Limit-Cycles and Chaos in the Current Through a Quantum Dot
Carlos López-Monís, Clive Emary, Gerold Kiesslich, Gloria Platero, Tobias Brandes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Limit-Cycles and Chaos in the Current Through a Quantum Dot
Carlos L´opez-Mon´ıs , Clive Emary , Gerold Kiesslich , Gloria Platero , and Tobias Brandes Instituto de Ciencia de Materiales de Madrid, CSIC,Sor Juana In´es de la Cruz 3, Cantoblanco, 28049 Madrid, Spain Institute f¨ur Theoretische Physik, TU Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany (Dated: 16 December 2011)We investigate non-linear magneto-transport through a single level quantum dot coupled toferromagnetic leads, where the electron spin is coupled to a large, external (pseudo-)spin via ananisotropic exchange interaction. We find regimes where the average current through the dot dis-plays self-sustained oscillations that reflect the limit-cycles and chaos and map the dependence ofthis behaviour on magnetic field strength and the tunnel coupling to the external leads.
I. INTRODUCTION
Single-electron transport through nanostructures hasdeveloped into a powerful spectroscopic tool for probingcorrelations, quantum coherence, and interactions withthe environment on a microscopic level . Some re-cent examples include experiments with semiconductorquantum dots that have provided detailed insight intolevel structures , Coulomb and spin-blockade effects ,phonon emission , or the statistics of individual electrontunnel events .In this paper, we propose the time-dependent, av-erage current of electrons through a single level quan-tum dot as probe for classical non-linear dynamics andchaos . Specifically, we consider electronic magneto-transport through a quantum dot containing two spin-split levels with an anisotropic coupling between the elec-tron spin and an external, classical magnetic momentor pseudo-spin. In order to have a spin-polarized cur-rent through the quantum dot, we consider ferromagneticleads (see Ref. [11] and references there in).Previous works have analyzed the anisotropic inter-action between two spins in a closed system under anexternal magnetic field , showing either regular (in-tegrable) or chaotic (nonintegrable) classical orbits. Theresults presented here demonstrate that the signatures ofnon-linear dynamics and classical chaos of the closed sys-tem also persist in the non-equilibrium regime, where theadditional coupling to the electronic reservoirs leads to aneven richer dynamics that can be probed, e.g., by vary-ing the magnetic field and the tunnel rates. In particular,one finds a transition from a regime with damped currenttransients and a constant current, to a situation wherethe current displays self-sustained regular limit-cycle os-cillations, or chaotic behaviour. Limit-cycles in transporthave also been found recently in theoretical calculationsin mesoscopic systems coupled to mechanical degrees offreedom .Experimental inspiration for our model comes from thehyperfine interaction in quantum dots. The interaction ofelectron spins in quantum dots with surrounding nuclearspins is usually viewed as simply giving rise to spin re-laxation and decoherence . Recently, however, trans-port experiments through semiconductor double quan- tum dots have shown non-linear current behaviour whichhas been attributed to hyperfine interaction inducing adynamical nuclear spin polarization . The feedbackbetween electron and nuclei spin polarization gives rise tonontrivial features in the current, including self-sustainedoscillations . In this setting, the large spin of our modelrepresents an effective description of the collective nu-clear spin system and the electronic part provides aminimal model for investigating the effects on transportof coupled spin-spin dynamics.A further potential realization of the large spin in ourmodel is a magnetic impurity in a quantum dot. Severalrecent works have considered the influence of such an im-purity on the transport properties through the dot .In this context, our model can be viewed as the large-spin counterpart of the previously studied models andin particular the spin-1/2 impurity model of Refs. [26]and [28]. This possibility is also closely related to trans-port through single molecular magnets , for whichour large spin would map to a magnetic atom and theisolated levels of our quantum dot to molecular orbitals.We mention that our study of classical chaos in a quan-tum dot with coupling to an external pseudo-spin is alsocomplementary to previous studies of intrinsic quantum chaos of, e.g., ballistic quantum dots. Those latter sys-tems are often analyzed with statistical tools such as ran-dom matrix theory .The outline of this paper is as follows. In Section II,we introduce the model Hamiltonian and the equations ofmotion. Section III presents results and a classificationof various non-linear regimes in the form of a map inparameter space, and we conclude with a brief discussionof the experimental relevance of our finding in SectionIV. II. MODELA. Hamiltonian
We investigate a quantum dot (QD) with a single or-bital level, coupled to an emitter (left electron lead),a collector (right electron lead) and to a large spin ˆ J (Fig. 1a). An external magnetic field B z is applied in z -direction which splits the QD spin levels (Fig. 1b). TheHamiltonian for this model isˆ H = ˆ H F A + ˆ H J + ˆ V . (1)Here, ˆ H F A is the Fano-Anderson model for the QD cou-pled to the leads, which is exactly solvable; ˆ H J is theHamiltonian for the free motion of the large spin due theexternal magnetic field; and ˆ V is the coupling between adot electron and the large spin. These individual Hamil-tonians read:ˆ H F A = X σ ǫ d ˆ d † σ ˆ d σ + B z ˆ S z + X lkσ ǫ lkσ ˆ c † lkσ ˆ c lkσ + X lkσ (cid:16) γ lk ˆ c † lkσ ˆ d σ + h.c. (cid:17) , (2a)ˆ H J = B z ˆ J z , (2b)ˆ V = X i = x,y,z λ i ˆ S i ˆ J i , (2c)where ǫ d is the energy of the QD level, ˆ d † σ / ˆ d σ cre-ates/annihilates a spin- σ electron in the dot, ˆ S i is the i -th component of the electron spin operator in secondquantization, ˆ J i is the i -th component of large spin op-erator, and λ i is the coupling between the i -th compo-nents of the electron and