantum dot energy levels, we now pay some attention to effect of tuning energy level of quantum on spin tunneling properties . In Fig. 3a and b, we plot the spin current as a function of quantum dot energy level in the absence and presence of laser field for different values of Zeeman splitting. When the value of Zeeman splitting is zero, the laser field induces two peaks in the spin current spectrum, one peak in the negative values of quantum dot energy level due to the tunneling through the energy channel E-1σ, which is located at the quantum dot energy level equal to -2eV, the second peak in the positive values of quantum energy level due to the tunneling through the channel E+1σ, which is located at the quantum dot energy level equal to 0.1 eV,see Fig. 3a green and blue solid lines, these features confirm our results in Figs. 1 and 2, in the case of absence Zeeman splitting. As the  value of Zeeman splitting increases to 0.2 eV, the laser field induces four peaks in the spin current spectrum, two peaks in the negative values of quantum dot energy level due to the tunneling through the channels E±1σ located at the quantum dot energy level equal to-2 eV, and -4 eV, respectively, the second two peaks in the positive values of quantum dot energy level due to the tunneling through the channel E±1σ, which is located at the quantum dot energy level equal to 0.1 eV and 3 eV, see Fig.3b green and blue solid lines, these features confirm our results in Figs. 1 and 2. 

CONCLUSION
The quantum dot system is very often directly serve as manipulating and controlling devices. They motivate us to investigate the spin tunneling properties in accurately controlled conditions. So the results discussed in our study are very important for future spintronic applications especially Photon- based spin electronics and quantum computing devices. The role of laser field in spintronic properties through quantum dot embedded between two normal leads is formulated based on the nonequilibrium Green’s function model. An external magnetic field, the intra-dot Coulomb correlation on quantum dot and their coupling with the leads are incorporates in this study. The occupation number on quantum dot is solved self-consistently, then it is utilized to formulate the spin current tunneling through the system. The spin current and spin differential conductance shows three classified multiple channels due to split energy level of quantum dot to three multiple energy levels. These classified multiple channels are enhanced and induced the photon-assisted (FAT) peaks in total differential conductance spectrum. The positions of these peaks related to the peaks in differential conductance spectrum of the channels. An interesting results is the observation of  photon-assisted (FAT) peak that can be generated in the Coulomb blockade effect region around spin bias equal zero, which is belong to one of the three opened channels in this region, which can be accurately determined by manipulating and tuning of  the energy level of quantum dot and the external magnetic field. This peak clarifies that the resistor behaviors of spin current in this region. This results can be utilized experimentally to estimate the behavior the spin current, differential conductance from the behavior of the opened channels with spin bias variation, depending on the energy level of quantum and the external magnetic field measuring and vice versa. 

     
CONFLICT OF INTEREST
The authors declare that there is no conflict of interests regarding the publication of this manuscript.