Non-linear effects in resonant tunnelling through a quantum dot

Non-linear effects in resonant tunnelling through a quantum dot

PERGAMON Solid State Communications 111 (1999) 217–221 Non-linear effects in resonant tunnelling through a quantum dot E.S. Rodrigues a,*, E.V. Anda...

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Solid State Communications 111 (1999) 217–221

Non-linear effects in resonant tunnelling through a quantum dot E.S. Rodrigues a,*, E.V. Anda b, P. Orellana c, H.N. Nazareno d b

a Instituto de Fı´sica, Universidade Federal Fluminense, Av. Litoraˆnea s/no., Gragoata´, Nitero´i RJ, 24210-340 Brazil Departamento de Fı´sica, Pontifı´cia Universidade Cato´lica do Rio de Janeiro, Caixa Postal 38071, Rio de Janeiro, 22452-970 Brazil c Departamento de Fı´sica, Universidad Cato´lica del Norte, Angamos 0610, Casilla 1280, Antofagasta, Chile d International Centre of Condensed Matter Physics, UNB, P.O. Box 04667, Brası´lia, DF, 70900-900 Brazil

Received 21 September 1998; accepted 12 March 1999 by C.E.T. Gonc¸alves da Silva

Abstract We study resonant tunneling transport properties of a quantum dot connected to two leads. The flowing electrons are supposed to interact strongly at the quantum dot. The system is represented by an Anderson impurity Hamiltonian. The transport properties are characterized by the well-known Coulomb blockade properties. However, superimposed to them, this one-dimensional (1D) system possesses novel non-linear current static bi-stability behavior and other time-dependent instability phenomena, probably current and charge oscillations. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: D. Electron–electron interactions; D. Electronic transport; D. Quantum localization; D. Tunnelling

1. Introduction Transport properties have been extensively studied due to their interesting device application and very rich phenomenology. Since the first observation of resonant tunneling in a three-dimensional (3D) double barrier structure by Chang et al. [1], these devices have been found to present non-linear properties, which are reflected in the observation of multistabilities in the I–V characteristic curve. Besides a peak due to simple resonant transmission, intrinsic static bistability in the negative differential resistance region has been measured [2,3]. These properties have been explained as a simple electrostatic effect due to the Coulomb interactions among the flowing charges inside the well [4]. The phenomenon can be thought to be produced by the pinning of the bound state at the well due to the e–e repulsion and the rapid leakage of * Corresponding author. E-mail address: [email protected] (E.S. Rodrigues)

the electronic charge accumulated between the barriers when the applied potential is just taking the device out of resonance [4,5]. Besides this static effect the Coulomb interaction produces a dynamic one, which appears as time dependent currents going along the system. Recently we were able to show that the addition of a magnetic field parallel to the current induces self-sustained intrinsic current oscillations in an asymmetric double barrier structure [6]. These oscillations were attributed to the enhancement due to the external magnetic field of the non-linear dynamic coupling of the current to the charge trapped in the well. These results showed that the system bifurcates as the field is increased, and may transit to chaos at large enough fields. Recently in a I–V characteristic measurement on a similar system some evidence of these dynamical properties were obtained [7]. In recent years, electron channels have been fabricated with lateral quantum-confined energy levels sufficiently separated one from the other as to be

0038-1098/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00149-0


E.S. Rodrigues et al. / Solid State Communications 111 (1999) 217–221

able to study properties of a one-dimensional (1D) system. When a quantum dot is connected through these electron channels to a battery, the system exhibits pronounced periodic oscillations of the conductance when the state-of-charge of the dot is modified by changing the gate potential. This is the consequence of a Coulomb blockade of an electron when it tries to go into a quantum island, which is already occupied by another electron [8,9]. Although the static bistability phenomena found in standard 3D resonant tunneling devices have not been detected in 1D systems where the Coulomb blockade is dominant, it is evident that both phenomena derive from the same Coulomb interaction acting between the flowing charges [8]. In this communication we develop a selfconsistent theory of resonant tunneling in quasi-1D systems, which predicts that, besides the normal Coulomb blockade behaviour in the region of negative differential resistance, the system develops static bi-stability properties as in 3D. Moreover, we show that, while the phenomena of hysteresis persists, the system bifurcates and is capable of further bifurcations as the gate potential applied to the dot is increased, leading eventually to true chaos. This fact emphasizes the absence of stationary solutions for the current in certain regions of the parameter space, indicating the existence of time-dependent phenomena, probably current and charge oscillations.