the large spin, ˆ c † lkσ /ˆ c lkσ cre-ates/annihilates an electron with momentum k and spin σ in lead l ∈ { L, R } , and γ lk is the coupling between theQD and the l -th lead. Coulomb interaction in the QDis neglected, and thus double occupation is allowed. Theflip-flop processes due to the spin-spin interaction are theorigin of the non-trivial dynamics that will be shown inthe next section. Much of the interesting dynamics foundoccurs at low magnetic fields, in particular, in a regimewhere the coupling between the electron and the largespin dominates Zeeman splittings ( B z ≪ λ ). Thus, inthis regime we believe that different g -factors will not bequalitatively important, meaning the energy mismatchbetween the Zeeman splittings will not lead to suppres-sion of the flip-flop processes. Therefore, for simplicity,we assume identical g -factors for the electron spin andthe large spin, and absorb the Bohr magneton and the g -factors into the definition of B z .The classical counterpart of the closed system ( γ lk = 0)is, for zero external magnetic field ( B z = 0), a completelyintegrable system for arbitrary λ i , while the isotropicmodel ( λ x = λ y = λ z ) is also completely integrable forfinite external magnetic fields. However, in presence of afinite magnetic field, an anisotropic coupling between theelectron spin and the large spin, makes the model non-integrable and can lead to a chaotic spin dynamics .Therefore, in this work we take the coupling between theelectron spin and the large spin to be anisotropic, andfor simplicity we will focus on the choice λ x = λ z = λ, λ y = 0 . (3)Finally, the spin-dependent rates of the contact barriersare chosen so that only spin-up electrons can tunnel out Γ L Γ R J SL R λ (a)(b) FIG. 1. (Colour online.) Scheme and setup of the investigatedsystem. a) An electron spin ˆ S (blue arrow) in a QD is cou-pled via the exchange interaction λ with a large spin ˆ J (redarrow). The QD is attached to ferromagnetic electron reser-voirs (brown regions), allowing electrons to tunnel throughthe QD. The large spin is isolated. b) The spin-dependentrates of the contact barriers are chosen so that a spin downelectron is always trapped in the QD, while spin up electronscan tunnel through it (see details in the text). The large spininteracts with the spin of the electron trapped in the QD, al-lowing its spin to flip and, hence, escape form the QD intothe right lead. of the QD (Fig. 1b), and B z ≫ k B T where T is the tem-perature of the leads and k B is Boltzmann’s constant. Inthis regime, current can flow only through the spin-uplevel of the QD. When an electron enters the spin-downlevel, it remains trapped until a spin-flip process (dueto the interaction with the large spin) produces a transi-tion from the spin-down to the spin-up level, allowing thetrapped electron to escape the QD. Notice that becausewe have taken identical g -factors for both the electronspin and the large spin, the spin-flip transition from theQD spin-down to the spin-up level conserves energy andthe energy that the electron absorbs in the spin-flip isemitted by the large spin. B. Equations of Motion
The equation of motion (EOM) for the expectationvalue of an operator ˆ A is ddt h ˆ A i = 1 i ~ h [ ˆ A, ˆ H ] i + * ∂ ˆ A∂t + . (4)Using this formula, we derive the EOM of each operatorin Eq. (2a), Eq. (2b) and Eq. (2c).We first observe that the length of the large spin j = | ˆ J | is a conserved quantity since [ˆ J , ˆ H ] = 0. Next, due tothe interaction ˆ V , the EOMs do not close and lead to aninfinite hierarchy of equations that needs to be truncated.In order to do so, we use a factorization approximationby invoking a mean-field approximation for ˆ V → ˆ V MF ,ˆ V MF = X i = x,y,x λ i (cid:16) ˆ S i h ˆ J i i + ˆ J i h ˆ S i i − h ˆ S i ih ˆ J i i (cid:17) (5)which neglects the term δ ˆ S i δ ˆ J i with δ ˆ S i = ˆ S i − h ˆ S i i and δ ˆ J i = ˆ J i − h ˆ J i i , i.e. the quantum fluctuations of theelectron spin and the external spin. We expect this to bea good approximation when j ≫ J can essentially be treated as a classical object due to itsinteraction with other environmental degrees of freedom.Furthermore, as in the semiclassical approximation weneglect quantum fluctuations of the large spin, we haveno spin decay, meaning the large spin is a constant ofmotion.We furthermore neglect terms proportional to γ lk λ i ,namely, second order transitions due to the coupling ofthe large spin with the contacts. This is a good approxi-mation in the infinite bias regime. For the electron leads,we perform the usual Born-Markov and flat band approx-imations and consider them to be in thermal equilibrium.Moreover, we consider the infinite bias regime, namely µ L → ∞ and µ R → −∞ , respectively (see Appendix Aand B for details). The resulting EOMs read ddt h ˆ n σ i = λ h ˆ J x ih ˆ S y i ( δ σ ↑ − δ σ ↓ ) − Γ h ˆ n σ i + Γ Lσ ddt h ˆ S x i = − (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ S y i − Γ h ˆ S x i ddt h ˆ S y i = − λ h ˆ J x ih ˆ S z i + (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ S x i − Γ h ˆ S y i ddt h ˆ S z i = λ h ˆ J x ih ˆ S y i − Γ h ˆ S z i + 12 (Γ L ↑ − Γ L ↓ ) ddt h ˆ J x i = − (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J y i ddt h ˆ J y i = − λ h ˆ S x ih ˆ J z i + (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J x i ddt h ˆ J z i = λ h ˆ S x ih ˆ J y i , (6)where ˆ n σ = ˆ d † σ ˆ d σ , Γ σ = Γ Lσ + Γ Rσ with σ = ↑ , ↓ , andΓ Lσ and Γ Rσ are the tunnelling rates through the leftand right contact barriers, respectively. We have takenΓ ↑ = Γ ↓ = Γ for simplicity. In order to have current onlythrough the spin-up level we take Γ R ↓ = 0. Therefore,spin-up electrons are allowed to tunnel through the QD,whereas spin-down electrons become trapped in it.The EOM for the total number of electrons in the QD( ˆ N = ˆ n ↑ + ˆ n ↓ ) is independent of both the electron andthe large spin components and is exactly solvable (seeAppendix A). Thus, as 2 ˆ S z = ˆ n ↑ − ˆ n ↓ , the level occupa- PSfrag replacements B z /λ Γ / λ DampedDampedand self-sustained Self-sustainedoscillationsoscillationsoscillations(I) (II) (III) (a)10203040 0 . . . . PSfragreplacements B z /λ Γ / λ (b)0 .
