Enhanced photon-assisted spin transport in a quantum dot attached to ferromagnetic leads
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Enhanced photon-assisted spin transport in a quantum dot attached to ferromagneticleads
Fabr´ıcio M. Souza, Thiago L. Carrara, and E. Vernek
Instituto de F´ısica, Universidade Federal de Uberlˆandia, 38400-902, Uberlˆandia, MG, Brazil (Date: June 20, 2018)We investigate real-time dynamics of spin-polarized current in a quantum dot coupled to fer-romagnetic leads in both parallel and antiparallel alignments. While an external bias voltage istaken constant in time, a gate terminal, capacitively coupled to the quantum dot, introduces aperiodic modulation of the dot level. Using non equilibrium Green’s function technique we find thatspin polarized electrons can tunnel through the system via additional photon-assisted transmissionchannels. Owing to a Zeeman splitting of the dot level, it is possible to select a particular spin com-ponent to be photon-transferred from the left to the right terminal, with spin dependent currentpeaks arising at different gate frequencies. The ferromagnetic electrodes enhance or suppress thespin transport depending upon the leads magnetization alignment. The tunnel magnetoresistancealso attains negative values due to a photon-assisted inversion of the spin-valve effect. I. INTRODUCTION
Time-dependent transport in quantum dot system(QDs) has received significant attention due to a vari-ety of new quantum physical phenomena emerging intransient time scale. A few examples encompass chargepump and photon-assisted tunneling transport.
Forinstance, a double dot junction sandwiched by leads canbe used to pump electrons uphill from a lead with lowerchemical potential to a lead with higher chemical poten-tial, in contradiction to the usual dc-regime. This wasachieved by applying a sinusoidal gate voltage amongthe dots. Photon-assisted tunneling can occur when anoscillating gate potential or laser field is applied in aQD or a metallic central island coupled to source anddrain terminals.
Time-dependent regime also leadsto zero-bias charge or spin pumping when a minimumset of two parameters of the system (e.g., gate poten-tial and tunneling rate in a QD system) are time modu-lated independently. This is the case, for instance, of thenon-adiabatic charge and spin pumping through interact-ing quantum dots and quantum pumping in graphene-based structures. Transient charge and spin dynamics in an interactingQD driven by step pulse or sinusoidal gate voltages re-vealed distinct charge and spin relaxation times. Anexquisite behavior that has been predicted theoreticallyis the self-sustained current oscillations in a quantum dotsystem driven out-of-equilibrium by a fast switching onof the bias voltage, contrasting to the expected steadystate behavior. This phenomena has been attributed todynamical Coulomb blockade. It is in the fascinating area of spintronics that time-dependent quantum transport reveals its prolific poten-tiality in producing spin polarized currents. For instance,a double dot structure driven by ac-field in the presenceof magnetic field turn out to be a robust spin filtering andpumping device. By applying oscillating gates (radiofrequency) in an open quantum dot in the presence ofZeeman field, an adiabatic spin pump was generated.
The current ringing that arises in a quantum dot systemwhen a bias voltage is suddenly switched on develops spindependent beats when the dot level is Zeeman split. Coherent quantum beats in the current, spin current andtunnel magnetoresistance of two dots coupled to threeferromagnetic leads were also reported recently. Addi-tionally, spin spikes take place when a bias voltage isabruptly turned off in a system of a QD attached to fer-romagnetic leads.
The study of quantum transport of spin polarized elec-trons in the presence of time varying fields was greatlymotivated by the development of experimental tech-niques. These techniques allow for the coherent control ofthe complete dynamics (initialization-manipulation-readout) of single electron spins in quantum dots. Particu-larly, some of those used to coherently manipulate spinstates are based on time dependent gate voltages.
