Inelastic electron transport through Quantum Dot coupled with an nano mechancial oscillator in the presence of strong applied magnetic field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Inelastic electron transport through Quantum Dot coupled withan nano mechancial oscillator in the presence of strong appliedmagnetic field.
M. Imran ∗ and K. Sabeeh Department of Physics, Quaid-i-Azam University, Islamabad, Pakistan.
B.Tariq
Department of Physics, Quaid-i-Azam University, Islamabad,Pakistan. National Center For Physics, Islamabad, Pakistan.
M. Tahir
Department of Physics, University of Sargodha, Sargodha, Pakistan. (Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate)
Abstract
In this study we explain the role of applied magnetic field in inelastic conduction properties of aQuantum Dot coupled with an oscillator . In the presence of strong applied magnetic field coulombblockade effects become weak due to induced Zeeman splitting in spin degenerate eigen states ofQuantum Dot.By contacting Quantum Dot by identical metallic leads tunneling rates of spin downand spin up electrons between Quantum Dot and electrodes will be symmetric. For symmetrictunneling rates of spin down and spin up electrons onto Quantum Dot, first oscillator get excitedby spin down electrons and then spin up elctrons could excite it further. Where as average energytransferred to oscillator coupled with Quantum Dot by spin down electrons will further increase byaverage energy transferred by spin up electrons to oscillator. Here we have also discussed that withincreasing Quantum Dot and electrodes coupling strength phononic side band peaks start hidingup, which happens because with increasing tunneling rates electronic states of Quantum Dot startgettting broadened. . Introduction In recent years, much attention has been focused on the concept and realization of nano-electromechanical systems (NEMS)[1–8] as a new generation of quantum electronic devices.A large number of new experimental techniques have been developed to fabricate and per-form experiments with NEMS in the quantum regime. Examples of high-frequency mechani-cal nano-structures that have been produced are nano-scale resonators[9, 10], semiconductorquantum dots or single molecules[11–16], cantilevers[18, 19], vibrating crystal beams[9], andmore recently graphene sheets[20] and carbon nanotubes[21, 22]. These devices are expectedto open up a number of future applications including nanomechanical transport effects, sig-nal processing which could be used in fundamental research and perhaps even form the basisfor new forms of mechanical computers. Many theoretical methods and models have beendesigned in order to account for the behavior of different types of NEMS system and tomake predictions and proposals for future experiments.In general, there are two different theoretical formulations that can be used to study thequantum transport in nanoscopic systems under applied bias. Firstly, a generalized quan-tum master equation approach[23–32] and secondly, the nonequilibrium Green’s functionformulation[33–35]. The former leads to a simple rate equation, where the coupling betweenthe dot and the electrodes is considered as a weak perturbation and the electron- phononinteraction is also considered very weak. In the latter case one can consider weak and in-termediate electordes to system and electron-phonon coupling. The nonequilibrium Green’sfunction technique is able to deal with a very broad variety of physical situations relatedto quantum transport at molecular levels[36, 37]. It can deal with strong non-equilibriumsituations and very small to very large applied bias. In the early seventies, the nonequilib-rium Green’s function approach was applied to mesoscopic transport[38–40] by Caroli et al.,where they were mainly interested in inelastic transport effects in tunneling through oxidebarriers. This approach was formulated in an elegant way[41–43] by Mier et al, where