Non-equilibrium triplet blockade in parallel coupled quantum dots

J Fransson

Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44, Stockholm, Sweden and Physics Department, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden

Email: jonasf@lanl.gov

Received 5 April 2006
Published 17 July 2006

From fundamental aspects of spin and charge correlations, the two-level system in a double quantum dot (DQD) has recently become highly attractive. It has been demonstrated that spin correlations lead to Pauli spin blockade in serially coupled quantum dots (QDs), where the current is suppressed because of spin triplet correlations [1]-[4], something which may be applied in spin-qubit readout technologies [5]. Pauli spin blockade has also been reported for general DQDs with more than two electrons [6]. Recently, the Pauli spin blockade with nearly absent singlet-triplet splitting has been employed in studies of hyperfine couplings between electron and nuclear spins [7]-[12]. Besides being present in serially coupled QDs, it is relevant to ask whether an analogue of the Pauli spin blockade is obtainable in parallel QDs.

The purpose of this paper is to demonstrate that parallel coupled QDs, see figure 1, exhibit regimes of non-equilibrium triplet blockade. Here only one of the QDs is tunnel coupled to the external leads while the second QD functions as a perturbation to the first QD. Important quantities in order to find the non-equilibrium triplet blockade regime is that the QDs are coupled through charge interactions, e.g. interdot Coulomb repulsion and exchange interaction, and weakly through tunnelling. In the absence of interdot exchange interaction, there may be regimes of usual Coulomb blockade in a finite bias voltage range around equilibrium.

In the presence of a sufficiently large ferromagnetic interdot exchange interaction, the triplet states |σ_A|σ_B, σ  =  ↑,↓ (one electron in each QD with equal spins) and acquire a lower energy than the lowest two-electron singlet (the singlet states being superpositions of the Fock states ). Then, the triplet naturally becomes the equilibrium ground state with a unit occupation probability, provided that the two-electron triplet state has a lower energy than all other states. The triplet persists in being fully occupied for bias voltages smaller than the energy separation between the triplet and singlet states, although transitions between the one-electron states and the singlets may open for conduction. However, for larger bias voltages, this low bias triplet blockade is lifted as the transitions between the triplet and the one-electron states become resonant with the lower of the chemical potentials of the leads. At this lifting, the current through the system is mediated via transitions between the two-electron singlets and the one-electron states.

The non-equilibrium triplet blockade regime is entered at bias voltages such that transitions between three-electron states and, at least, one of the singlet states become resonant, see figure 1, while transitions between the three-electron states and the triplet lie out of resonance. At those conditions, an electron can enter the DQD from the lead with the higher chemical potential, through transitions between the singlet and the three-electron states. Transitions from the triplet to the three-electron states are suppressed since the bias is lower than the energy barriers between those states. However, electron tunnelling from the three-electron states in the DQD to the lead with the lower chemical potential is supported through transitions from those states to the triplet, since the tunnelling barrier to this lead is sufficiently low. Thereto, the probability for such transitions are about unity whereas the probability for transitions between the three-electron states and the singlet is at most a half. Finally, charge that end up in the triplet through this process is trapped in this state because of the negligible probability for transitions between the triplet and the one-electron states.

It is noticed that a finite (ferromagnetic) interdot exchange interaction is not a necessary condition for the existence of a non-equilibrium triplet blockade regime. Nevertheless, a ferromagnetic exchange yields a larger degree of freedom in the variation of the interdot tunnelling and, also, allows a higher temperature.

For quantitative purposes, consider two single-level QDs (_A,_B, spin-degenerate) with intradot charging energies (U_A,U_B), which are coupled by interdot charging (U '), exchange (J≥0), and tunnelling (t) interactions. Specifically, the DQD is modelled by [13]-[15] , where , σ,σ '  =  ↑,↓, i  =  A,B, are the spins of the two levels. In analogy with the Pauli spin blockade in serially coupled QDs [1, 4], it is required that the lowest one-electron states, the triplet, and the two lowest singlets are nearly aligned, and that the lowest three-electron states lie below the equilibrium chemical potential μ. Hence, E_T≈ E_S1≈ E_S2≈min_{n  =  1}⁴{E_1n} < min_{n  =  1}⁴{E_3n} < μ < E₄, where E_T and E_Sn, 1,2, are eigen- energies for the triplet and the two lowest singlet states, respectively, whereas E_1n, E_3n and E₄ are the energies for the one-, three- and four-electron states, respectively. This requires that μ - _B≈Δ, U '≈Δ, and U_A≈2Δ≤ U_B, where Δ  =  _B - _A. The inequality U_A≤ U_B points out that the QDs do not have to be identical, merely that the charging energy of the second QD should be lower bounded by the charging energy of the first. It should be emphasized, however, that the presence of the second QD is essential in order to obtain the effect discussed in this paper. Finally, weakly coupled QDs, e.g. ξ  =  2t/Δ  1 implies that the energies for the lowest one- and three-electron states acquire their main weight on QD_A. This condition yields a low (large) probability for transitions between the triplet and the lowest one-electron (three-electron) states.