In the present work we consider a single level quan-tum dot coupled to left and right ferromagnetic leads inthe presence of a static bias voltage and sinusoidal gatevoltage. The oscillating gate potential introduces addi-tional photon-assisted conduction channels that can betunned via a dc-gate field to lie within the conductionwindow of the system. In the presence of Zeeman split-ting, produced by an applied magnetic field, the contribu-tion from the photon-assisted channels becomes differentfor spins up and down, resulting in photon-assisted spinpolarized currents. It is worth mentioning that this effecttakes place even in the absence of ferromagnetic leads.However, when the leads are ferromagnetic and paral-lel aligned, the resonant current peaks are amplified forone spin component and suppressed for the other. Thusthe photon-assisted current-polarization is enhanced. Wealso calculate the tunnel magnetoresistance (TMR) as afunction of the gate frequency, which exhibits a variety ofpeaks and dips, having even a changed of sign, dependingon the gate frequency.The paper is organized as follows: in Sec. II we presentthe theoretical model and describe the formulation basedon nonequilibrium Green’s function technique and inSec. III we show and discuss the numerical results. Fi-
FIG. 1. (color online) Energy diagram for the system con-sidered. A quantum dot is coupled to a left and to a rightferromagnetic electron reservoirs via tunneling barriers. Thedot level is Zeeman split. A capacitively coupled gate terminalintroduces a periodic perturbation of the dot level. This mod-ulation induces additional photon-assisted channels (dashedlines) for spin polarized transport. nally, in Sec. IV we present our concluding remarks.
II. MODEL AND THEORETICALFORMULATION
For concreteness, the energy profile of our system isillustrated in Fig. 1 and is described by the Hamiltonian, H = H L + H R + H D ( t ) + H T , where H L ( R ) = X k σ ǫ k σL ( R ) c † k σL ( R ) c k σL ( R ) , (1)describes the free electrons in the ( L ) or the right ( R )lead, in which c k σL ( R ) [ c † k σL ( R ) ] is the operator that an-nihilates [creates] an electrons in the lead L ( R ) with mo-mentum k , spin σ and energy ǫ k σL ( R ) . We consider astatic source-drain applied voltage ( eV SD = µ L − µ R )which drives the system out of equilibrium, breaking theleft/right symmetry of the Hamiltonian. The time de-pendence of our Hamiltonian is fully accounted via thedot Hamiltonian, H D ( t ) = X σ ǫ σ ( t ) d † σ d σ , (2)where ǫ σ ( t ) = ǫ d ( t ) + σE Z /
2, with ǫ d ( t ) being the time-dependent dot level and E Z a Zeeman splitting of the dotlevel due to an external magnetic field. Here we use σ =+ and σ = − for spins up and down, respectively. The operator d σ ( d † σ ) annihilates (creates) one electron withspin σ and energy ǫ σ ( t ) in the dot. In practice the timedependence in the dot level is controlled by an oscillatinggate voltage V g ( t ), such that ǫ d ( t ) = ǫ + eV g ( t ), where ǫ is the dc component of the energy and eV g ( t ) = ∆cos( ωt )oscillates with amplitude ∆ and frequency ω . Finally H T = X k ση ( V c † k η d σ + V ∗ d † σ c k ση ) , (3)describes the tunnel coupling between the leads and thedot, with a constant coupling strength V and allows forcurrent to flow across the QD.To calculate the time dependent spin polarized currentwe employ the Keldysh Green’s function formalism thatallows for an appropriate approach to our nonequilibriumtime-dependent situation. Starting from the current def-inition I ησ ( t ) = − e h ˙ N σ i = − ie h [ H, N σ ] i , where N σ is thetotal number of particle operator for spin σ (here we take ~ = 1), the current can be written as I ησ ( t ) = 2 e Re (X k V G <σ,kση ( t, t ) ) , (4)where G <σ,kση ( t, t ′ ) = i h c † kση ( t ′ ) d σ ( t ) i . Using the equationof motion technique and taking analytical continuation to obtain G <σ,kση ( t, t ′ ) one finds to the current the follow-ing I ησ ( t ) = − e Γ ση Im n Z dǫ π Z t −∞ dt e − iǫ ( t − t ) × [ G rσσ ( t, t ) f η ( ǫ ) + G <σσ ( t, t )] o , (5)where f η ( ǫ ) is the Fermi distribution function of the η -thlead, and Γ ησ = 2 π | V | ρ ησ gives the tunneling rate betweenlead η and dot for spin component σ . ρ ησ is the densityof states for spin σ in lead η . In the present model weassume constant density of states (wide-band limit). Theferromagnetism of the electrodes is modeled by consid-ering Γ ησ = Γ (1 ± p η ) where + ( − ) stands for spin up(down), p η is the polarization of lead η -th and Γ thetunneling rate strength. The quantity Γ is fixed alongthe paper, so all the other energies will be expressed inunits of Γ . We consider both parallel (P) and antipar-allel (AP) alignments of the lead polarizations. In the Pcase we assume majority down population in both leads,while in the AP configuration we take majority downpopulation in the left lead and majority up populationin the right lead. In terms of the parameters p η we have p L = p R = p = − . p L = − p R = p = − . Taking the time average of the currentwe find h I Lσ ( t ) i = − e Γ Lσ Γ Rσ Γ Lσ + Γ Rσ Z dǫ π [ f L ( ǫ ) − f R ( ǫ )]Im h A σ ( ǫ, t ) i , (6)where h A σ ( ǫ, t ) i = ∞ X n = −∞ J n ( ∆ ω ) g Rn,σ ( ǫ, ω ) , (7) FIG. 2. (color online) Color map of the total transmissioncoefficient T ( ǫ, ω ) = T ↑ ( ǫ, ω ) + T ↓ ( ǫ, ω ) as a function of fre-quency and energy in the parallel alignment ( p L = p R = − . ω , T ( ǫ, ω ) develops additional photon-assisted peaks that allows off-resonant spin transport. Themain two central peaks correspond to the Zeeman split levels ǫ ↑ = ǫ + E Z / ǫ ↓ = ǫ − E Z /
2. The satellite peaks aregiven by ǫ ( n ) ↑ = ǫ + E Z / ± nω and ǫ ( n ) ↓ = ǫ − E Z / ± nω ,with n = 1 , , , ... . For increasing ω , the satellite peaks tendto vanish and the system recovers its original two levels ǫ ↑ and ǫ ↓ . The horizontal dashed lines delimit the conduction win-dow [ µ R , µ L ]. Units: Energy in units of Γ and ω = Γ / ~ .Parameters: ǫ = − , E Z = 4Γ , ∆ = 5Γ , µ L = 1Γ , µ R = 0. with J n being the n-th order Bessel function and g Rn,σ ( ǫ, ω ) = [ ǫ − ǫ σ − nω + i Γ Lσ +Γ Rσ ] − . Here we used thefact that ǫ σ ( t ) = ǫ σ + ∆ cos( ωt ), with ǫ σ = ǫ + σE Z / h I Lσ ( t ) i = e Z dǫ π T σ ( ǫ )[ f L ( ǫ ) − f R ( ǫ )] . (8)Here we define T σ ( ǫ ) = Γ Lσ Γ Rσ ∞ X n = −∞ J n ( ∆ ω )( ǫ − ǫ ( n ) σ ) + ( Γ σ ) , (9)where ǫ ( n ) σ = ǫ σ + nω . Eq. (9) shows that the harmonicmodulation of the dot level yields to photon-assistedpeaks in the transmission coefficient. In addition to this,here we have the spin splitting of these peaks and theferromagnetic leads, that results in an enhanced spinphoton-assisted transport.A further simplification can be made in Eq. (8) byconsidering the low temperature regime, where the Fermi ω/ω -40-20020 E n e r gy l e v e l s ( Γ ) µµ LR n=0n=1n=2n=3 n=-1n=-2n=-3 Zeeman split µ R is fixed Used in Figs.4,5,6 L µ in Fig. 7(a) µµ in Fig. 7(b)in Fig. 7(c) along the figures LL FIG. 3. (color online) Multiplet structure developed in thepresence of an oscillating gate frequency. The black lines cor-respond to spin up while the gray lines to spin down. Thelevels are shifted linearly with the gate frequency, following ǫ ( n ) σ = ǫ σ ± nω , n = 1 , , , ... The up and down levels are Zee-man split. The horizontal dashed lines correspond to the left( µ L ) and to the right ( µ R ) chemical potentials. The channels ǫ ( n ) ↑ and ǫ ( n ) ↓ attain resonance within the conduction