05 0 .
10 0 .
15 0 .
20 0 . . . . . . . FIG. 2. a) Parameter space with three regions describing thebehaviour of solutions of EOMs (Eq. (6)). Boundaries be-tween the regions are obtained analytically from Eq. (14).Region I: damped oscillations; region II: both damped andself-sustained oscillations; and region III: self-sustained os-cillations only. b) Numerically-obtained small-Γ region inthe mixed region II. In the dark-coloured region, dampedoscillations are obtained. In the light-colored region: self-sustained oscillations. Initial conditions: h ˆ S x i t =0 = 1 / h ˆ S y i t =0 = h ˆ S z i t =0 = 0, h ˆ J x i t =0 = h ˆ J y i t =0 = j/ h ˆ J x i t =0 = j/ √ tions can be obtained through the following expression: h ˆ n σ ( t ) i = 12 (cid:18) h ˆ N (0) i e − Γ t + Γ L ↑ + Γ L ↓ Γ (cid:0) − e − Γ t (cid:1)(cid:19) + ( δ σ ↑ − δ σ ↓ ) h ˆ S z ( t ) i (7)which relates the level occupation with the z -componentof the electron spin. If the coupling between the elec-tron and the large spins is isotropic ( λ x = λ y = λ z ), itis straightforward to see that in the stationary limit thespins decouple, and the well known Fano-Anderson solu-tion is obtained (see Appendix C). In contrast, we showbelow that the situation is drastically different for theanisotropic case where the stationary solutions for theEOMs depend on the coupling between the spins.The average electron current h ˆ I i through the QD issolely due to a decay at rate Γ R ↑ from the spin-up QDlevel into the right lead, h ˆ I ( t ) i = e Γ R ↑ h ˆ n ↑ ( t ) i , (8)where e denotes the electron charge. In the long-timelimit of the current can be written as (see Eq. (7)) h ˆ I ( t ) i e Γ R ↑ = 12 Γ L ↑ + Γ L ↓ Γ + h ˆ S z ( t ) i . (9)Henceforth, for convenience we take Γ L ↑ = Γ R ↑ = Γ / III. REGIONS IN PARAMETER SPACE
The stationary solutions of the EOMs, Eq. (6), can beobtained analytically and we find eight fixed points. Twoof these fixed points, however, always have a finite imag-inary component, and as they have no physical meaning,we leave them out of the subsequent analysis. The re-maining fixed points serve to divide the parameter spaceof the model into distinct regions, as shown in Fig. 2.Introducing the notation P = (cid:16) h ˆ S x i , h ˆ S y i , h ˆ S z i , h ˆ J x i , h ˆ J y i , h ˆ J z i (cid:17) , (10)the six relevant fixed points are P ± = (cid:18) , , − , , , ± j (cid:19) , (11a) P II , ± = (cid:18) , B , − B z λ , Γ B z B , ±B , − B z λ (cid:19) , (11b) P II , ± = (cid:18) , −B , − B z λ , − Γ B z B , ±B , − B z λ (cid:19) , (11c)where B = − s j − (cid:18) λ B z − (cid:19) (cid:18) Γ λ (cid:19) − (cid:18) B z λ (cid:19) , (12a) B = − s B z λ (cid:18) − B z λ (cid:19) . (12b)For certain values of B z , Γ and λ , the quantities B and B ((12a) and Eq. (12b)) can have finite imaginary com-ponents and therefore, points P II , ± and P II , ± only havephysical meaning in the region of parameter space where B and B are real. Fig. 2a shows a projection of the3-dimensional parameter space on the Γ versus B z planefor a fixed λ . This diagram is divided in three regions.In region I, B is a pure imaginary number, and hence, P II , ± and P II , ± are nonphysical, and P ± the only phys-ical fixed points. In region II, B and B are both real,and all six fixed points are physical. In region III, B is purely imaginary, and again P ± are the only physicalfixed points. Points P ± are thus physical solutions forthe EOMs Eq. (6) in all three regions, whereas the fixedpoints P II , ± are physical only in region II. The boundariesbetween the regions are obtained by solving the equations B = 0 and B = 0, namely B = 0 ⇒ Γ = s j − ( B z /λ ) / − B z /λ B z λ , B = 0 ⇒ B z λ = 14 , (13)and these two equations give the lines plotted in Fig. 2a. A. Region I
In order to obtain the time evolution of the electronand large spin components and the electronic currentthrough the QD, the EOMs Eq. (6) are solved numeri-cally. Fig. 3 shows the time evolution of the electron spinand the large spin components, and the current throughthe QD in region I of the parameter space. All exhibit
PSfrag replacements h I i / e Γ R ↑ h ˆ J i i / j h ˆ S i i λt xxyyzz (a)(b)(c)30002500200015001000500 − − . . . − . − . . . FIG. 3. (Colour online.) Time evolution in region I of a)the electron spin components, b) the large spin components,and c) the current through the QD obtained by solving nu-merically the EOMs of Eq. (6). In this region, the solutionsexhibit a slow damped behaviour. In the stationary limit, thelarge spin is completely polarized in the direction parallel tothe external magnetic field, and a QD electron trapped in thespin-down state. Current is due only to tunnelling throughthe spin up level and in the stationary limit tends to a con-stant value of 5 /
8. The parameters here are B z /λ = 0 . /λ = 9, with initial conditions h ˆ S x i t =0 = 1 / h ˆ S y i t =0 = h ˆ S z i t =0 = 0, h ˆ J x i t =0 = h ˆ J y i t =0 = (5 / √ √ − / h ˆ J z i t =0 = (5 / √ p √ completely damped oscillations. In the previous discus-sion, we have seen that in region I, P ± Eq. (11a) are theonly physical fixed points. Depending on the choice of pa-rameters and initial conditions, the system will evolve to P + or P − . For the parameters and initial conditions cho-sen in Fig. 3, the system evolves towards the fixed point P + . In this case, the large spin becomes completely po-larized in the direction parallel to the external magneticfield (Fig. 3b), and a spin-down electron remains trappedin the QD (Fig. 3a) and the interaction between the elec-tron spin and the large spin is no longer effective. Spin-up electrons, however, can still tunnel through the QD(Fig. 3c), and in the stationary limit the current becomes(see Eq. (9)) h ˆ I ( t ) i e Γ R ↑ = 12 . (14)In region I, then, the coupling of the two spin systemswith the external leads results in complete damping ofthe transient oscillations of the electron and the largespin components and the current. A finite, fully spin-polarized electron current flows through the QD that inthe stationary limit is not influenced by the interactionwith the large spin. B. Region II