In general there are 16 eigenstates of , labelled {|N,n,E_Nn} denoting the nth state of the N-electron (N  =  1, ...,4) configuration at the energy E_Nn [4]. In diagonal form, the DQD is thus described by . Taking the leads to be free-electron like metals and the (single-electron) tunnelling between the DQD with rate v_kσ, the full system can be written as [4]

where is the matrix element for the transition |N,nN + 1,n '|. The operator d_Aσ signify that electrons tunnel from molecular like orbitals in the DQD through QD_A to the leads, which appropriately describes the physical tunnelling processes.

Following the procedure in [4], the occupation of the eigenstates are described by a density matrix ρ  =  {|N,nN,n|}. In the Markovian approximation (sufficient for stationary processes) one thus derives that the equations for P_Nn≡|N,nN,n| to the first order in the couplings between the DQD and the leads, can be written as

where P _- 1n  =  P_5n≡0. Here, denote the energies for the transitions |N,nN + 1,n '|, while . Also, f ⁺ _L/R(ω)  =  f(ω - μ_L/R) is the Fermi function at the chemical potential μ_L/R of the left/right (L/R) lead, and f ^- _L/R(ω)  =  1 - f ⁺ _L/R(ω). Effects from off-diagonal occupation numbers |N,nN,n '|, which only appear in the second order (and higher) in the couplings, are neglected since these include off-diagonal transition matrix elements to the second order (or higher) which generally are small for ξ  1.

Since the low bias triplet blockade can be found for weakly coupled QDs whenever J > 0 is sufficiently large, the following derivation focus on the non-equilibrium blockade. The non-equilibrium blockade discussed here is driven by opening transitions between the two- and three-electron states. For simplicity, assume that the bias voltage V  =  (μ_L - μ_R)/e is applied such that μ_L/R  =  μ± eV/2. Then for |eV| < 7Δ/4, k_BT < 0.01U_A, and ξ < 0.2, which is sufficient for the present purposes, only the population numbers P_1n, n  =  1,2, N_T  =  P_2n/3, n  =  1,2,3, P₂₄, P₂₅ and P_3n, n  =  1,2, are non-negligible. The other populations are negligible since the corresponding transition energies lie out of resonance. Because of spin-degeneracy it is noted that P_1n  =  N₁/2, n  =  1,2, and P_3n  =  N₃/2, n  =  1,2, which reduces the system to five equations for the population numbers. As discussed in the introduction, the non-equilibrium blockade arises when transitions between a singlet and the three-electrons state are resonant. Therefore, the bias voltage is tuned into the regime where μ_L lies around these transition energies, e.g.^Note1  , n  =  1, ...,5, n '  =  1,2 (here eV  >  0, the case eV  <  0 follows by symmetry of the system). For such biases it is clear that f_L ⁺ (Δ_2n,1n ')  =  f_R ^- (Δ_2n,1n ')  =  1, n  =  1, ...,5, n '  =  1,2, and that f_R ⁺ (Δ_3n ',2n)  =  0, n  =  1, ...,5, n '  =  1,2. It is also clear that the charge accumulation in the triplet is lifted for biases that supports transitions from the triplet to the three-electron states, hence the bias voltage has to be such that f_L/R ⁺ (Δ_3n ',2n)≈0, that is Δ_3n ',2n  =  E_3n '  -  E_T  >  μ_L + k_BT, n  =  1,2,3, n '  =  1,2. Thus, the equations for the population numbers can be written as

Here, (n '  =  1,2, n  =  4,5) , , , and are the matrix elements for the relevant transitions. The above relations are due to spin-degeneracy, e.g. Δ_2n,11  =  Δ_2n,12 and Δ_31,2n  =  Δ_32,2n, n  =  1, ...,5. Using equations (3), charge conservation ( ) thus implies that

Now, the matrix elements L_n²,Λ_n², n  =  4,5, are finite and bounded, however, L₄²,2Λ₅²→1 and L₅²,Λ₄²→0 as ξ→0, hence the last term in equation (4) is at most 1/2 for weakly coupled QDs since p→0, ξ→0, in the considered bias regime (see discussion below). However, the ratio 2p(κ/β)² is finite for all ξ and J > 0, while it diverges as ξ→0 for J  =  0, see the main panel in figure 2. For weakly coupled QDs, one thus finds that N_T≈1/(1 + [1 + 2p(κ/β)²] ^- 1)≈1 whenever 2p(κ/β)²  1. The inset in figure 2 illustrates a subset in (t,J)-space where this ratio is larger than 10². At this condition, the boundary is approximately given by J(t)  =  J₀ - 15t²[1 + (10t)²].

Figure 2. Variation of the ratio 2p(κ/β)2 as function of J for different t at constant Δ, U ', UA/B. The inset shows the region in (t,J)-space where 2p(κ/β)2 > 102.

