Coherent electronic properties of coupled two-dimensional quantum dot arrays

Superlattices and Microstructures, Vol. 20, No. 4, 1996

Coherent electronic properties of coupled two-dimensional quantum dot arrays R. Kotlyar, S. Das Sarma Department of Physics, University of Maryland, College Park, Maryland 20742, U.S.A.

(Received 20 May 1996) In this paper we review our recent study of coherent electronic properties of coupled twodimensional quantum dot arrays using numerical exact-diagonalization methods on a Mott– Hubbard type correlated tight-binding model. We predict the existence of a novel kind of persistent current in a two-dimensional isolated array of quantum dots in a transverse magnetic field. We calculate the conductance spectrum for resonant tunneling transport through a coherent two-dimensional array of quantum dots in the Coulomb Blockade regime. We also calculate the effective two-terminal capacitance of an array coupled to bias leads. c 1996 Academic Press Limited

Key words: Collective Coulomb Blockade, persistent current.

1. Introduction By tuning the tunnel barriers between the individual dots of a linear chain of semiconductor quantum dots, which have been electrostatically defined in a 2D electron gas [1], it is, in principle, possible to achieve various theoretically interesting [2] coupling regimes where interdot tunneling, Coulomb interaction, and intradot interlevel energies compete with each other. The transition from the single dot Coulomb Blockade (CB) conductance oscillations to their splitting into minibands due to coherent tunneling in the array can be identified with the formation of an ‘artificial molecule’ [2–4]. The suppression of conductance due to the Mott–Hubbard spin-polarization transition, which has no analogy within the classical charging model, has been predicted [3] to conclusively test the formation of a coherent many-body ‘molecular’ state. In this presentation we briefly review our extensive recent work [5] on the electronic properties of coupled and coherent two-dimensional (2D) quantum dot arrays using the Mott–Hubbard Hamiltonian approach. Our motivation is to investigate the characteristic signature of the formation of an artificial 2D quantum dot molecule in experimentally observable electronic properties. Recent fabrication of coherent quantum dot structures in several laboratories motivates our numerical investigation of the 2D L x × L y quantum dot arrays in the Collective Coulomb Blockade (CCB) regime [2]. We discuss the existence of an equilibrium persistent current in 2D arrays and contrast the situation with persistent current results in 1D rings. In general, the transport current-carrying states can be probed by measuring the low bias CCB conductance spectrum, which we calculate within a linear response theory. Finally, we also present our results on the two-terminal effective capacitance of a 2D quantum dot array. In Section 2, we define and describe our coherent 2D Mott–Hubbard model for the coupled quantum dot array, and in Sections 3, 4 and 5 we present our numerical results for the persistent current, the conductance, and the capacitance, respectively. A detailed discussion of our other results will be given elsewhere [5]. 0749–6036/96/080641 + 09 $25.00/0

c 1996 Academic Press Limited

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N Fig. 1. Typical equilibrium persistent current vs. the number N of excess electrons in 2D quantum dot arrays with field-independent single-particle levels εi↑ = 1, εi↓ = εi↑ + 0.031 with the long-range interaction Vi j set to zero. The response is suppressed in the interacting (U = 1 meV) 3 × 3 array in both the clean and disordered (W = 1) systems (dashed and dash-dotted lines respectively) by orders of magnitude at half-filling, i.e. where electron filling n in the array is n = N /2L = 0.5, compared to the clean (solid line) and the disordered (W = 1, dotted line), non-interacting system response due to the Mott–Hubbard metal-insulator transition. In the inset the non-interacting persistent current in the 2D 3 × 3 array (solid line) is compared with the corresponding 1D result in the 9-site ring and the 8-site ring (the dotted and dashed lines, respectively).