In region II, the EOMs Eq. (6) exhibit both dampedand self-sustained oscillatory solutions, depending on thechoice of parameters and initial conditions. Fig. 2b showsthe part of region II where the self-sustained oscillationsare found. This behaviour can be seen for all intensi-ties of the external magnetic field in region II, but onlyfor small values of coupling Γ with the leads. Compar-ing Fig. 2a and Fig. 2b we see that most values for B z and Γ in region II lead to damped oscillations. Further-more, although we have given analytical expressions forthe boundaries between the different regions Eq. (13),we have not found an expression for the boundary be-tween the regions inside region II where self-sustainedand damped oscillations are found. Fig. 2b has been ob-tained by solving the EOMs Eq. (6) in region II. As canbe seen, the boundary between both regions is fuzzy incontrast with the ones obtained between regions I, II andIII Eq. (13). Moreover, Fig. 2b shows small “islands” inthe oscillatory region, where damped solutions are ob-tained.
1. Damped Oscillations
Fig. 4a, Fig. 4b and Fig. 4c, show the time evolution ofthe electron and the large spin components, and the cur-rent in region II with parameters B z and Γ such that theyall exhibit damped oscillations. Previously we have seenthat in region II all the six fixed points are physical. Forthe parameters and initial conditions chosen in Fig. 4a,Fig. 4b and Fig. 4c, the system evolves towards the fixedpoint P II , . The large spin becomes almost completelypolarized in the y -direction (Fig. 4b), perpendicular tothe external magnetic field. and the current becomes(see Eq. (9)) h ˆ I i e Γ R ↑ = 34 − B z λ . (15)Thus, the stationary current increases if either the exter-nal magnetic field decreases or the coupling between thespins increases. Since in region II B z /λ < /
4, the cou-pling between the electron and the large spins enhancesthe current through the QD, compared with the currentobtained in region I (Eq. (14)). Nevertheless, the result ofcoupling the two spins to the leads stills yields completedamping of both spin oscillations, as in region I.
2. Self-Sustained Oscillations and Chaos
We shall now focus on the small region in regionII where self-sustained oscillatory solutions are found (Fig. 2b). Fig. 4d, Fig. 4e and Fig. 4f, show the timeevolution of the electron and the large spin components,and the current through the QD. The chosen values of B z and Γ lead to complicated, but periodic, undampedoscillations. Fig. 5a shows the Fourier spectrum of thecurrent time evolution of Fig. 4f in the long-time limit.The spectrum exhibits peaks at well defined frequencies,which clearly confirms the periodic behaviour of the cur-rent. Furthermore, in non-linear systems, self-sustainedoscillations are a signature of limit-cycles and in Fig. 5aand Fig. 5b we plot the electron and the large spin tra-jectories in phase space, projected on the x - z plane, inthe long-time limit. These figures show that the spintrajectories are precisely limit-cycles. For all the ini-tial conditions chosen, the system always converges tothem. Finally, Figs. 4g, 4h and 4i show that decreasingΓ turns the periodic self-sustained oscillations chaotic. Inthis case, the Fourier spectrum of the current, shown inFig. 5b, is uniformly distributed through all frequencies,which is a clear signature of chaos. Fig. 5c and Fig. 5dshow the electron and large spin trajectories in the long-time limit, where it can be seen that they perform com-plicated non-periodic paths. In this area of region II,the coupling between the interacting spins and the leadsdoes not produce damping of the spins as in the previouscases. Moreover, the electron current through the QDcaptures the complicated dynamics due to the interac-tion between the electron and the large spin, as seen inFig. 4f and Fig. 4i. C. Region III
Fig. 8a, Fig. 8b and Fig. 8c show the time evolution ofthe spin components and current for typical parametersin region III. They all exhibit periodic self-sustained oscil-lations. Fig. 8 shows the different limit-cycles performedby the electron spin in phase space, projected in the y - z plain, when the value of the external magnetic field is in-creased. The trajectories found for the large spin in thelong-time limit suggest that this behaviour can be under-stood by means of an effective model in which the largespin simply acts on the QD electrons as an ac magneticfield in x -direction with amplitude B ac ( t ) = λj √ B z t ) − sin( B z t )) . (16)The EOMs for this effective model are (see Appendix Dfor details) ddt h ˆ S x i = − B z h ˆ S y i − Γ h ˆ S x i ddt h ˆ S y i = B z h ˆ S x i − B ac ( t ) h ˆ S z i − Γ h ˆ S y i ddt h ˆ S z i = B ac ( t ) h ˆ S y i − Γ h ˆ S z i − Γ4 . (17)Thus, in this region the six autonomous non-linear equa-tions Eq. (6) can be approximated by a set of three non-autonomous linear equations Eq. (17). The agreement PSfrag replacements h I i / e Γ R ↑ h ˆ J i i / j h ˆ S i i λtλtλtxyz (a)(b)(c) (d)(e)(f) (g)(h)(i) 500500490480470460450 4003002001005040302010 − − . . . − . − . . . . . FIG. 4. Results for three different parameter sets are shown, each of which gives rise to very different system behaviour. Figuresa), b), c) show fast damping behaviour ( B z /λ = 0 .