Using the transport equation derived in [16], identifying and , the current in the considered regime is given by

This expression clearly shows that a large value of 2p(κ/β)² yields a suppression of the current, that is, at the formation of a unit occupation in the triplet state. For biases such that μ_L < min_nn '{Δ_3n,2n} is follows that f_L ⁺ (Δ_3n ',2n)≈0 ⇒ p≈0, which accounts for a lifting of the triplet blockade where the current is ~2p(κ/β)² larger than in the blockaded regime.

The non-equilibrium triplet blockade depends on the interplay between J and t. A reduced t leads to a strong localization of the odd number states in either of the QDs, which for Δ > 0 leads to that the lowest odd number states are strongly localized on QD_A. Then, the probability for transitions between the triplet, and the one-/three-electron states is small/large (β→0/κ→1).

The singlets, on the other hand, are expanded in terms of the Fock states with weights that are slowly varying functions of t, however, strongly dependent on J. A negligible J yields that the two lowest singlets are almost equally weighted on the states and |↑↓_A|0_B. Increasing J > 0 redistributes the weights such that the lowest singlet (|2,4) acquires an increasing weight on |↑↓_A|0_B, whereas the second singlet (|2,5) becomes stronger weighted on . Hence, for a finite J > 0 and t→0, this redistribution leads to that transitions between the lowest one-electron states and |2,4 (|2,5) occur with an enhanced (reduced) probability, e.g. L₄²→1 (L₅²→0), and oppositely for transitions between the singlets and the three-electron states, e.g. Λ₄²→0 (Λ₅²→1/2). This implies that p→0 as t→0 while p(κ/β)² remains almost constant. This constant, however, becomes larger (smaller) for smaller (larger) J.

The typical variation of the triplet state occupation number N_T, calculated from equation (2), as function of the bias voltage and the equilibrium chemical potential for 0 < J < J₀ - 15t²[1 + (10t)²] and t/(k_BT) < 2 is plotted in figure 3(a)). Here, varying the equilibrium chemical potential mimics the effect of applying an external gate voltage V_g by means of which the levels of the DQD are shifted relatively the equilibrium chemical potential. The extended diamond marks the region where the occupation of the triplet is nearly unity and where the transport through the DQD is blockaded. The calculated current is displayed in figure 3(b)), from which it is legible that the triplet blockade regime is subset of a larger domain of a nearly vanishing current through the DQD. The two diamonds within the low current regime are caused by a lifting of the triplet blockade (see the introduction), where the current is mediated by transitions between the one-electron states and the singlets.

As is seen in figure 3, shifting μ in the range _B + (Δ - J,2Δ) causes an extension of the low bias triplet regime since the transitions between the triplet and the one-electron states become resonant at higher biases. On the other hand, the non-equilibrium triplet blockade is shifted to lower biases since μ lies closer to the transition between the three-electron states and the singlets. The two blockade regimes merge into single as |μ - Δ_3n ',2n| < |μ - Δ_21,1n '|, n  =  4,5, n '  =  1,2, e.g. for μ - _B(3Δ/2,2Δ), see figure 3. Shifting μ in the interval _B + (Δ/2,Δ - J) removes the low bias blockade since the one-particle states become the equilibrium ground state. The non-equilibrium blockade is shifted to lower biases, here caused by transitions between the one- and two-electron states which tend to accumulate the occupation in the triplet.

While the case Δ > 0 is considered here, the non-equilibrium blockade is also found in the opposite case, e.g. Δ < 0 and μ - _A≈Δ. In this case, however, the system has to be gated such that only the four-electron state lies above μ, whereas the charge accumulation of the triplet state is governed by the same processes as described here.

It should be noted that higher order effects, as well as singlet-triplet relaxation, have been neglected in the equation for the population probabilities P_Nn. However, in many aspects the situation discussed here corresponds to the experiment reported in [1], hence the effect considered should be measurable under much the same conditions. Therefore, as in the case of the serially coupled DQD, the higher order effects give contributions that are at least two orders of magnitude smaller than the second order contributions. Therefore, these can be neglected in the present study. On the same basis as in the description of the serially coupled DQD [4], the singlet-triplet relaxation may be neglected here.

The conditions required for the existence of non-equilibrium triplet blockade, concerning the intra- and interdot charge interactions for weakly coupled QDs, have been experimentally obtained for serially coupled QDs [1]-[3]. The additional requirement, i.e. a ferromagnetic interdot exchange interaction which is larger than the interdot tunnelling and the thermal excitation energy, is accessible within the present state-of-the-art technology [3, 12, 17, 18].

Acknowledgments

Support from Carl Trygger's Foundation is acknowledged. The Institute of Physics and Deutsche Physikalische Gessellschaft is gratefully acknowledged for covering the publications costs.

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Notes

Note1  There is one (spin-degenerate) transition energy (Δ_3n,26,n  =  3,4) in the vicinity of μ which is only relevant for eV≥ U since min_n = 1⁴{Δ_26,1n - μ} U/2.