Time Dependent Current Oscillations Through a Quantum Dot E. S. Rodrigues†, E. V. Anda‡ and P. Orellana†† † Instituto de F´ısica, UFF, Av. Litorˆanea s/no, Gragoat´a, Niter´oi RJ, Brasil, 24210-340. ‡ Departamento de F´ısica, PUC-RJ, Caixa Postal 38071, Rio de Janeiro, Brasil, 22452-970. †† Departamento de F´ısica, Universidad Cat´olica del Norte, Angamos 0610, Casilla 1280, Antofagasta, Chile. 0 0 I. ABSTRACT 0 2 Time dependent phenomenaassociatedto chargetransportalongaquantumdotinthe chargequantizationregime n is studied. Superimposed to the Coulomb blockade behaviour the current has novel non-linear properties. Together a with static multistabilities in the negative resistance region of the I-V characteristic curve, strong correlations at the J dot give rise to self-sustained current and charge oscillations. Their properties depend upon the parameters of the 3 quantum dot and the external applied voltages. 1 ] PACS numbers: 73.40.Gk, 85.30.Mn l e - r t II. INTRODUCTION s . t a Electron correlation effects on the transport properties of a quantum dot (QD) under the influence of an external m potential have been the subject of many experimental and theoretical studies in recent years. This device although - possesses thousands of atoms behaves as if it were an artificial atom due to its discrete energy spectrum. The d enormous advantages of this artificial atom are that its properties can be modified continuously by adjusting the n o external potentials applied to it [1]. Unlike a real atom it can be experimentally isolated and its transportproperties c obtained studying the flow of charges through it. The quantization of charge and energy plays the major role in the [ transportpropertiesof a QD. The conductance exhibits oscillations as a function of the externalpotential which can be explained in terms of a transport mechanism governed by single-electron tunneling and Coulomb blockade effect 1 v due to the e-e interaction inside the dot[2]. 0 ItiswellestablishedthattheCoulombinteractionproducesnon-lineareffectsin3Ddoublebarrierheterostructure, 0 observed as intrinsic static bistability in the negative differential resistance region of the I-V characteristic curve of 2 the device[3]. This propertyhasbeenexplainedasanelectrostaticeffectdueto the e-einteractionamongthe charges 1 inside the well. Besides this static behavior, theoretical studies have proposed that the Coulomb interaction can 0 produce time dependent oscillations[4]. More recently current oscillations promoted by an external magnetic field 0 applied in the direction of the current were predicted[5]. These last results showed that the system bifurcates as the 0 / field is increasedand may transitto chaos at large enoughfields. Recently very interesting experimental evidences of t a the existence of these oscillations induced by an external magnetic field, have been given[6]. m Although Coulomb interaction phenomena have been a matter of great concern in the last years, to the best of - our knowledge, there have not been experimental studies reporting neither static bistabilities nor current and charge d oscillations through a QD. In the search for stationary behavior, we were able to theoretically predict a bistability n similar to the one that appears in the I-V characteristic curve of a 3D double barrier heterostructure. We obtained o as well evidences of the non-existance of stationary solutions in some regions of the parameter space for the currents c : flowing through the QD[7]. v In this communication letter we address the study of currents going through a QD connected to leads under the i X effect of an external potential. We report the existence of static hysteresis behavior of the current in the negative resistance region. What is more important, we find novel time dependent phenomena which appear as self-sustained r a current and charge oscillations with properties depending on the internal parameters of the system and the external appliedpotentials. These time dependentphenomenaarederivedfromthe highly correlatedelectronsinside theQD. It has been shown that the physics associatedto a QD connected to two leads can be readily understood in terms of a single magnetic impurity Anderson Hamiltonian, where the impurity is the quantum dot[2]. The leads can be representedbya1Dtight-bindingHamiltonianconnectingthe dotto particlereservoirscharacterizedby Fermilevels E and E . The difference E −E gives the bias voltage applied to the system when