theyhave shown an exact time dependent expression for the non-equilibrium current throughmesoscopic systems. In this model an interacting and non-interacting mesoscopic systemwas placed between two large semi-infinite leads. In most of the theoretical work on NEMSdevices since the original proposal, the mechanical degree of freedom has been describedclassically/semiclassically[23, 44] or quantum mechanically[24–26, 45, 46]using the quantum2aster or rate equation approach. In the original proposal, the mechanical part was alsotreated classically, including the damped oscillator, and assuming an incoherent electron tun-neling process. This approach is based on a perturbation, weak coupling and large appliedbias approximations, whereas the Keldysh nonequilibrium Green’s function formulation cantreat the system leads and electron-phonon coupling with strong interactions[47]for bothsmall and large applied bias voltage. The transport properties have been described anddiscussed semi-classically/classically but need a complete quantum mechanical description.A theory beyond these cases is required in order to further refine experiments to investigatequantum transport properties of NEMS devices. In the quantum transport properties ofthese devices; the quantized current can be determined by the frequency of the quantummechanical oscillator, the interplay between the time scales of the electronic and mechanicaldegrees of freedom, and the suppression of stochastic tunneling events due to matching ofthe Fermionic and oscillator properties.In the present work, we consider a spin dependent electron transport through a quantumdot connected to two identical metallic leads via tunneling junctions. A single nanoelec-tromechanical oscillator is coupled with quantum dot and gate voltage is used to tune thelevels on the dot. The application of strong magnetic field induce Zeeman splitting in spindegenrate eigen states of quantum dot. As a result spin down states moves lower and spinup states move higher than the degenrate spin eigen states of quantum quantum dot, andthus offers the different channels of conductance for spin up and spin down electrons. Inthe presence of strong applied magnetic field the coulomb blockade effects will be weak dueto Zeeman splitting and we theoratically included it by mean field approximation, which isquite resonable approximation for tackling weak interactions. Although electron transportthrough mesoscopic systems in the presence of Zeeman splitting has been an active area ofresearch [57–59] . In our calculation the inclusion of the oscillator is not perturbative whichenable us to predict strong electron phonons coupling effects in NEMS system. Hence, ourwork provides an exact analytical solution to the current-voltage characteristics, conduc-tance, coupling of leads with the system, and it includes both the right and left Fermi- levelresponse regimes. However, we have used wide-band approximation[48–50], where the cou-pling between leads and quantum dot is taken to be independent of energy. This providesa way to perform transient transport calculations from first principles while retaining theessential physics of the electronic structure of the quantum dot and the leads.3 . Model Hamiltonian
Our mesoscopic system consists of a Quantum Dot(QD) coupled with an Oscillator toinclude the role of phonons effects in conduction through a QD. Application of externalapplied magnetic field induce Zeeman splitting in spin degenerate eigen states of QD. Thisconstitute microscopic part of mesoscopic system.To incoporate coulomb blockade effects inQD we use mean field approximation which is useful for weak interaction. As in presenceof strong magnetic field coulomb blockade effects will be small because of Zeeman splitting.Hamiltonian of the present microscopic system would be, H QD + Oscillator = X σ ( ǫ σ + 12 µ B gσ.B ) d † σ d σ + X σ α σ ( a + a † ) d † σ d σ + ω ( a † a + 12 ) (1)The first term represents