the energy of the localized state within the well, e i the diagonal matrix element which incorporate the potential profile of the heterostructure (the two barriers and the well) and the potential induced by the external applied electric field. The parameter U is the local electronic repulsion of the charges inside the well localized at site 0 and t represents the nearest-neighbor hopping matrix element. The electron–electron interaction will be treated within the context of the Hubbard approximation [10], which is adequate to treat the Coulomb blockade and the non-linear effects derived from the Coulomb interaction. However, it will not be capable of describing the low-lying excitations associated to the Kondo effect, which in principle the system would have at temperatures below the Kondo temperature. On a tight-binding basis, the stationary state of energy Ek can be written as X k wk s ˆ ai;s fis …r†; …2† i

where f is is a Wannier state localized at site i of spin s and the coefficients akis obey the non-linear difference equations …1i ⫺ Ek ⫹ 2t†aki;s ˆ …aki⫺1;s ⫹ aki⫹1;s †t; …10 ⫺ Ek ⫹ 2t ⫹ Vp ⫹ U†ak0;s ˆ

2. The model A theoretical understanding of these properties should consider that the system is in a non-equilibrium situation under the effect of many-body interactions. A 1D Anderson Hamiltonian microscopically describes the system where the quantum dot plays the role of the impurity [8,9]. The leads are represented by a 1D tight-binding Hamiltonian connecting the dot to two reservoirs characterized by Fermi levels El and Er. The difference El – Er corresponds to the potential drop from left to right along the sample. The nearestneighbour Hamiltonian can be written as, X ⫹ X UX Hˆt c is c js ⫹ 1i nis ⫹ Vp n0s ⫹ n n ; 2 s 0s 0s is 具ij典s …1† where Vp is the external gate potential which controls

i 苷 0; …3†


ak1;s †

! Un⫺ s 1⫹ ; ⫺ Ek ⫹ 2t ⫹ Vp


i ˆ 0; where Ek ˆ 2t cos ka the dipersion relation of a 1D chain, n⫺ s ˆ 1 ⫺ 具n0;s 典 and 具n0;s 典 the number of electrons with spin s at the well site and is calculated from the tight-binding amplitudes at the well, according to the following equation: X k 2 兩a0s 兩 ; …5† 具no;s 典 ˆ k⬍kfl

where the sum over k covers all occupied electrons states of the system with energy below the Fermi level, incident from the emitter side. In order to study the solutions of the equation [2] we assume a plane wave incident from the left with an intensity I, with a partial reflection amplitude R. The waveform at

E.S. Rodrigues et al. / Solid State Communications 111 (1999) 217–221


Fig. 1. I–V characteristic curve for a system with U ˆ 0.005t, Ef ˆ 0.008t and the emitter and the collector barrier width 0.8t and 1.6t, respectively, for four different values of the gate potential: (a) Vp ˆ ⫺ 0.4t; (b) Vp ˆ ⫺ 0.5t; (c) Vp ˆ ⫺ 0.7t; and (d) Vp ˆ ⫺ 1.0t.

the far right is a simple plane wave with intensity given by the transmission probability T. Taking this to be the solution at the far right and left edges of the system, we can write

wk …r† ˆ Ieikr ⫹ Re⫺ikr wk …r† ˆ Teikr

r q L;

r p 0;

…6† …7†

where L is the length of the active part of the system in units of the lattice parameter. The solution of equations [3,4] can be obtained through an adequate iteration of them from right to left. For a given transmitted amplitude, the associated reflected and incident amplitudes may be determined by matching the iterated function to the proper plane wave at the far left, equation [6]. Owing to the


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presence of the non-linear term, the equations [3,4] have to be solved self-consistently. With this purpose, we define a second pseudo-time-like iteration in the following way. Initially equations [3,4] are solved ignoring the non-linear term and for energies up to the Fermi energy. The coefficients thus obtained correspond to a solution for non-interacting electrons. They are used to construct the non-linear term for the next solution. The procedure is continued, using for the non-linear term solutions corresponding to the previous iteration. This procedure defines a spatialpseudo-temporal map that is linear in space and non-linear in time. Maps, non-linear in space and time have been studied in other contexts. In particular, the coupled map model for open flow was found to exhibit spatial chaos with temporal periodicity [11]. A similar behaviour has been found studying resonant tunneling properties of 3D double barrier devices under the effect of an external magnetic field along the current [6]. As we shall show, in our case chaos develops in the iteration of time, as the self-consistent loop is developed. Once akj;s are known, the current at site j is numerically obtained from Jj ˆ

he Zkf Im…aki;s …aki⫹1;s ⫺ aki;s †† dk; m* 0


where kf is the Fermi momentum of the emitter and m* the electronic effective mass m* ˆ ប2 =2ta2 ; where a is the lattice parameter.