window[ µ L , µ R ] for certain frequencies, which differ for each spin com-ponent. Units: Energy levels in units of Γ and ω = Γ / ~ .Parameters: ǫ = − , E Z = 4Γ , ∆ = 5Γ , µ L = 1Γ , µ R = 0. functions are approximated by step functions. In thisregime, the integral in Eq. (8) is carried out in the range[ µ L , µ R ], thus resulting in h I Lσ i = I σ Φ σ , (10)where I σ = e Γ Lσ Γ Rσ Γ Lσ +Γ Rσ is the resonant current without mod-ulated gate voltage andΦ σ = ∞ X n = −∞ J n ( ∆ ω )[Θ σLn ( ω ) − Θ σRn ( ω )] /π, (11)with Θ σηn ( ω ) = arctan[2( µ η − ǫ σ − nω ) / Γ σ ]. In whatfollows we present our numerical results to the spin po-larized transport. III. NUMERICAL RESULTS
Figure 2 shows the sum T = T ↑ + T ↓ as a functionof ω and energy in the case of polarized leads with par-allel magnetizations. As ω increases, a multiplet struc-ture takes place in the transmission coefficient [Eq. (9)].The two central peaks in T ( ǫ, ω ) correspond to ǫ ↑ and ǫ ↓ ,while the lateral peaks are related to ǫ σ ± nω . Due tothe Zeeman splitting, the whole pattern for T ↑ is shiftedupward while T ↓ is moved downward. The highest of thepeaks are strongly affected by the frequency. For the n-thpeak its amplitude is given by 4Γ Lσ Γ Rσ J n (∆ /ω ) / Γ σ . Forsufficiently large ω , the additional photon-assisted peaksare suppressed, remaining only the two central peaks.The broadening difference for up and down spin channelscomes from the ferromagnetism of the electrodes that areparallel aligned, with majority down population in bothsides ( p L = p R = − . T σ ( ǫ, ω ) with respect to the con-duction window. Fig. (3) shows the channels ǫ ↑ + nω and ǫ ↓ + nω for n = 0 , ± , ± , ± ǫ σ + nω attains the conduc-tion window (CW) interval [ µ L , µ R ]. Due to the Zeemansplitting, each spin component crosses µ L or µ R at dif-ferent frequencies, thus resulting in a frequency selectivespin transfer between the leads. In Fig. (3) we indicateby up and down arrows the corresponding crossing of theCW for spins ↑ and ↓ , respectively. In the present studywe focus on the off-resonant regime, where the dot levels ǫ ↑ and ǫ ↓ are below the CW. In this case only photon-assisted electrons can tunnel through the system. In or-der to match this condition we adopt to the numericalparameters the following values: ǫ = − , E Z = 4Γ ,∆ = 5Γ , µ L = 1Γ and µ R = 0. Later on we willalso look at distinct parameters in order to explore therobustness of our main results. In experiments we findtypically Γ ∼ µeV . So to the parameters as-sumed we have E Z ∼ µeV . This Zeeman energysplit is reasonable for semiconductor quantum dots in thepresence of magnetic fields ∼ Additionally, forthese values we find ω = Γ ~ ∼
150 GHz. So the presenttheoretical effects could be observed for gate frequenciesaround 1.5 THz [ ω ∼ ω , see Fig. (4)]. Alterna-tively, if Γ is reduced to ∼ µ eV, we obtain gatefrequencies around ω ∼ ω ∼
15 GHz, which is quitefeasible experimentally. Our currents will be given inunits of I = e Γ / ~ , which is in the range I ∼ ∼ µ eV - 100 µ eV. Since our spin resolvedphoton-assisted currents are typically ∼ − I , we havepA currents, which could be measured with picoamperemeasurement technologies.Comparing Fig. (3) to Fig. (2) one can note thateven though n = 3 and n = 2 attain resonance with-ing [ µ L , µ R ], their corresponding transmission amplitudeare very low, which makes the transport weak via thosechannels. In contrast, the n = 1 channels, for up anddown spins, have a higher transmission amplitude, whichmakes the spin transfer via these channels more appre-ciable.Fig. (4) shows the up and down components of the I σ / Ι I σ / Ι ω/ω I σ / Ι n=1n=1n=2n=2n=3 nonmagnetic leadsFM leads - parallelFM leads - antiparallel (a)(b)(c) FIG. 4. (color online) Spin resolved currents against gate fre-quency for leads (a) nonmagnetic and (b)-(c) ferromagnetic(black lines for spin up and gray lines for spin down). Inpanels (b) and (c) we show the parallel and antiparallel align-ments, respectively. Both up and down currents show peakscorresponding to the crossing of ǫ ( n ) ↑ and ǫ ( n ) ↓ illustrated inFig. (3). The highest peak for each spin component comesfrom the resonance of the levels ǫ (1) ↑ and ǫ (1) ↓ , withing the con-duction window. In the parallel alignment the majority downpopulation in both leads turns into an amplification of thedown current. In the antiparallel case, though, the currentsare very similar to the nonmagnetic case. Units: I = e Γ / ~ and ω = Γ / ~ . Parameters: ǫ = − , E Z = 4Γ ,∆ = 5Γ , µ L = 1Γ , µ R = 0, p = − . current against gate frequency. Three cases are consid-ered: (a) nonmagnetic leads, ferromagnetic leads in the(b) parallel and (c) antiparallel alignments. In all thethree cases two major peaks are found ( n = 1). Satel-lite peaks for high order channels ( n = 2 ,
3) are alsoseen. Each peak emerges whenever a channel ǫ σ + nω enters the conduction window [indicated by ↑ and ↓ ar-rows in Fig. (3)]. Due to the Zeeman splitting, the res-onance for spin up arises in lower frequencies than thatfor spin down. Additionally, the interplay between Zee-man splitting and the amplitude of the transmission co-efficient results into a higher peak for spin up than forspin down in the case of nonmagnetic leads. Furtheramplification of the spin down current peak is observedwhen the leads are made ferromagnetic. In Fig. 4(b)we present I ↑ and I ↓ for leads parallel aligned, with amajority down population in both sides. This meansthat we assume for the polarization parameter a nega-tive value, with p L = p R = p = − .
4. This implies thatΓ
L,R ↓ > Γ L,R ↑ ( ρ L,R ↓ > ρ L,R ↑ ), which favors more the spindown electrons to tunnel through the system, thus in-creasing the spin down current peak. In the antiparallelcase, where we have a majority down population in theleft lead and a majority up population in the right lead(Γ L ↓ > Γ L ↑ and Γ R ↓ < Γ R ↑ for p L = − p R = p = − . p becomes more negative, thespin down tunneling rates Γ L ↓ and Γ R ↓ are strengthened,while Γ L ↑ and Γ R ↑ are diminished in the parallel case.This amplifies the peak for spin down current while sup-presses the peak for spin up, thus making the currentmore down polarized. Eventually, for larger enough | p | the I ↓ current dominates over I ↑ for all gate frequen-cies, Conversely, in the antiparallel configuration, as p be-comes more negative, both I ↑ and I ↓ are suppressed. Onemay note that when we pass from P to AP alignment,the major spin up peak has its width broadened. Thiscan also be seen by calculating the width of this peak:Γ ↑ = Γ L ↑ + Γ R ↑ = 2Γ (1 − | p | ) and Γ ↑ = Γ L ↑ + Γ R ↑ = 2Γ for P and AP, respectively. In particular, this broadeningeffect makes the total antiparallel current slightly higherthan the parallel current ( I P < I AP ), and this resultsinto a negative magnetoresistance, as we will see next.In Fig. (6) we show the total current in both P andAP alignment and the tunnel magnetoresistance, definedaccording to T M R = I P − I AP I AP , (12)where I P/AP = I P/AP ↑ + I P/AP ↓ . The highest currentpeak is related to the resonance of the channel ǫ (1) ↑ withthe CW. The second highest peak is due to spin downresonance, ǫ (1) ↓ ≈ µ R . The spin valve effect is clearlyseen for almost all frequencies, i.e., I P > I AP . However,for frequencies around 10 ω , we observe I P < I AP for p = − . I σ / Ι I σ / Ι ω/ω I σ / Ι (a)(b)(c) p=-0.4p=-0.6p=-0.8 solid line: Pdashed line: AP FIG. 