2, Γ /λ = 0 .
7, dark region in Fig. 2b). In the long-time limit the large spin isalmost completely polarized in the direction perpendicular to the external field, but unlike in region I, the spin down electron canescape from the QD into the right lead due to the interaction with the large spin. Figures d), e), f) show periodic self-sustainedoscillations ( B z /λ = 0 .
1, Γ /λ = 0 .
16, light region in Fig. 2b), which is a signature of limit-cycles in phase space (see Fig. 5).Figures g), h), i) show chaotic self-sustained oscillations ( B z /λ = 0 .
1, Γ /λ = 0 . h ˆ S x i t =0 = 1 / h ˆ S y i t =0 = h ˆ S z i t =0 = 0, h ˆ J x i t =0 = h ˆ J y i t =0 = (5 / √ √ − / h ˆ J z i t =0 = (5 / √ p √ between the solutions obtained with this effective modeland the full EOMs is very good.In region III, the coupling between the two spin systemleads to self-sustained oscillations which are visible in theelectron current through the QD, as shown in Fig. 8c. IV. CONCLUSIONS
We have studied electron transport through a quan-tum dot coupled to ferromagnetic leads, in which theelectron spin interacts with a large spin while an exter-nal magnetic field is applied. We have found that themotion of the electron spin, the large spin and the cur-rent through the QD strongly depend on the coupling be-tween spins. When the electron spin and the large spinare isotropically coupled, the large spin becomes com-pletely polarized and decouples from the electron spin.Conversely, when the electron spin and the large spin areanisotropicaly coupled, we have found that their motionand the current through the QD can either behave asin the isotropic case or show self-sustained oscillations which, furthermore, can be periodic or chaotic. Switch-ing between different behaviours can be obtained by vary-ing either the strength coupling with the leads or theintensity of the external magnetic field.We foresee two possible experimental realisations ofthe large spin of our model. The first is as an effectivemodel on a hyperfine bath. Here a semi-classical treat-ment may be justified by considering that the number ofnuclei spins in semiconductor QDs interacting with anelectron spin is very large (e.g., for GaAs QDs there aretypically 10 -10 nuclei spins). Situations in which thehyperfine interaction is anisotropic have been discussedin Refs. [35–37]. The second realisation is that our largespin represents the spin of a magnetic impurity of a dopedsemiconductor or a magnetic atom in a single molecularmagnet. While in this case the spin may not be so large,mean-field analyses such as pursued here can still provideuseful information, e.g. Ref. [38].From the theoretical point of view, it would be inter-esting to investigate how the features of this semiclassicaltreatment are reflected in a quantum master equation ap-proach, in which the electron and the large spin are both PSfrag replacements h I i ν / e Γ R ↑ h I i ν / e Γ R ↑ ν/λ (a)(b)0 . . . . . . . . . . . . . . . . . FIG. 5. Fourier spectra of the non-damped current time evo-lutions shown in Fig. 4 in the long-time limit. Figs. a) andb) show the Fourier transform of Fig 4f and 4i, respectively,where ν is the frequency. Figure a) shows peaks at well de-fined frequencies, meaning that behaviour of the current isperiodic. However, figure b) shows a uniform frequency dis-tribution, which is a signature of chaotic dynamics. PSfrag replacements h ˆ S z i h ˆ S z i h ˆ J z i / j h ˆ J z i / j h ˆ S x i h ˆ S y i h ˆ J x i /j h ˆ J x i /j (a) (b)(c)(d) − . − . − . − . − . − . − . − . − . − . − . − . − . . . . . . . . . . . . . . − FIG. 6. Electron spin (left figures) and large spin (right)trajectories projected on a two dimensional plane for the non-damped solutions in region II (light region in Fig. 2a). Figuresa) and b) show the formation of a limit-cycle as seen in thetime evolution plots, Fig. 4d, Fig. 4e, and Fig. 4f ( B z /λ =0 .
1, Γ /λ = 0 . B z /λ = 0 .
1, Γ /λ = 0 . treated as quantum objects. This opens a path to inves-tigate the quantum/classical divide in a nonequillibriumcontext. PSfrag replacements h I i / e Γ R ↑ h ˆ J i i / j h ˆ S i i λt xyz (a)(b)(c) 5040302010 − − . . . − . − . . . FIG. 7. In this region the solutions exhibit periodic self-sustained oscillations, which are reflected in the current. Thecorresponding limit-cycles are shown in Fig. 7). The pa-rameters chosen here are B z /λ = 1 .
0, Γ /λ = 10. The ini-tial conditions are h ˆ S y i t =0 = 1 / h ˆ S x i t =0 = h ˆ S z i t =0 = 0, h ˆ J x i t =0 = h ˆ J y i t =0 = 3 p / h ˆ J z i t =0 = − PSfrag replacements h ˆ S z i h ˆ S z i h ˆ S z i h ˆ S y i h ˆ S y i (a)(b)(c) (d)(e)(f) − . − . − . − . − . − . − . − . − . . . . . . . . . . − . . −
00 000
FIG. 8. Electron spin trajectories projected in the h ˆ S y i - h ˆ S z i plane in region III. The solutions of the EOMs given in Eq. (6)are periodic self-sustained oscillations (Fig. 7). Figures a)-f) show the different limit-cycles obtained when varying theexternal magnetic field. Γ /λ = 1. ACKNOWLEDGMENTS
The authors would like to thank S. Kohler, A. Metel-mann, R. S´anchez and G. Schaller for helpful discussions.We acknowledge financial support through Grant No.MAT2008-02626 (MICINN), from FPU grant (C. L´opez- Mon´ıs), from ITN under Grant No. 234970 (EU) andthrough Grant No. DE2009-0074 (DAAD-MICINN).