connected to a battery. The l r l r Hamiltonian is given by H = ǫ n +v c† c + (ǫ +V ) n +Un n . (1) i iσ iσ jσ o g oσ o↑ o↓ Xiσ <Xij>σ Xσ 1 The diagonal matrix element ǫ is site dependent because it takes into account the potential profile and the two i barriers. The gate voltage V is supposed to be effective only on the QD. Since we are not interested in very low g temperaturephysics,asforinstancetheKondoeffect,weuseadecouplingprocedure[8]tosolvetheequationofmotion for the one particle operator. Although this is a simple approximation, it is adequate to treat high correlations at the dot, responsible for Coulomb blockade phenomena[2]. The non-linear behavior stems from the dependence of the solution upon the self-consistent calculated charge content of the dot. As we are not interested in studying magnetic properties, we omit the subindex σ corresponding to the spin. Using the set of Wannier functions localized at site i, φ (r), we can write the time dependent state ϕ (r,t) as, i k ϕ (r,t)= ak(t)φ (r) (2) k i i Xi The probability amplitude foran electronto be atsite j in a time dependent state ϕ (r,t) is givenby the equation k of motion, dak i¯h j =ǫ ak+v(ak +ak ) (i6=0) (3) dt j j j−1 j+1 dak,α i¯h o =(ǫα+V )ak,α+nαv(ak +ak) (4) dt o g o −1 1 where α = ±; n+ = n; n− = 1−n; ǫ+ = U; ǫ− = 0 and n is the number of electrons per spin at the dot that is o o calculated from the probability amplitudes by the equation, kf n=2 |ak |2 dk (5) Z o 0 where ak =ak,++ak,− and k is the Fermi wave vector of the emitter side. We have assumed for simplicity that the o o o f collector is not doped so that the charge that goes towards the emitter is restricted to the electrons that have been reflected on the barriers. The time dependent equations are solved by discretizing equations (3) and (4), and using a half-implicit numerical method, which is second-order accurate and unitary[9]. The solution of these equations require the knowledge of the wave function at −∞ and +∞. We suppose that an incident electron of wave vector k is described by the function, ak =(Ieikxj +R e−ikxj)e−iǫt/h¯ (x ≤−L) (6) j j j ak =T eik′xje−iǫt/h¯ (x ≥L) (7) j j j where the system is explicitly defined between −L and L and k′ is the wave vector of the transmitted particle k′ = (2m(El−Er)+k2). The incidentamplitude Iisassumedtobe spatiallyconstant. Instead,farfromthebarriers, h¯2 the epnvelope function of the reflected and transmitted waves Rj and Tj are supposed to be weakly dependent on site j. This permits to restrict the dependence to the linear term, which results to be an adequate approximation providedthe time steptakentodiscretize equations(3)and(4)isless thancertainlimit valuethatdepends uponthe parametrization of the system. A maximum value of 0,3fs was adequate to guarantee numerical stability up to 200 ps. To eliminate spurious reflections at the boundary it was necessary to take systems of the order of 2L=400 sites. In the numerical procedure, the Wannier intensities obtained for one bias are used as the starting point for the next bias step. Once the Wannier intensities are known the current is calculated from[9], J ∝2 kf Im(ak∗(ak −ak) dk (8) j Z (cid:20) j j+1 j (cid:21) o It is important to emphasize that the system we study has to be in the Coulomb blockade regime. Charge quanti- zation is obtained satisfying the relation t′2 < U where t′ is an effective coupling constant between the QD and the W leads controlledby the barriersparametersandW is the leads bandwidth. In this casethe energy spacingU between the states with N and N +1 electrons is greater than the width of the resonant levels within the dot. We study the system defined by an emitter barrier of 1.4eV and a collector barrier of 2.4eV both of a thickness of 5 lattice parameters and an electronic repulsion U = 20meV. This configuration represents a GaAs structure in 2 the charge quantization regime. A similar effective coupling constant can be obtained assuming lower and thicker barriers. We have chosen the case with less number of sites in order to reduce the computational effort. In order to enhance the charge content of the dot and consequently the non-linear effects the second barrier is taken to be higher than the first. The Fermi level lies 30 meV above the bottom of the emitter conduction band. With these parameters and zero gate voltage and bias the localized level at the dot lies well above the Fermi level, so that the system is completely out of resonance. For small values of