2. Model The system we model is a finite two-dimensional square array of L x × L y = L quantum dots at zero temperature in the non-ballistic regime, where intra- or/and inter-dot electron correlations and coherent electron tunneling are of comparable magnitudes. The parameters we use in our calculations correspond to GaAs quantum dots of ∼ 104 nm2 area and a typical interdot spacing a of ∼ 200 nm. The single-particle intradot level spacing, 1 = h¯ ω0 ∼ 0.3 meV, is of the same order as the tunneling coupling strength t between the dots, e.g. t ∼ 0.1 meV. With our choice of parameters (t ∼ 1) the arrays are in the Collective Coulomb Blockade regime where the Coulomb Blockade in the individual dots is destroyed [2, 3]. The intradot charging energy U is ∼ 1 meV. Within the Mott–Hubbard type model approximation [2, 3] the Hamiltonian of the 2D quantum dot array in an external magnetic field can be written as X X U X 2 X Vi j † † εiα (B)ciα ciα + ρˆi + (tα,i j eiφi j ciα c jα + H.c.), (1) ρˆi ρˆ j − H= 2 2 i,α i ij hi, ji,α Z e AE · lEi j , with AE as the magnetic vector potential. (2) where φi j = h¯ i j P † † The indices i, j denote the spatial positions of dots; ρˆi = α ciα ciα is the density operator, where ciα (ciα ) is a creation (annihilation) operator for an electron on the ith dot in a state α. We use the capacitance matrix formalism to set Vi j = (C −1 )i j , where Cii = C g + Ni C, Ci j = −C, i j indices correspond to the neighboring dots [3, 6, 7]. The capacitance C g of the dot with respect to the nearest gate approximates the 2 intradot electrostatic charging energy of each dot U = Ce g , the capacitance C approximates the electrostatic coupling between electrons on the ith dot and the Ni neighboring dots. We use the tight-binding approximation

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Fig. 2. Conductance vs. chemical potential µ through arrays of 9 quantum dots (9 × 1 (A); 3 × 3 (B), (C)) with field-independent single-particle levels εi↑ = 1, εi↓ = εi↑ + 0.031 and the long-range interaction Vi j = 0; T = 35 mK, t1↑ = t1↓ = 0.1 meV, t2↑ = t2↓ = 0.12 meV. The symmetry about the half-filling in the 9 × 1 chain in (A) is broken in (B) in the 3 × 3 array in the interacting U = 1 meV systems. The resonant transport through the non-interacting 3 × 3 array U = 0 is shown in (C).

keeping only the nearest neighbor tunneling. The magnetic field B enters through the tunneling amplitudes ti j modified [cf. eqn (2)] by the Peierls phase factors [8], and through the intradot single-particle field dependence entering εiα which is treated in the harmonic confinement approximation [9, 10]. The magnetic field also defines the Zeeman energy in the usual way. We consider a single spin–split level per dot unless otherwise stated: α = 1, 2, and εiα ≡ εα = h¯ [(ωc /2)2 + ω02 ]1/2 + (−1)α ge µ B B/2. We neglect all intradot correlation effects (other than the effect of the Hubbard U ) which have to be included in the model for fields larger than B ≈ [ge (ge + 2)]−1/2 (1/µ B ), when the single-particle levels start to cross. Our goal here is to investigate the effect of quantum coherence (i.e. finite values of tα ) on the CB physics in two-dimensional arrays. The 2 magnetic field required to spin polarize the array is determined by gµ B B ∼ Ut . We perform our calculations by doing an exact diagonalization of eqn (1) in the basis of the total number of electrons N in the array and E We use the Lanczos tridiagonalization method for our exact the total spin Sz in the direction of the field B. diagonalization of eqn (1), and carry out a minimization over Sz to find the stable ground state for a given N [11]. Our work is thus a 2D generalization of the 1D theory presented in Refs [2, 3]. Our 2D persistent current results are, however, qualitatively new since there can be no persistent current in an isolated 1D system without periodic boundary conditions.

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Fig. 3. The magnetic field dependence of the conductance spectrum in the 3 × 3 non-interacting (A) and interacting (B) arrays for the parameters of Fig. 2 for one half of the magnetic period, i.e. the flux through the elementary cell varies from 0 to φ = 0.5φ0 , with φ0 = h/e.

