INSTITUTE OF PHYSICS PUBLISHING

NANOTECHNOLOGY

Nanotechnology 14 (2003) 152–156

PII: S0957-4484(03)53810-7

Spin-polarized pumping in a double quantum dot E Cota1,2 , R Aguado1 , C E Creffield1 and G Platero1 1 2

Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, Madrid, Spain Centro de Ciencias de la Materia Condensada—UNAM, Mexico

Received 20 September 2002, in final form 31 October 2002 Published 10 January 2003 Online at stacks.iop.org/Nano/14/152 Abstract We study the pumping of spin-polarized electrons in a double quantum dot system with up to two electrons per dot, via an applied AC field and a constant magnetic field. The behaviour of the current through the double-dot system is studied as a function of the AC field and coupling to the leads, using a Markov master equation approach for the time evolution of the reduced density matrix. For up to two electrons in the system, we find that the formation of a spin-triplet state blocks the current through the device, and analyse possible solutions. When we incorporate three-and four-particle states, with up to two opposite spin electrons per dot, we find a regime where the pumping of spin-polarized electrons is realized through double occupancy states in each dot. This property is robust against spin-relaxation and decoherence processes which are taken into account phenomenologically. Finally we study the effects of applying a pulsed AC field and the possibility of the resolution of Rabi oscillations.

1. Introduction Understanding and controlling the behaviour of single electron spins in nanostructures has become the subject of intense investigation due to its relevance to quantum information processing. In particular, the pumping of electrons in mesoscopic devices can be realized by applying an AC field with a tunable frequency. As the decoherence time in these devices is typically long, any transport controlled in this way would be truly quantum coherent; this is particularly exciting due to its possible applications in the field of quantum computation. Experimental work by Nakamura et al [1] demonstrated the possibility of the observation and quantum control of Rabi oscillations in an electronic two-level system with the application of short voltage pulses via a gate electrode. Theoretical studies of electron pumping and spin filters in single and double quantum dot (QD) systems, where only single electron states in each dot take part in the dynamics, have been carried out [2, 3], and the current as a function of time has been calculated for pulsed ESR fields, allowing Rabi oscillations to be resolved. A review of the present experimental status of transportation studies in double-dot systems was published recently [4]. In this paper we study the AC pumping of electrons in a double-dot system with up to two particles per dot in

the system, and thus we include states in which the dots are doubly occupied with opposite spin electrons. By using master equations for the time evolution of the reduced density matrix (RDM), we find that the spin is an important factor that modifies the pumping mechanism. This complements recent work [5] where we considered up to two electrons in the system and discussed the spin blockade of the device due to the formation of the spin-triplet state, similarly to experimental observations of dc transport in double QDs by Ono et al [6]. Intrinsic relaxation and decoherence mechanisms are taken into account phenomenologically through the T1−1 (relaxation rate) and T2−1 (decoherence rate) terms in the corresponding density matrix rate equations [7] and are found to have small effects over the range of values of the relevant parameters that are considered.

2. Model Consider a double QD system connected to leads by tunnel barriers, described by the Hamiltonian Hˆ = Hˆ S + Hˆ T + Hˆ leads + Vˆ S R , where Hˆ S represents the double-dot system, with one spin-degenerate single-particle level per dot, Hˆ T is the interdot tunnelling Hamiltonian, and Vˆ S R is the coupling between the system and leads (reservoir). Explicitly

 j nˆ j σ + VL R nˆ L nˆ R + Ui nˆ i↑ nˆ i↓ (1) Hˆ S =

0957-4484/03/020152+05$30.00 © 2003 IOP Publishing Ltd Printed in the UK

j ∈{L ,R}σ

i∈{L ,R}

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Spin-polarized pumping in a double QD

2.1. Markovian master equation We use a closed and time-local equation for the RDM: ρˆ S(I ) (t) = Tr R {ρˆ (I ) (t)} where ρˆ (I ) is the total density matrix in the interaction representation and the trace is carried out over the degrees of freedom of the reservoir, R. With the usual assumptions (Markov approximation) one arrives at the time evolution equation for the RDM elements (details of the derivation can be found in the literature [7, 8]). This master equation can be written in the Schr¨odinger picture as

Figure 1. Schematic representation of the double QD in the pumping configuration and showing Zeeman splitting on both dots. E S1 and E S2 are the energies of the doubly occupied states in each dot. In the case N
2, we impose the condition E S1 > µ L so that only one particle is allowed in dot L. In the pumping configuration, for N
4, the condition is E S1 < µ L , E S2 > µ R while E S2 − z < µ R . The spin (↓) polarized current is obtained either through the sequence |↓↑, ↑ → |↑, ↓↑ → |↑, ↑ → |↓↑, ↑ or |↓↑, ↑ → |↑, ↓↑ → |↓↑, ↓↑ → |↓↑, ↑ which involve states of double occupation on both dots.