3. Numerical results We study a model which consists of an emitter and collector barrier taken to have a height of 0.8t and 1.6t, respectively, and of width 5a. The collector barrier is assumed to be wider than the emitter in order to accumulate charge inside the dot and enhance the non-liner effects. The Fermi energy is supposed to be Ef ˆ 0.008t and U ˆ 0.005t. The Fermi energy assumed corresponds approximately to a doping concentration of the leads of 10 20 cm ⫺3. The U value could represent the GaAs dot of radius r ⬃ ˚ which, for the barriers taken, possesses a loca600 A lized energy spacing D q Dd, where Dd is the width of the dot resonances. This guarantee that the system is in the Coulomb blockade regime and that the

Hamiltonian given by Eq. (1) is adequate to describe it. The normalization of the wave function is taken so that each site could have up to a maximum of two electrons. The map is linear in space so that the normalization involves a simple multiplication of all coefficients by a constant. As electron density has rather long-range oscillations, we made sure that the sample was long enough to make finite size effects negligible. This is guaranteed by taking a sample of N ⬎ 100 sites. The border conditions (6) and (7) implicitly incorporates the rest of the system. The non-linear behavior of the system is controlled by the gate potential applied at the quantum dot. For small Vp as shown in Fig. 1(a), the non-linear effect due to the e–e interaction is almost negligible although it produces small oscillations of the current. Owing to the parameters adopted to study the system the two electrons state localized at the well participates in the resonant peak appearing in Fig. 1(a). This implies that the Coulomb peak aligns with the populated states of the emitter when the one-electron resonance is still within the emitter conduction band. However, as this resonance is near the bottom of the band, it gets out of resonance before the Coulomb peak could be reflected as a distinct resonance in the I–V characteristics. Its influence appears as a clear asymmetry in the curve of Fig. 1(a). When the gate potential is greater than a threshold value, which depends upon the parameters of the system, a completely novel feature starts to develop where two periods and two fixed points are encountered for certain voltages, creating a bubble-like solution shown in the Fig. 1(b). This pattern reflects the absence of stationary solutions for the current. Increasing the gate potential still further as in the case of Fig. 1(c), there are two stationary solutions for the current, as the voltage is increased when the dot is charged and as the voltage is decreased when the dot is with less charge. In either case, the self-consistent solution converges to a stable fixed point after some finite number of iterations. From this view-point we obtain a similar behavior for this 1D system to the very well-known bi-stability in 3D. Similarly, for 3D tunneling, the choice of an asymmetric structure (the emitter barrier greater than the collector one) and an increase of the Fermi level, augment the charge content of the well and consequently the width of the hysteresis loop. However, differently to the

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traditionally 3D static bi-stability a complicated structure appears in the I–V characteristics with a multiplicity of static bi-stabilities on the top of the curve. This behavior is shown in Fig. 1(c). It is noticeable that for this gate potential the bubble like solution undergoes further bifurcation and finally a bifurcation cascade leading to a chaotic region as appears in the figure. The bifurcation and the chaotic solutions are obtained for values of the applied voltage outside the region of static bi-stability, showing that although both phenomena are derived from the non-linearities introduced by the local Coulomb repulsion, they correspond to two different effects. As the potential gate is raised still further, the bubble and chaotic area increases and has a superposition with the bi-stable electrostatic region, which augments as well. For greater values of the applied potential, the chaotic region eventually unfolds to a solution of finite periodicity or to a unique self-consistent one. The chaotic or the finite periodicity solutions in reality underline the non-existence of a self-consistent timeindependent solution for the system. A similar behavior obtained for the 3D system suggests that these time-dependent instability phenomena are probably current and charge oscillations. 4. Conclusions We have shown that a quantum dot connected to two leads and under the effect on an externally applied potential have, superimposed to the Coulomb blockade oscillations of the current, electrostatic bi-stable solutions and dynamical instabilities, which eventually become chaotic. Although bi-stabilities in the current are a well-known phenomenon in 3D resonant


tunneling, to the best of our knowledge, neither its existance has been proposed nor has it been measured in a 1D system. Static bi-stabilities and general dynamical instabilities, although arising from the same Coulomb interaction, can appear in different regions of the parameter space reflecting the fact that they are different phenomena. A full-time dependent study of this problem is under way. Acknowledgements This work was partially supported by the Brazilian Agencies CNPQ and Finep, and DGICT-UCN, FONDECYT Grants 1980225 and 1960417 (Chile).

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