3. Persistent current We consider an array of coupled quantum dots placed in the arrangement of the single-electron capacitance spectroscopy (SECS) experiments by Ashoori et al. [12]. Then the number of excess electrons in the array, which is controlled by the gate voltage, can be fixed. In a transverse magnetic field the SECS traces the field-dependence of the ground state energy of the array. For our typical 2D array of the semiconductor quantum dots the phase coherence length lφ ≈ 50 µm is larger than the characteristic size of the system. In this situation our coherent model (without any phase-breaking) should apply, and it predicts the existence of an orbital magnetization in the multiply-connected 2D sample due to the existence of persistent currents in the system [13, 14]. If we define the plane of the square lattice as the (x − y) plane, and the z-direction is taken to be the direction of the applied field, then the magnetic moment per unit cell has only a z-component given [5] by the negative of the derivative of the Hamiltonian of eqn (1) with respect to the magnetic flux φ, mz ≡ −

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We calculate the expectation value of eqn (3) in the ground state of the disordered 2D array. We assume a random disorder, modeled by a uniform distribution of width W centered around the on-site energies εiα . The energy shift between spin ‘up’ and spin ‘down’ states is left unchanged. Our typical results for the persistent current in a 2D quantum dot array are shown in Fig. 1. The topology of a 2D quantum dot array is, by definition, ‘multiply connected’, and all possible closed paths contribute to the resultant current between any two sites in a two-dimensional array, leading to new qualitative features in the 2D orbital magnetization as compared to the corresponding one-dimensional ‘singly-connected’ situation. We find that the orbital magnetization in the two-dimensional array is larger in magnitude than in the corresponding one-dimensional ring with the same number of quantum dots. The quantitative effect of short-range electron–electron interactions on the two-dimensional persistent current is weaker than in one dimension. Localization effects due to random disorder are also weaker in two dimensions as compared to one-dimensional results [15]. Persistent current experiments [16] based on orbital magnetization measurements in coherent 2D quantum dot arrays should, in principle, be able to verify our predictions.

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4. Conductance The coherent states discussed in Section 2 can be probed in the low bias CCB conductance spectrum by varying the electrochemical potential µ of the gate electrode. We consider an array of L x × L y dots weakly coupled to the right and to the left leads through the tunneling matrix elements connecting the electrodes to the adjacent dots. Then in the limit 1E  k B T  h¯ 0, where 1E is the level spacing in the array and 0 is the largest rate of tunneling out of the array through the right or the left junction, the two-terminal conductance is given within the linear response theory by the same formula as the formula derived for a single dot [17] or for a linear chain of dots [3, 4]: e2 X 0rN 0lN f 0 (E N0 − E N0 −1 ), where h N 0rN + 0lN 2 X X r,l † = 2π ti,nα ci,nα |N − 1i ρ r,l (E N0 − E N0 −1 ). hN |

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In eqn (5) ρ r,l (E N0 − E N0 −1 ) is the density of states in the leads, and in eqn (4) f 0 (E N0 − E N0 −1 ) is the derivative of the Fermi–Dirac distribution function. The inner sum in the tunneling rates in eqn (5), derived using Fermi’s Golden Rule, is over all adjacent dots ir,l coupled to the right or the left lead through the r,l . By adjusting the various matrix elements the system can be made to couple tunneling matrix elements ti,nα to the leads through tunneling from individual dots. In Figs 2 and 3 we show some representative results for our calculated coherent transport properties in 2D quantum dot arrays. The linear response regime in our calculated Coulomb Blockade transport properties of the 2D array shows an intricate magnetic field induced modulation of the conductance peaks, which is not seen in transport through the Landau band of a single dot. The coherent states, associated with the many-body Hofstadter [18] spectrum in an isolated two-dimensional array, lead to splitting of the conductance peaks due to resonant tunneling through each individual many-body state. Our calculated coherent transport features can, in principle, be verified in Coulomb Blockade transport experiments [1] carried out in coherent 2D quantum dot arrays. Currently existing multidot structures are, however, too small (only 2 or 3 coupled dots) for the observation of coherent two-dimensional effects in conductance properties predicted here.