the gate potential equations (3) and (4) have a stationary solution after a transient, for the whole interval of bias voltage V, which drives the system into and off resonance. Increasing the gate potential the system enters into a completely different regime. As the bias voltage required to drive the system into resonance is small, because the resonantlevel is now near the Fermi level, the second barrier is not significantly reduced by the bias voltage and is able to trap a large amount of electronic charge inside the dot. As a consequence, the non-linear effect is enhanced giving rise to a region in the I-V characteristiccurve where there is no stationary solution. In this parameter region the current has self-sustained oscillations while for greater values ofthe bias voltagethe systempossessescomplex electrostaticmultistabilities whichcanbe obtained, depending upon the initial condition, augmenting or diminishing the bias voltage. These static multistabilities are very well known phenomenaof3Dsystems,althoughtheygenerallyassumeabistablestructure[3]. ThesituationfortheQDdescribed is illustrated in Fig. 1 where the bubble shows the maximum and minimum of the current oscillations, while for a greatervalueofthe biasvoltagethe electrostaticinstabilities areclearlyrepresented. The non-stationarysolutionslie outside the static multistable region for small values of V . However, for larger values of V , there is a superposition g g of the regions where both effects are active. The static largemultistability is related to the dropof the resonantlevel below the bottom of the conduction band, while the dynamic oscillations appear when the resonantlevel aligns with the Fermi level. The size and location of the oscillating behavior in the I-V space depends upon the gate voltage. Its size increases with V . The non-linearities are enhanced by the gate potential, constituting an external tunable g parameter through which the oscillating phenomena can be controlled. 0.98 0.96 1.0 0.94 0.92 0.90 0.8 2.603 2.616 2.629 u.) a. nt ( 0.6 urre C 0.4 0.2 0.0 2.48 2.59 2.70 V (volts) FIG. 1. Current versus bias voltage, I-V characteristic curve. The bubble indicates the maximum and minimum of the oscillating current amplitude. The inset shows a complex structure of static bistable behaviour. Dashed line shows the uncharged QD solution. We havetakendifferentvalues ofthe biasV to study explicitly the time dependent currentgoingalongthe device. g For V =0.2 volts the system possesses a stationary solution for all values of the bias voltage. For a greater value of g thegatepotentialV =0.86voltsandV =5.6volts,weshowthecurrentandthechargeasafunctionoftimeinthree g different locationsofthe system: inthe emitter andcollectorside before andafter the barriers,inFig. 2band, inside the dot, in Fig. 2c. We see that after a transientof some ps, the system begins to oscillate in a self-sustained way. It isimportanttonote thedifferentbehaviorsoftheoscillationsdepending uponthe sitewherethe currentisevaluated. Theoscillationsarelargeintheregionbeforetheemitterbarrierandinsidethewell,whileafterthechargehaspassed through the device the more opaque collector barrier damps the oscillation in a very significant way. Although not perfectlyperiodic,theoscillationshavesomeregularity. Thefrequencyspectrumiscontinuouswiththepredominance of two and some times three well define frequency regions. The origin of this behavior is not completely clear to us. However, the existence of more than one frequency region has to be attributed to the amount of non-linearity in the system. A less non-linear situation, obtained for instance in the near vicinity of the bifurcation point where time dependent behavior appears in the I-V characteristic curve of Fig. 1, shows small amplitude oscillating currents of almost only one frequency. Very remarkably, superimposed to the oscillating currents, we exhibit in the inset of Fig. 3 2c very high frequency oscillations greatly enhanced inside the well and almost non-existent outside it. We attribute them to the electrons going back and forth inside the well with a period corresponding to the time taken by the electrons at the neighborhood of the Fermi energy being successively reflected by the barriers. C a h a r 0.5 0.8 eg u( n o sti 0.0 0.6 )e f 1.0 b u.) nt (a. 0.5 e r r u C 0.0 0.42 c 1.0 0.38 34.20 34.25 0.5 0.0 0 20 40 60 80 Time (ps) FIG.2. CurrentandchargeversustimeforVg =0.86voltsandV =5.6volts. a)Charge(dashedline)andcurrent(fullline) at