Appendix A: Derivation of the equations of motion
In this appendix, we summarize the steps in the derivation of the EOMs (Eq. (6)). We start with the Hamiltonianof Eq. (1) and, for later convenience, we shift the reservoir frequencies P lkσ ǫ lkσ ˆ c † lkσ ˆ c lkσ → P lkσ ( ǫ lkσ + µ l )ˆ c † lkσ ˆ c lkσ where µ l is the chemical potential of lead l . Under the mean-field approximation considered in this work (see Eq. (5)),the closed set of EOMs obtained for the time evolution of operators in the Hamiltonian (Eq. (1)) are then computedto be: i ddt h ˆ d † σ ˆ d σ ′ i = λ (cid:16) δ σ ′ ↑ h ˆ d † σ ˆ d ↓ i + δ σ ′ ↓ h ˆ d † σ ˆ d ↑ i − δ σ ↑ h ˆ d †↓ ˆ d σ ′ i − δ σ ↓ h ˆ d †↑ ˆ d σ ′ i (cid:17) h ˆ J x i + 12 (cid:16) δ σ ′ ↑ h ˆ d † σ ˆ d ↑ i − δ σ ′ ↓ h ˆ d † σ ˆ d ↓ i − δ σ ↑ h ˆ d †↑ ˆ d σ ′ i + δ σ ↓ h ˆ d †↓ ˆ d σ ′ i (cid:17) (cid:16) λ h ˆ J z i + B z (cid:17) − X l, k (cid:16) γ lk h ˆ c † lkσ ˆ d σ ′ i − γ ∗ lk h ˆ d † σ ˆ c lkσ ′ i (cid:17) (A1) i ddt h ˆ c † lkσ ˆ d ↑ i = λ h ˆ J x ih ˆ c † lkσ ˆ d ↓ i + 12 (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ c † lkσ ˆ d ↑ i + X l ′ , k ′ γ ∗ l ′ k ′ h ˆ c † lkσ ˆ c l ′ k ′ ↑ i − ( ǫ lkσ + µ l ) h ˆ c † lkσ ˆ d ↑ i − γ ∗ lk h ˆ d † σ ˆ d ↑ i i ddt h ˆ c † lkσ ˆ d ↓ i = λ h ˆ J x ih ˆ c † lkσ ˆ d ↑ i − (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ c † lkσ ˆ d ↓ i + X l ′ , k ′ γ ∗ l ′ k ′ h ˆ c † lkσ ˆ c l ′ k ′ ↓ i − ( ǫ lkσ + µ l ) h ˆ c † lkσ ˆ d ↓ i − γ ∗ lk h ˆ d † σ ˆ d ↓ i and ddt h ˆ J x i = − (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J y i ddt h ˆ J y i = − λ h ˆ S x ih ˆ J z i + (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J x i ddt h ˆ J z i = λ h ˆ S x ih ˆ J y i (A2)where we have used the choice λ x = λ z = λ and λ y = 0. Since the EOMs for the large spin components have alreadythe desired form (see Eq. (6)), hereinafter we shall focus on the time evolution of the electron operators (Eq. (A1)).Under the Born approximation the leads are assumed to be in thermal equilibrium for all time, h ˆ c † lkσ ˆ c l ′ k ′ σ ′ i = f lσ δ ll ′ δ σσ ′ δ ( k ′ − k ) , (A3)with f lσ the equilibrium Fermi-Dirac distribution for spin- σ electrons in lead l : f lσ = f ( ǫ lkσ ) = 1 e ( ǫ lkσ ) /k B T + 1 . (A4)Applying the Laplace transform, h ˆ A i s ≡ R ∞ e − st h ˆ A i t dt , to Eq. (A1) we obtain: is h ˆ d † σ ˆ d σ ′ i s = λ (cid:16) δ σ ′ ↑ h ˆ d † σ ˆ d ↓ i s + δ σ ′ ↓ h ˆ d † σ ˆ d ↑ i s − δ σ ↑ h ˆ d †↓ ˆ d σ ′ i s − δ σ ↓ h ˆ d †↑ ˆ d σ ′ i s (cid:17) h ˆ J x i s + 12 (cid:16) δ σ ′ ↑ h ˆ d † σ ˆ d ↑ i s − δ σ ′ ↓ h ˆ d † σ ˆ d ↓ i s − δ σ ↑ h ˆ d †↑ ˆ d σ ′ i s + δ σ ↓ h ˆ d †↓ ˆ d σ ′ i s (cid:17) (cid:16) λ h ˆ J z i s + B z (cid:17) − X l, k (cid:16) γ lk h ˆ c † lkσ ˆ d σ ′ i s − γ ∗ lk h ˆ d † σ ˆ c lkσ ′ i s (cid:17) + i h ˆ d † σ ˆ d σ ′ i (A5a)and is h ˆ c † lkσ ˆ d ↑ i s = λ h ˆ J x i s h ˆ c † lkσ ˆ d ↓ i s + 12 (cid:16) λ h ˆ J z i s + B z (cid:17) h ˆ c † lkσ ˆ d ↑ i s + f lσ δ σ ↑ γ ∗ lk − ( ǫ lkσ + µ l ) h ˆ c † lkσ ˆ d ↑ i s − γ ∗ lk h ˆ d † σ ˆ d ↑ i s (A5b) is h ˆ c † lkσ ˆ d ↓ i s = λ h ˆ J x i s h ˆ c † lkσ ˆ d ↑ i s − (cid:16) λ h ˆ J z i s + B z (cid:17) h ˆ c † lkσ ˆ d ↓ i s + f lσ δ σ ↓ γ ∗ lk − ( ǫ lkσ + µ l ) h ˆ c † lkσ ˆ d ↓ i s − γ ∗ lk h ˆ d † σ ˆ d ↓ i s (A5c)where h A i denotes the expectation value of operator ˆ A at time t = 0, and where we have taken h ˆ c † lkσ ˆ d σ ′ i = 0. Aftersome algebra, Eq. (A5b) and Eq. (A5c) become: h ˆ c † lkσ ˆ d ↑ i s = γ ∗ lk ( f lσ δ σ ↑ − h ˆ d † σ ˆ d ↑ i s ) ǫ lkσ + µ l − ( λ h ˆ J z i s + B z ) + is + λ h ˆ J x