5. Two-terminal capacitance We now consider the quantum dot array of Section 2 in the Coulomb Blockade regime when the tunneling between the system and the bias leads is suppressed. This regime can be realized experimentally in a.c. measurements if the frequency of the a.c. bias voltage exceeds the largest rate of tunneling out of the array through either of the junctions. We assume an ideal voltage source, which is able to sustain a constant voltage difference between the bias leads. The array with a fixed number N of excess electrons is coupled capacitively to the bias leads. In the presence of electrostatic equilibrium throughout the combined system of the leads and the array, the minimum of the Gibbs free energy determines the equilibrium charge on the leads and the charge distribution inside the array. A novel feature of the tunneling-coupled coherent quantum dot array is that the competition among the applied electric field, the charging energy, and the tunneling can lead to the polarization of the array (depending on the strength of the applied bias voltage), giving rise to a quantum correction of the classical effective capacitance Ceff of the system [19]. In the limit of the capacitances Cr,l of the junctions being much less than the capacitance C g of the dots (with respect to the gate electrode), i.e. Cr,l  C g , we study the simple configuration depicted in Fig. 4. We neglect interdot capacitance matrix elements, and the dots are assumed to be connected only through tunneling. These simplifications should not affect the qualitative structure of quantum corrections to the capacitance, but when taken into account they rescale the bias voltage and reduce somewhat the magnitude of the correction. The operator for the Gibbs free

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QM Fig. 4. The quantum correction Ceff (N ) to the effective capacitance of the linear 2 × 1 (A), 4 × 1 (B), 6 × 1 (C) chains of quantum QM (N ) for arrays with different dots vs. the bias voltage V . The parameters are the same as in Fig. 1. Each curve in the figure gives Ceff QM number N of excess electrons from 1 to L. The Ceff (N ) curves for n = 0.5, n = (L − 1)/2L, and n = (L − 2)/2L electron fillings in QM (N ) curves for n < (L − 2)/2L the arrays are shown with the solid, the dashed, and the dashed-three dot lines respectively. The Ceff

electron fillings in the arrays are shown with the dotted lines. The charge distribution, ρ(xi ) = hN |ρ(x ˆ i )|N i, for V = 1 mV is ρ(1) = 1.5, ρ(L x ) = 0.5, ρ(xi 6= 1, L x ) = 1 for the half-filled case. A labeling convention for the coordinates of quantum dots in an array is given in a typical configuration depicted in the inset.

energy for the studied configuration is L 1X V2 e2 2 eV G = Cg L y [1 − L y ] + ρˆ − 4 2 i=1 Cii i 2

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effective two-terminal capacitance of a coherent quantum dot array gives rise to the Coulomb oscillations in the capacitance versus the applied source–drain voltage due to the spatial charge polarization in the array. The effective capacitance shows a non-trivial magnetic field dependence due to the existence of persistent currents in the two-dimensional array in the presence of electrostatic equilibrium. Our calculated quantum corrections to the two-terminal capacitance of a 2D quantum dot array can, in principle, be measured by experiments similar to recent SECS experiments by Ashoori et al. [12].

6. Summary In this article we have briefly reviewed some of our recent numerical results on the persistent current, the conductance, and the two-terminal capacitance of coherent 2D and 1D quantum dot arrays using the Mott–Hubbard many-body approach. We focus on the regime where the interdot tunneling, the intradot level separation and the Coulomb charging energies are all comparable in magnitudes. We find a large number of interesting (and somewhat complicated) effects and discuss possible experiments where our predicted effects can, in principle, be observed. Direct experimental observation of coherent many-body effects in quantum

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dot arrays will be a verification of the recently introduced artificial molecule concept in 2D semiconductor structures. It should, however, be emphasized that the existing coupled quantum dot systems which can be fabricated with the current technology may be too small (only 2 or 3 coupled coherent dots) for the observation of 2D Mott–Hubbard correlation effects discussed in our paper. Thus a direct experimental observation of 2D artificial molecular effects may have to await substantial improvement in quantum dot array fabrication techniques so that 2D coherent arrays of reasonable sizes (e.g. 2 × 2, 3 × 3) are available. Acknowledgements—This work is supported by the US-ONR. The authors thank C. A. Stafford for helpful discussions during the early part of this work.

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