Hˆ T = −t L R


† (cˆ Lσ cˆ Rσ + h.c.)

(2)

σ




Hˆ leads =

lk dˆ† lkσ dˆlkσ

(3)

l∈{L ,R}kσ

Vˆ S R =




† (γlσ dˆlkσ cˆlσ + h.c.).

(4)

l∈{L ,R}kσ

Here,  j is the confinement energy level in the j th QD, VL R is the inter-dot Coulomb interaction and Ui is the intra-dot Coulomb interaction energy for dot i . The amplitude t L R is the tunnelling matrix element between the dots, cˆ†j σ and cˆ j σ are the creation and annihilation operators for electrons in dot j with spin σ , nˆ j σ = cˆ†j σ cˆ j σ is the electron number operator at † dot j with spin σ . dˆlkσ and dˆlkσ are the creation and annihilation operators for electrons in reservoir l with momentum k, spin σ and energy lk , and γlσ are transition amplitudes to and from lead l, which can be spin dependent. Hˆ S may also include a Zeeman coupling term with splitting z proportional to a constant magnetic field Bz acting on both QDs, as shown schematically in figure 1. In addition we consider an external AC field acting on the dots such that  L(R) = ±0 /2 →  L(R) (t) = ±(0 + ˜ cos ωt)/2

(5)

where ˜ and ω are the amplitude and frequency, respectively, of the applied field and 0 =  L −  R is the energy difference between single particle energy levels in the absence of the AC field. In the particle number representation, our basis consists of 16 states, corresponding to 0, 1 and 2 extra electrons (with opposite spin) on each QD, i.e., |1 = |0, 0, |2 = |↑, 0, |3 = |↓, 0, |4 = |0, ↑, |5 = |0, ↓, |6 = |↑, ↑, |7 = |↓, ↓, |8 = |↑, ↓, |9 = |↓, ↑, |10 = |↑↓, 0, |11 = |0, ↑↓, |12 = |↑↓, ↑, |13 = |↑↓, ↓, |14 = |↑, ↑↓, |15 = |↓, ↑↓, |16 = |↑↓, ↑↓.

i ˆ ρ(t) ˙ s  s = −iωs  s ρ(t)s  s − φs  |[ Hˆ T , ρ(t)]|φ s h ¯

Wsm ρmm − Wks ρss (s = s  )  m = s k = s +  −γs  s ρs  s (s = s  )

(6)

where φs are eigenfunctions of HS with energy E s(S) . The first line in equation (6) represent reversible dynamics (‘coherent’ effects), in terms of the transition frequencies ωs  s = (E s  − E s )/¯h , while the following terms describe relaxation (irreversible dynamics) in the open system. In the last line, γs  s is a so-called non-adiabatic parameter whose real part is responsible for the time decay of the off-diagonal density matrix elements (coherences), i.e., it is the decoherence rate due to interactions with the reservoir. These parameters can be written in terms ratesWi j (see below)  of the transition  by Re {γs  s } = ( k=s Wks + k=s  Wks  )/2. To account for intrinsic decoherent processes acting even in the isolated system, we add a term T2−1 to γs  s for terms involving spin-flips. Typically, T2 ≈ 100 ns [9], and is at least an order of magnitude smaller than T1 , the spin relaxation time. This time is of the order of microseconds [10] and is given by (W↑↓ + W↓↑ )−1 , where W↑↓ and W↓↑ are spin-flip relaxation rates, such that W↑↓ / W↓↑ ≈ exp(z /kT ). These spin-relaxation rate terms are taken into account in the evolution equations (6) for the diagonal elements of the RDM. In the sequential tunnelling regime, where one electron can tunnel from the lead to the dot and back, the number of electrons in each dot fluctuates between N and N  = N ± 1. The transition rates, Wmn , from states |n to |m are given by Wmn = lσ {|n|clσ |m|2 fl (ωmn − µl )δ N  ,N +1 + |m|clσ |n|2 (1 − fl (ωnm − µl ))δ N  ,N −1 }

(7)

where fl () is the Fermi function in lead l with chemical potential µl and temperature T . lσ ∝ νσ |γlσ |2 are transition amplitudes. Here, the first term (∝ f ) describes electrons coming onto dot l, from lead l, while the second (∝ (1 − f )) describes electrons leaving from dot l onto lead l, both at a frequency given by the system’s transition energy ωmn . In addition to these transition rates due to tunnelling between the system and leads, we include spin-flip processes as mentioned above. Thus, for example, the equation for the time evolution of state |14 = |↑, ↓↑, which has an important role in the transport through our system, is 2t L R Im{ρ12,14 } h¯ + (W14,6 ρ6 + W14,8 ρ8 + W14,11 ρ11 + W↑↓ ρ15 + W14,16 ρ16 )