i s λ h ˆ J z i s + B z +2( is + ǫ lkσ + µ l ) + 2 γ ∗ lk λ h ˆ J x i s ( h ˆ d † σ ˆ d ↓ i s − f lσ )( λ h ˆ J z i s + B z ) − is + ǫ lkσ + µ l ) + λ h ˆ J x i s (A6a) h ˆ c † lkσ ˆ d ↓ i s = γ ∗ lk ( f lσ δ σ ↓ − h ˆ d † σ ˆ d ↓ i s ) ǫ lkσ + µ l + ( λ h ˆ J z i s + B z ) + is − λ h ˆ J x i s λ h ˆ J z i s + B z − is + ǫ lkσ + µ l ) + 2 γ ∗ lk λ h ˆ J x i s ( h ˆ d † σ ˆ d ↑ i s − f lσ )( λ h ˆ J z i s + B z ) − is + ǫ lkσ + µ l ) + λ h ˆ J x i s . (A6b)We now consider the infinite bias limit and set, for the left lead, µ L → ∞ , and for the right, µ R → −∞ . In this limit,the denominator of the first term in Eq. (A6a) becomes ǫ lkσ + µ l + i + , with positive infinitesimal 0 + , and the secondterm is seen to be of the order µ − l and thus negligiable compared with the first term (of order µ − l ). Equations (A6a)and (A6b) thus become: h ˆ c † lkσ ˆ d σ ′ i s = γ ∗ lk ǫ lkσ + µ l + i + ( f lσ δ σσ ′ − h ˆ d † σ ˆ d σ ′ i s ) . (A7)This result allows us to rewrite the summation that appears in Eq. (A5a) as: X lk (cid:16) γ lk h ˆ c † lkσ ˆ d σ ′ i s − γ ∗ lk h ˆ d † σ ˆ c lkσ ′ i s (cid:17) = 12 π X l Z ∞−∞ dǫ " Γ lσ ( ǫ ) ǫ + µ l + i + − Γ lσ ′ ( ǫ ) ǫ + µ l − i + ( f ( ǫ ) δ σσ ′ − h ˆ d † σ ˆ d σ ′ i s )with the lead- and spin-dependent rates Γ lσ ( ǫ ) = 2 π ρ lσ ( ǫ ) | γ l ( ǫ ) | . (A8)with ρ lσ ( ǫ ) density of states of the l -th lead. We assume these rates to be energy-independent, Γ lσ ( ǫ ) = Γ lσ (flat bandapproximation). Using the Sokhatsky-Weierstrass theorem,1 x ± i + = P x ∓ iπδ ( x ) , upon evaluation of the Fermi functions at µ L = ∞ and µ R = −∞ , we obtain X l, k (cid:16) γ lk h ˆ c † lkσ ˆ d σ ′ i s − γ ∗ lk h ˆ d † σ ˆ c lkσ ′ i s (cid:17) = i X l (Γ lσ + Γ lσ ′ ) h ˆ d † σ ˆ d σ ′ i s − i Γ Lσ δ σσ ′ . (A9)Replacing the previous expression in Eq. (A5a) gives: h ˆ n σ i s = λ h ˆ J x i s h ˆ S y i s ( δ σ ↑ − δ σ ↓ ) − Γ h ˆ n σ i s + Γ Lσ s h S x i s = −h ˆ S y i s (cid:16) λ h ˆ J z i s + B z (cid:17) − Γ h ˆ S x i s + h ˆ S x i s h ˆ S y i s = − λ h ˆ S z i s h ˆ J x i s + h ˆ S x i s (cid:16) λ h ˆ J z i s + B z (cid:17) − Γ h ˆ S y i s + h ˆ S y i s h ˆ S z i s = λ h ˆ J x i s h ˆ S y i s − Γ h ˆ S z i s + 12 (Γ L ↑ − Γ L ↓ ) (A10)where Γ σ = Γ Lσ + Γ Rσ for σ = ↑ , ↓ , although we have assumed for simplicity Γ ↑ = Γ ↓ = Γ; and the identities:ˆ S x = 12 (cid:16) ˆ d †↑ ˆ d ↓ + ˆ d †↓ ˆ d ↑ (cid:17) ˆ S y = 12 i (cid:16) ˆ d †↑ ˆ d ↓ − ˆ d †↓ ˆ d ↑ (cid:17) ˆ S z = 12 (cid:16) ˆ d †↑ ˆ d ↑ − ˆ d †↓ ˆ d ↓ (cid:17) (A11)0have been used. Finally, inverse Laplace transforming Eqs. (A10) yields the EOMs (6) for the occupation and thespin components of the electron in the QD.The EOM for the total occupancy of the QD is obtained by summing the EOMs of the spin-up and spin-downoccupations, ddt h ˆ N i = − Γ h ˆ N i + Γ L ↑ + Γ L ↓ . (A12)Notice that this EOM is independent of the electron and large spins, moreover, it is exactly solvable giving: h ˆ N ( t ) i = h ˆ N (0) i e − Γ t + Γ L ↑ + Γ L ↓ Γ (cid:0) − e − Γ t (cid:1) . (A13)
1. Heuristic derivation