5. (color online) Spin resolved currents against gatefrequency for different polarization p , in both parallel andantiparallel configurations (black lines for spin up and graylines for spin down). In the plots p is negative, which meansthat both leads have majority spin down population in theP case and majority down (up) population in the left (right)lead for the AP case. In the P alignment when p increases (inmodulus) the tunneling rates between the dot and the leadsenlarge for spin down and reduce for spin up. This resultsin an amplification of I ↓ and a suppression of I ↑ as observed.In the AP alignment both I ↑ and I ↓ are suppressed as | p | increases. Units: I = e Γ / ~ and ω = Γ / ~ . Parameters: ǫ = − , E Z = 4Γ , ∆ = 5Γ , µ L = 1Γ , µ R = 0. near the position of the spin up peak the TMR is fullysuppressed, while near to the peak of the spin down cur-rent it is enlarged for all values of p . This indicates thatthe magnetoresistance is mainly dominated by spin downtransport.Since from the experimental point of view the quan-tities V SD , ǫ d and E Z are relative easily tunable ex-perimental parameters, we analyze how these quantitiesaffect the present results, aiming to providing a better I / I I / I ω/ω I / I T M R T M R T M R dashed line: APsolid line: P (a)(b)(c)p=-0.4p=-0.6p=-0.8 TMR<0
FIG. 6. Total current ( I ↑ + I ↓ ) in both P and AP alignmentand TMR as a function of the gate frequency. From panels(a) to (c) the left and right leads polarization are enlarged( p = − . − . − . p = − . I P < I AP ).Units: I = e Γ / ~ and ω = Γ / ~ . Parameters: ǫ = − , E Z = 4Γ , ∆ = 5Γ , µ L = 1Γ , µ R = 0. guide for future experimental realizations. For instance,using the same set of parameters adopted previously[ p L = − . p R = − . p R = +0 . ǫ = − ,∆ = 5Γ , E Z = 4Γ ], in Fig. (7) we show how the spinresolved currents evolve when the source-drain voltageincreases in both parallel and antiparallel configurations.The bias voltages considered are (a) eV SD = 5Γ , (b)10Γ and (c) 15Γ . These values were also indicated bydashed lines in Fig. (3). By comparing Fig. 7(a) to Fig. I σ / Ι I σ / Ι ω/ω I σ / Ι TMRTMRTMR (a)(b)(c)eV SD =5 Γ eV SD =10 Γ eV SD =15 Γ dashed line: APsolid line: P FIG. 7. (color online) Spin resolved currents against gate fre-quency for both parallel and antiparallel configurations (blacklines for spin up and gray lines for spin down). From panels(a) to (c) we increase the bias voltage [(a) eV SD = 5Γ , (b) eV SD = 10Γ , (c) eV SD = 15Γ ], thus enlarging the conduc-tion window [dashed lines in Fig. 3]. Fig. 7(a) is similar toFig. 5(a), except by the width of the peaks that are slightlyenlarged by the CW. When eV SD increases even further [pan-els (b)-(c)] the peaks turn even more broaden. In the insetswe show the TMR for each bias voltage. For eV SD = 5Γ the TMR presents a negative value around ω ≈ . ω . For eV SD = 10Γ and eV SD = 15Γ the TMR becomes positivefor all frequencies. Units: I = e Γ / ~ and ω = Γ / ~ . Pa-rameters: ǫ = − , E Z = 4Γ , ∆ = 5Γ , µ L − µ R = eV SD , p = − . V SD increases. This feature can be understood looking at thedifferent conduction windows (dashed lines) in Fig. (3).As µ L − µ R becomes larger, the frequency range in whichthe photon-assisted channels remains inside the CW be-comes wider. In the insets of Fig. (7) we show the TMRagainst frequency for each V SD considered. Notice thatfor eV SD = 10Γ and eV SD = 15Γ [panels (b) and (c),respectively] the TMR is positive for all frequencies.In order to obtain a clear picture on the set of pa-rameters needed to obtain a negative TMR, in Fig. (8)we plot in a color map of the total transmission coeffi-cient difference, ∆ T = T P − T AP , against gate frequency(horizontal axis) and energy