two discrete energy levels in QD, which orginates because ofmagnetic field induce Zeeman splitting. d † σ , d σ ( a † a ) create and annihilate an electron in state | σ > on the dot (create and annihilate a phonon in state | n > on the oscillator) . Here ǫ σ ,µ B , g, σ, B, α σ and ω are energy levels of QD electronic-state with spin σ ,Bohar magneton,Lande g factor,Pauli spin matrix, applied magnetic field, QD and oscillator coupling andoscillator vibrations frequency. Second term represents oscillator QD coupling and last termrepresents oscillator energy spectrum.In first term of hamiltonian ( σ = 1) for spin up electrons and ( σ = −
1) for spin downelectrons, ǫ ↓ = ǫ o , ǫ ↑ = ǫ o + U (2)where U represents coulomb repulsion between spin down and spin up electrons.To pass current through this sytem we employ left/right electrodes which constituesmacroscopic part of of our system. Hamiltonian of left/right electrodes is H Leads = X k,σ ǫ k,σ,ν c † k,σ,ν c k,σ,ν . (3)Here ǫ k,σ,ν represents electrodes electronic states with wave vector k , spin σ ,and electrodes υ (left/right). c † k,σ,ν ( c k,σ,ν ) is electron creation (annhilation) operator in electrode υ .Hopping of electrons between electrodes and QD is defined by the following Hamiltonian,4 Hopping = X k,σ ( T k,σ,ν d † σ c k,σ,ν + hermitian − congugation ) (4)Here T k,σ,ν represents electron hopping amplitudes between QD and electrodes.We first solve our microscopic system Hamiltonian. Our approach include electron-phonon interaction exactly (non-perturbatively).To diagonalize microscopic system Hamiltonain , we employ Lang-Firsovtransformation[51]. ˜ H QD + Oscillator = Exp [ S ] H QD + Oscillator
Exp [ S † ] (5)where S = X σ ( a † − a ) d † σ d σ (6)After diagonalization ˜ d † σ = d † σ Exp [ α σ ω ( a † − a ) (7)˜ d σ = d σ Exp [ − α σ ω ( a † − a ) (8)˜ a † = a † − α σ ω d † σ d σ ,˜ a = a − α σ ω d † σ d σ (9)Therefore, ˜ H QD + Oscillator = X σ ( ǫ σ + 12 µ B gσ.B − ∆ σ ) d † σ d σ + ω ( a † a + 12 ) (10)Where ∆ σ = α σ ω Now the eigen function of the diagonalized Hamiltonian in k-space ( eigen function ofharmonic oscillator remain same in real and Fourier’s space) would be | oσn > = 1 √ X σ A n Exp [ − k σ H n [ k σ ] Exp [ − ik σ x σ ] | σ > (11) | uσn > = 1 √ X σ A n Exp [ − k σ H n [ k σ ] | σ > (12)5 oσn > is state of occupied QD with electron and | uσn > is the state of un-occupied QDwith electron.Here x σ represents displacement of oscillator due to occupancy of electron in QD. x σ = ∆ σ ω and H n [ k σ ] are usual Hermite polynomials.Now the amplitude of the occupied and un-occupied QD electronic state would be, A mnσ = < oσm | uσn > (13) A mnσ = 1 √ π m + n +2 n ! m ! Z dk σ Exp [ − k σ ] H m [ k σ ] H n [ k σ ] Exp [ ik σ x σ ] (14) A mnσ = r n − m − m ! n ! Exp [ − x σ ix σ | n − m | L | n − m | m [ x σ L | n − m | m [ x σ ] represents associated Lagurre’s polynomials.After diagonalization tunneling Hamiltonian H Hopping will become,ˆ H Hopping = X k,σ ( ˆ T k,σ,ν d † σ c k,σ,ν + hermitian − congugation ) (16)Where (cid:16) ˆ T k,σ,ν = T k,σ,ν Exp [ α σ ω ( a † − a )] (cid:17) C. Current from the mesoscopic system
Current from the ( ν ) electrode to the QD can be calculated by taking time derivative ofoccupation number operator of ( ν ) electrode. J ν ( t ) = − e (cid:28) ∂∂t N ν (cid:29) = ie h [ N ν , H ] i (17)where (cid:16) N ν = P k,σ c † k,σ,ν c k,σ,ν (cid:17) and (cid:16) H = ˜ H QD + Oscillator + H Leads + ˆ H Hopping (cid:17) .Therefore, J ν ( t ) = ie X k,σ (cid:16) ˆ T k,σ,ν D c † k,σ,ν ˜ d σ E − ˆ T † k,σ,ν D ˜ d † σ c k,σ,ν E(cid:17) (18) J ν ( t ) = i e Im[ X k,σ ˆ T k,σ,ν D c † k,σ,ν ˜ d σ E ] (19)Now we define electrode and QD coupled lesser Green’s function,6 A single level QD in the absence of magnetic field has spin degenerate levels. As theapplied voltage from electrodes become equal to QD energy levels, then spin up and spindown electrons start tunneling through the QD. Application of applied magnetic field toQD induce Zeeman splitting. As a result spin up level moves higher than unsplitted levelsand similarly spin down level moves lower than unsplitted levels. Now as applied voltagefrom electrodes is increased first spin down energy level resonates with applied bias andthen spin up energy level resonates with applied bias. In Fig.