thedot as a function of time, showing thetime lag between them. b) Current at the emitter (full line) and at thecollector (dashed line). c) Current at thedot. Inset shows high frequency oscillations. Changing the gate potential the dot enters into resonance when its local level aligns with the Fermi level. The current increases abruptly and, due to the trapping effect of the collector barrier, the dot charge per spin n assumes itsmaximumvalueafteratime. Asaconsequence,thestrengthofthe resonantlevel,whichduetomanybodyeffects depends upon the dot charge as (1−n), diminishes and the flow of electrons going into the dot reduces. As time evolves, the total amount of charge inside the dot drops because the leaking of charge through the second barrier becomesgreaterthanitsentrancethroughthefirstone. Thereductionofchargeincreasesthestrengthoftheresonant level, which increases the flowing current starting a new cycle. As we have numerically verified, for a configuration outside the Coulomb blockade regime, by diminishing for instance the first barrier, the oscillations are damped out andthe systemreachesa stationaryregime. Inthis case,asthe systemis driveninto resonance,the currentincreases and the dot charge reacts almost instantaneously with no significant lag time between them. Togenerateself-sustainedcurrentoscillationsthesystemhastobeinaparameterregionwherethereisnodynamic accommodationbetweenthe currentandthe charge inside the dot. It requires anabrupt entrance in resonancewhen theexternalpotentialismodified. Thisisthecaseofthechargequantizationregimewhenthechannelfortheelectron to go through is provided by the very sharp resonant peak inside the dot. It is interesting to compare the current and the charge oscillations inside the well in this case. Although the frequencies and the general properties of both oscillating quantities are the same, there is a clear lag time τ of the dot charge in relation to the current. The dephasing between chargeand currentwhich disappears with more transparentbarriersis illustrated in Fig. 2a. It is clear that the lag τ depends upon the time scale controlling the entrance of the charge into the resonantstate, which is called the buildup time. For opaque emitter barriers, as in our case, the buildup time is found to be essentially independent of the barrier parameters. The buildup process is essentially determined by the spatial distribution of the extra chargein conditions to enter into the well[10]. This is slightly different in eachoscillation,creating a rather irregular lag among the two quantities as time goes along, as exhibited in Fig. 2a. Due to computer time limitations it was not possible to develope a systematic investigation of the phenomenon. However,we have studied the system modifying some of its parametersin order to give support to the interpretation above. Thereisacleartendencyoftheoscillationmeanvaluefrequencytoincreasereducingtheopacityofthebarriers although it is more clearly dependent upon the secondbarrier. This is consistent with the interpretationgiven above as it is the collector barrier that controls the leaking time of the charge when the its entrance is abruptly reduced by non-linearity. In order to study the stability of the self-sustained oscillations we have calculated, for one particular case, the time evolution of the system up to a value of 200ps. The oscillations maintain their amplitude and regular shape. We have taken a charged and a discharged dot as two different initial conditions. The first condition creates oscillationsimmediatelyafterthecircuitisswitchedonwhileinthesecondthesystemgoesthroughatransientduring 4 which the dot is charged, as shown in Fig. 3. Although the oscillations have a similar behavior shifting one relative to the other by this chargingtime, there are differences which in fact increase in time as shown in the inset of Fig. 3. In certain regions of the parameter space the oscillations result to be very sensitive to initial conditions, suggesting the existence of chaotic phenomena. 1.0 uncharged dot charged dot 0.8 e) s of 0.6 nit u e ( rg0.4 a h 0.8 C 0.6 0.2 0.4 20 40 60 80 0.0 0 20 40 60 80 Time (ps) FIG.3. Charge at thedot as a function of time, V =3.13 volts and Vg =3.02 volts. Initial states, dot uncharged (full line) and dot charged (dashed line). Inset shows the two solutions where the dot uncharged initial state solution is shifted by the changing time. 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