ǫ (vertical axis), for two dif-ferent Zeeman splitting energies: (a) E Z = 4Γ and (b) E Z = 8Γ . Notice the appearance of negative values(dark regions) around the spin up peaks, when the upand down branches are not overlapping. The overlap canbe avoided by controlling the Zeeman splitting. Observethat for E Z = 8Γ [panel (b)] the spin up and spin downbranches are far enough from each other to ensure rela-tively large dark areas. This effect can be understood interms of the broadening of the photon-assisted peaks. Inthe P configuration, the spin down peaks at the transmis-sion coefficient are broader than the spin up ones, as canbe seen by the rates Γ ↓ = Γ L ↓ + Γ R ↓ = 2Γ (1 + | p | ) = 2 . and Γ ↑ = 2Γ (1 − | p | ) = 1 . ( | p | = 0 . E Z = 0) thespin down peaks lie above the spin up ones, thus mak-ing the total transmission coefficient dominated by thespin down component. When the magnetic alignment isrotated from P to AP, the width of the spin up peak in-creases and spin down diminishes, becoming both 2Γ .This facilitates T AP ↑ > T P ↑ and T AP ↓ < T P ↓ nearby eachtransmission peak. If the peaks are far enough from eachother ( E Z = 0) it is possible to obtain T = T ↑ + T ↓ ≈ T ↑ around the up peaks, leading to T AP ≈ T AP ↑ > T P ≈ T P ↑ [see Fig. 9(c) for clarity]. This is the main condition fora negative TMR. For instance, by tunning the set ofparameters such that the CW lies on the darker area ofthe map, the TMR becomes negative. By increasing theCW, we can eventually cover an energy range with morebright than dark areas, resulting in a positive TMR.In Fig. (9) we show the current vs. ω obtained forthe conduction window set as in Fig. 8(b)[dashed (green)lines] Since the CW is dominated by a dark region( T P < T AP ) we expect T M R <
0. The spin up and spindown currents in Fig. 9(a) reveal contrasting behavior.While the spin down current shows the same behavioralready seen in previous results [e.g., Fig. (5)], the spinup current oscillates and then it increases to a highersaturation value. This higher value of I ↑ in the large fre-quency limit can be understood by looking at the trans-mission coefficient in panels 9(b) and (c). Spin up andspin down transmissions coefficients are drawn in both Pand AP configurations. The vertical thin solid lines givethe border of the CW. Comparing the amplitude of T ↑ inside the CW for both ω ≪ ω and ω ≫ ω , we observethat T ↑ becomes amplified for larger frequencies, whichresults into higher spin up current in this limit. Con-versely, the spin down transmission coefficient is slightlyhigher in the CW for ω ≪ ω , resulting in a spin downcurrent a bit higher in the low frequency limit, comparedto its value in the high frequency regime.To better understand the oscillatory structure found FIG. 8. (color online) Two-dimensional map of the difference∆ T = T P − T AP against frequency ω and ǫ for two differentZeeman splitting: (a) E Z = 4Γ and (b) E Z = 8Γ . The darkregions indicate T AP > T P which results in negative TMR.As E Z increases [from panel (a) to (b)] the dark ares enlarge.This is so because the spin up and down channels becomeapart from each other, allowing to have T AP = T AP ↑ + T AP ↓ ≈ T AP ↑ > T P = T P ↑ + T P ↓ ≈ T P ↑ (broadening effect) around thespin up channels. By playing with the set of parameters onecan place the conduction window of the system in the darkerregion of the map, which results into a more negative TMR.Units: I = e Γ / ~ and ω = Γ / ~ . Parameters: ǫ = − . ,∆ = 5Γ , µ L = 1Γ , µ R = 0, p = − . in the spin up current, we plot in Fig. 9(d) the spin upparallel transmission coefficient T P ↑ vs. energy for various ω . As ω increases the transmission coefficient developsa variety of peaks mainly in