(1) we have showed differentialconductance as a function of applied bias from electrodes.Now we explain QD coupling with an oscillator in the presence of magnetic field. Atzero temperature oscillator will be in ground state.As spin down electron comes onto QD itgives energy to oscillator and oscillator moves to excited state. This explain the phononic10eaks appearance in differential conductance for spin down electrons. The main peak ofspin down electron will get shifted due to (∆) and amplitude of main peak of spin downelectron becomes smaller as only spin down electron could give energy to oscillator in zerotemperature. We have assumed strong dissipation effects with enviornment, which means asspin up or spin down electron leaves oscillator it comes to ground state. This rules out accu-mulation of energy in oscillator. When applied bias resonates with (cid:0) ǫ ↑ + ( m + ) + U − ∆ (cid:1) then spin up electron channel too get activated along with spin down electron channel. Here( m )represents number of phonons produced by spin down electron, and moreover spin upand spin down electrons have symmetric coupling with electrodes and QD (cid:0) Γ ↑ = Γ ↓ (cid:1) whichmeans both spin up and spin down electrons comes onto QD and leaves the QD in the sametime, and this is quite resonable as for identical metallic electrodes QD electrodes couplingstrength will be same for both spin up and spin down electrons, This could be changed byusing ferromagnetic leads[52] . Therefore spin up electron excites oscillator to even moreexcited state.So we get satellite peaks in differential conductance of spin up electrons. Herephononic peaks of spin up and spin down electrons is not same. This happens as excitedstates of occupied QD coupled with oscillator eigen states are not same for different valuesof excitation.See Fig(2).Effects of electrodes-QD coupling (Γ s ) in the presence of applied magnetic field andoscillator-QD coupling is of particular importance. As the tunneling rates of spin up andspin down electrons from electrodes to QD is increased then energy states of the QD getsbroadened. In Fig.(3) we have showed that for a fixed value of applied magnetic field andQD-oscillator coupling when electrodes-QD coupling are small than Zeeman splitted peaksand phononic side band peaks are clearly visible. But as we increase electrodes-QD couplingthen phononic side band peaks starts disappearing.Average energy transferred to the oscillator by spin down and spin up electrons is shownin fig(4). Here we could see that spin down electron curve lies lower than spin up curve,where as small steps are signature of phonons creation in averge energy versus applied biasplot. Spin up electron starts contributing to increase average energy of oscillator where spindown electrons ends up, and applied bias resonates with (cid:0) ǫ ↑ + ( m + ) + U − ∆ (cid:1) .11 Applied Bias H meV L D i ff e r e n ti a l C ondu c t a n ce dJ (cid:144) dVdJ ¯ (cid:144) dVdJ (cid:144) dV FIG. 2: Differential conductance plot with respect to applied bias .Here ǫ ↓ ( B ) = 0 . meV , ǫ ↑ ( B ) = 1 meV , U = 0 . meV ,Γ = 0 . meV , α = 0 . meV , ω = 0 . meV . Applied Bias H meV L D i ff e r e n ti a l C ondu c t a n ce G = G = G = FIG. 3: Differential conductance plot with respect to applied bias .Here ǫ ↓ ( B ) = 0 . meV , ǫ ↑ ( B ) = 1 meV , U = 0 . meV , α = 0 . meV , ω = 0 . meV . Applied Bias H meV L A v er ag e E n er gy E ¯ E FIG. 4: Average energy plot with respect to applied bias .Here ǫ ↓ ( B ) = 0 . meV , ǫ ↑ ( B ) = 1 meV , U = 0 . meV , α = 0 . meV , ω = 0 . meV . A. 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