The electronic part of our EOMs can be seen to make sense by considering a more intuitive derivation using rateequations for the QD occupations (Eq. (6)) when λ = 0. The QD states in the transport window are {| i , | ↑i , | ↓i , | i = | ↑↓i} . In the infinite bias regime electrons can tunnel into the QD from the left lead and tunnel out of the QDto the right lead, thus ˙ p = W R ↑ p ↑ + W R ↓ p ↓ − ( W L ↑ + W L ↓ ) p ˙ p ↑ = W L ↑ p + W R ↑ p − ( W R ↑ + W L ↑ ) p ↑ ˙ p ↓ = W L ↓ p + W R ↓ p − ( W R ↓ + W L ↓ ) p ↓ ˙ p = W L ↑ p ↑ + W L ↓ p ↓ − ( W R ↑ + W R ↓ ) p . (A14)where p i is the probability of finding an electron in state | i i . W lfi is the tunneling rate from the initial state | i i to thefinal state | f i through the l -th barrier. Using the conservation of total probability (Tr( ρ ) = 1) we get˙ p ↑ = W L ↑ (1 − p ↓ ) + ( W R ↑ − W L ↑ ) p − ( W L ↑ + W R ↑ + W L ↑ ) p ↑ ˙ p ↓ = W L ↓ (1 − p ↑ ) + ( W R ↓ − W L ↓ ) p − ( W L ↓ + W R ↓ + W L ↓ ) p ↓ ˙ p = W L ↑ p ↑ + W L ↓ p ↓ − ( W R ↑ + W R ↓ ) p . (A15)We now consider that W L ↑ = W L ↓ = Γ L ↑ and W L ↓ = W L ↑ = Γ L ↓ , and W R ↑ = W R ↓ = Γ R ↑ and W R ↓ = W R ↑ = Γ R ↓ , so˙ p ↑ = Γ L ↑ (1 − p ↓ ) + (Γ R ↓ − Γ L ↑ ) p − (Γ L ↑ + Γ R ↑ + Γ L ↓ ) p ↑ ˙ p ↓ = Γ L ↓ (1 − p ↑ ) + (Γ R ↑ − Γ L ↓ ) p − (Γ L ↓ + Γ R ↓ + Γ L ↑ ) p ↓ ˙ p = Γ L ↓ p ↑ + Γ L ↑ p ↓ − (Γ R ↑ + Γ R ↓ ) p . (A16)Finally, since ˙ p σ + ˙ p = Γ Lσ − (Γ Lσ + Γ Rσ )( p σ + p ) , (A17)and h ˆ n σ i = p σ + p , we arrive to h ˙ˆ n σ i = − Γ h ˆ n σ i + Γ Lσ (A18)where we have used that Γ Lσ + Γ Rσ = Γ.1 Appendix B: Isotropic model
The EOMs for the completely isotropic case λ x = λ y = λ z = λ are ddt h ˆ n σ i = λ (cid:16) h ˆ J x ih ˆ S y i − h ˆ J y ih ˆ S x i (cid:17) ( δ σ ↑ − δ σ ↓ ) − Γ h ˆ n σ i + Γ Lσ ddt h ˆ S x i = λ h ˆ J y ih ˆ S z i − (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ S y i − Γ h ˆ S x i ddt h ˆ S y i = − λ h ˆ J x ih ˆ S z i + (cid:16) λ h ˆ J z i + B z (cid:17) h ˆ S x i − Γ h ˆ S y i ddt h ˆ S z i = λ (cid:16) h ˆ J x ih ˆ S y i − h ˆ J y ih ˆ S x i (cid:17) − Γ h ˆ S z i + 12 (Γ L ↑ − Γ L ↓ ) ddt h ˆ J x i = λ h ˆ S y ih ˆ J z i − (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J y i ddt h ˆ J y i = − λ h ˆ S x ih ˆ J z i + (cid:16) λ h ˆ S z i + B z (cid:17) h ˆ J x i ddt h ˆ J z i = λ (cid:16) h ˆ S x ih ˆ J y i − h ˆ S y ih ˆ J x i (cid:17) . (B1)To find the solutions in the stationary limit we put to zero the time derivatives. Therefore, it can be seen right awaythat in the long time limit the quantum dot occupations decouple from the large spin components and becomes h ˆ n σ i = Γ Lσ Γ . (B2)Thus, the spin dynamics can not be observed in the current. Appendix C: Effective model for region III
In this appendix, we summarize the steps in the derivation of the effective EOMs (Eq. (17)) for region III of theparameter space (Fig. 2a). Applying the transformation h ˆS i = e − Γ t R ( t ) · h ˜S i and h ˆJ i = R ( t ) · h ˜J i with R ( t ) = cos( B z t ) − sin( B z t ) 0sin( B z t ) cos( B z t ) 00 0 1 , (C1)to the EOMs (Eq. (6)) they become: ddt h ˜ S x i = − λ n h ˆ J z ih ˜ S y i + h h ˜ J x i cos( B z t ) − h ˜ J y i sin( B z t ) i h ˜ S z i sin( B z t ) o ddt h ˜ S y i = λ n h ˆ J z ih ˜ S x i − h h ˜ J x i cos( B z t ) − h ˜ J y i sin( B z t ) i h ˜ S z i cos( B z t ) o ddt h ˜ S z i = λ h h ˜ J x ih ˜ S y i cos ( B z t ) − h ˜ J y ih ˜ S x i sin ( B z t ) + (cid:16) h ˜ J x ih ˜ S x i − h ˜ J y ih ˜ S y i (cid:17) sin( B z t ) cos( B z t ) i + 12 (Γ L ↑ − Γ L ↓ ) e Γ t ddt h ˜ J x i = − λe − Γ t n h ˜ S z ih ˜ J y i + h h ˜ S x i cos( B z t ) − h ˜ S y i sin( B z t ) i h ˆ J z i sin( B z t ) o ddt h ˜ J y i = λe − Γ t n h ˜ S z ih ˜ J x i − h h ˜ S x i cos( B z t ) − h ˜ S y i sin( B z t ) i h ˆ J z i cos( B z t ) o ddt h ˜ J z i = λe − Γ t h h ˜ S x ih ˜ J y i cos ( B z t ) − h ˜ S y ih ˜ J x i sin ( B z t ) + (cid:16) h ˜ S x ih ˜ J x i − h ˜ S y ih ˜ J y i (cid:17) sin( B z t ) cos( B z t ) i (C2)Since in the long-time limit d h ˜ J i i /dt →
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