ρ˙14 =

− (W6,14 + W8,14 + W11,14 + W16,14 + W↓↑ )ρ14

(8) 153

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where, for convenience we use the notation ρi = ρii . Here the first term represents the contribution due to interdot tunnelling, while the second and third terms are the contributions from transitions entering (+ sign) or leaving (− sign) the corresponding state. The solution of the master equation (6) is carried out numerically. We verify that the general property Tr {ρ} ˆ =1 for the RDM is satisfied at all times. The numerical solution of the RDM is used to calculate the dynamics of the current through the system, as the physical observable of interest here. For example, the current to lead R is given by I R (t) = W1,4 ρ4 + W1,5 ρ5 + W2,6 ρ6 + W3,7 ρ7 + W2,8 ρ8 + W3,9 ρ9 + (W4,11 + W5,11 )ρ11 + W10,12 ρ12 + W10,13 ρ13 + (W6,14 + W8,14 )ρ14 + (W7,15 + W9,15 )ρ15 + (W12,16 + W13,16 )ρ16 − ((W4,1 + W5,1 )ρ1 + (W6,2 + W8,2 )ρ2 + (W7,3 + W9,3 )ρ3 + W11,4 ρ4 + W11,5 ρ5 + W14,6 ρ6 + W15,7 ρ7 + W14,8 ρ8 + W15,9 ρ9 + (W12,10 + W13,10 )ρ10 + W16,12 ρ12 + W16,13 ρ13 )

(9)

ˆ ↑, with a similar expression for I L (t). Here, ρ4 = 0, ↑|ρ|0, ˆ ↓, etc. In the stationary limit, I L = I R and ρ5 = 0, ↓|ρ|0, in general, at any given time, the total current through the system is I (t) = (I L (t) + I R (t))/2. For simplicity, we take the temperature T = 0 and VL R = 0. Our conclusions are nevertheless valid for finite temperatures provided that T is sufficiently small compared to the relevant energy spacings. Also, the inclusion of VL R = 0 in the Hamiltonian (1) would change the energies of some states but our main findings would remain the same.

3. Results and discussion 3.1. The case with N = 0, 1, 2 electrons Consider the double-dot system with up to two electrons and tunnelling coupling t L R , initially prepared in state |4 = |0, ↑, connected to leads L and R, with couplings  L and  R . The chemical potentials of the leads are taken to be equal µ L = µ R = µ in all calculations (pumping configuration). With an applied AC field with frequency ω tuned in resonance with E 11 − E 9 (where |9 = |↓, ↑ and |11 = |0, ↓↑) it is possible to overcome the Coulomb repulsion and pump electrons through the doubly occupied state. Due to the AC field, the interdot coupling is renormalized and the effective Rabi frequency between the singly and doubly occupied states is given approximately by  R ≈ 2t L R J1 (˜/ω), where J1 is the Bessel function of order 1 [11, 12]. In order to investigate the role of spin polarization in the pumping mechanism we introduce a Zeeman splitting in the right dot due a local magnetic field. We see that the pumping mechanism starts to populate the state |11 = |0, ↓↑, however, the density matrix shows very clearly that the spin-triplet state |6 = |↑, ↑ dominates the dynamics very rapidly (figure 2(a)). The configuration of chemical potentials is such that the current through the right barrier can occur only through the process |↓, ↑ → |0, ↓↑ → |0, ↑ (i.e. only the pumped spin ↓ contributes to the current). As a consequence, the total current eventually goes to zero for ρ11 → 0 and ρ6 → 1. In other ˆ ↑ blocks words, the appearance of the triplet ρ6 = ↑, ↑|ρ|↑, 154

Figure 2. Density matrix and total current as a function of time, for the case N
2, with initial condition ρ4 = 1, for t L R = 0.1, µ = 10, U L = 20, U R = 12,  L =  R = 0.01 with Zeeman splitting z = 6 on the right dot and the AC field tuned to the resonance condition, ω = 15 and ˜ = ω. (a) For unpolarized leads, f = 1, the spin-triplet state ρ6 dominates the dynamics and the spin-blockade of the pumping mechanism takes place. (b) For fully polarized lead L , f = 0, the pumped current is finite. Time is in units of τ = 2π/ω.