the range between ǫ ↑ − and ǫ ↑ + 5Γ ( ǫ ↑ ± ∆). Since this energy interval containsthe CW, some of the photon-assisted peaks that arise forincreasing ω appear within the CW and eventually lieoutside it for large enough ω , followed by a new peak ω/ω I σ / I -15 -10 -5 0 5 10 15 Energy ( Γ ) T r a n s m i ss i on C o e ff i c i e n t -15 -10 -5 0 5 10 15 Energy ( Γ ) -15 -10 -5 0 5 10 15 Energy ( Γ ) T r a n s m i ss i on C o e ff i c i e n t -15 -10 -5 0 5 10 15 Energy ( Γ ) ω=2.5 =2.1 =2.1=1.7 =1.7=1.3 =1.3=0.9 =0.9=0.5 =0.5=0.1 =0.1 (a) ω=2.5 (b) (c)(d) (e) TMR<0 ω<<ω ω>>ω solid line: Pdashed line: APCW T AP >T P spin up spindown FIG. 9. (color online) (a) Spin resolved currents in both Pand AP alignments against gate frequency for the conductionwindow drawn in Fig. 8(b). Similar to previous figures, thespin down component reveals peaks at some particular fre-quencies due to the matches ǫ ( n ) σ = µ R . In contrast, the spinup current oscillates and then it tends to a relatively large sat-uration value for large ω . In the inset of Fig. 9(a) we show theTMR which acquires negative values in accordance to the CWin Fig. 8(b). Panels (b)-(c) present the spin resolved trans-mission coefficient for (b) small and (c) large ω in both P andAP configurations. The thin vertical lines denote the borderof the CW. In particular, observe in panel (c) that inside theCW we have T AP ↑ > T P ↑ . Since T P/AP ↓ is too small at the CW,the total transmission coefficient T P/AP is given essentially by T P/AP ↑ , so T AP = T AP ↑ + T AP ↓ ≈ T AP ↑ > T P = T P ↑ + T P ↓ ≈ T P ↑ around the peak. This results in a negative TMR. In pan-els (d)-(e) we plot (d) T P ↑ and (e) T P ↓ for different valuesof ω . The curves were vertically displaced for clarity. As ω increases additional peaks emerge mainly in the interval ǫ σ ± ∆ = ǫ σ ± . More specifically, in the spin up case theappearance and suppression of these additional peaks insidethe CW gives rise to the oscillatory patter of I ↑ in the lowfrequency limit. Units: I = e Γ / ~ and ω = Γ / ~ . Parame-ters: ǫ = − . , E Z = 8Γ , ∆ = 5Γ , µ L = 1Γ , µ R = 0, p = − . emerging inside it. This results in an oscillatory patternof the current. In panel 9(e) we show T P ↓ as function ofenergy for the same values of ω as in panel (d). Since ǫ ↓ + ∆ < µ R the photon-assisted peaks can only crossthe CW for particular frequencies, which gives rise tothe peaks observed. Finally, in the inset of Fig. 9(a) weplot the TMR, which presents very low negative values( ∼ − IV. CONCLUSION
We have studied spin polarized transport in a quantumdot attached to ferromagnetic leads in the presence of anoscillating gate voltage V g ( t ). A static source-drain biasvoltage is also applied in order to generate current. Theoscillating V g ( t ) gives rise to photon-assisted transportchannels that allow electrons to flow through the system.Due to a Zeeman splitting of the dot level, the photon-assisted contributions to the transport are distinct forspins up and down, providing an interesting way to ob-tain current polarization that can be controlled by gatefrequency. As the leads polarization is enlarged, with amajority down population in both leads (P alignment),the spin down photon-assisted current peak is enhanced,while the spin up peak is suppressed. Moreover, when therelative polarization alignment of the leads is switchedfrom P to AP, the width of the main spin up peak of thecurrent is broadened. This additional broadening effectresults in an opposite spin valve behavior ( I P < I AP )for gate frequencies around the spin up resonance. Asa result, a photon-assisted negative tunnel magnetoresis-tance is found. ACKNOWLEDGMENTS
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