the pumping characteristics of the system. This result is general, with ρ6 growing with time more or less rapidly as a function of  L ,  R and t L R . As we mentioned before, this spin blockade property has been observed very recently [6] in dc transport measurements on weakly coupled hydrogenlike QDs. This blockade can be overcome by reducing the density of spin-up electrons injected onto the system from the left lead on I (t). Indeed, as shown in figure 2(b), in the limit of fully spin-polarized injection from the left lead ( f = ν↑ /ν↓ = 0), the pumping of ↓ spins is efficient. The possibility of making QDs with spin-polarized injections has been reported recently [13]. As we mentioned, only spin-down electrons are pumped in this configuration. As a consequence, our scheme works similarly to previous proposals [2, 3], where the pumping occurs through states of single occupancy. 3.2. Adding N = 3- and 4-electron states We investigate an alternative solution to the spin blockade of our double-dot device by extending the basis to include threeand four-particle states, i.e., considering up to two opposite spin electrons on each dot. This is shown schematically in figure 1 where E S1 and E S2 are the energies of the doubly occupied states in each dot, with the same Zeeman splitting z on both dots and the frequency ω of the AC field tuned accordingly. Preparing the system initially in the state |12 = |↓↑, ↑ (or in the state |6 = |↑, ↑ which is immediately filled by a ↓ electron when E S1 < µ L ), we see (figure 3) that pumping of the ↓ spin is obtained in the regime where the chemical potential for taking ↓ electrons out of the right dot fulfils E S2 > µ R if, on the other hand, the chemical potential for taking ↑ electrons out of the right dot fulfils E S2 −z < µ R . It is important to emphasize that this pumping of spin-polarized (↓) electrons is realized with unpolarized leads. Such a spinpolarized current (I R in figure 3) is obtained either through the sequence |↓↑, ↑ → |↑, ↓↑ → |↑, ↑ → |↓↑, ↑ or

Spin-polarized pumping in a double QD

Figure 3. Density matrix elements and current to the right lead I R as a function of time in units of the period of Rabi oscillations τ0 , for t L R = 0.1, µ = 10, U L = 6, U R = 12,  L =  R = 0.01 with Zeeman splitting z = 6 on both√dots. When the AC field is tuned to the resonance condition, ω = ((E 14 − E 12 )2 + 4t 2 ) and ˜ = ω, there is a (↓) spin-dependent pumped current for unpolarized leads. The initial state is ρ12 = 1.

|↓↑, ↑ → |↑, ↓↑ → |↓↑, ↓↑ → |↓↑, ↑ which involve states of double occupation on both dots. 3.3. Spin relaxation and decoherence effects As mentioned in section 2.1 we include spin-relaxation effects, in addition to intrinsic decoherence effects of the electron spin due, for example, to the hyperfine interaction between electrons and nuclei. The results show little difference with respect to those in figure 3(a). For the parameter values chosen, corresponding to typical experimental values in QDs [10], our system is robust against intrinsic spin relaxation. This can be explained by realizing that, in the regime considered, the current is established involving states |12 = |↑↓, ↑ and |14 = |↑, ↓↑ where spin-flip mechanisms are irrelevant, due in particular to the condition W↑↓ W↓↑ . Spin decoherence is dominated by tunnelling since we are in the regime where  > 1/ T2 . 3.4. Pulsed AC field As shown experimentally [1] and theoretically [2, 3], by applying a short pulse via a gate electrode, Rabi oscillations can be resolved and observed through current measurements. This is an important example of the possibility of the observation and control of coherent quantum-state time evolution. Importantly, it is crucial for the system to be able to return to the initial (ground) state after the pulse has been turned off. In the case of N
2, we found, in a previous work [5], that the system cannot return to the initial state after removal of the pulse due to the formation of the spin-triplet state |6 = |↑, ↑. In order for the pumping mechanism to work ρ6 had to be made to vanish at all times, which was accomplished by using spinpolarized injections from the left lead. Here, we have established that including three- and four-particle states, the pumping mechanism is realized over the proper regime of parameter

Figure 4. (a) Time evolution of the density matrix with an AC-field pulse of duration τ P tuned to the resonance condition. (b) The time-averaged spin-polarized current to lead R in units of , as a function of pulse length τ P . The oscillations in the current reflect Rabi oscillations of ↓ spins within the double dot. The parameter values are the same as in figure 3.

values. We prepare the system initially in state |12 = |↑↓, ↑, apply the AC field in a pulse of duration τ P and then let the system evolve for a time ∼3τ P (figure 4(a)). We see, from the time evolution of the density matrix elements that the Rabi oscillations are clearly resolved for t < τ P , and that the system eventually regains the initial ground state (ρ12 → 1) after the field is turned off. Next, by applying pulses of different length τ P in sequence, we calculate the time-averaged current to the right lead I R (τ P ). The results (figure 4(b)) show that the Rabi oscillations of ↓ spins between the two QDs can be clearly resolved and observed through current measurements. We emphasize that in our present set-up, states of double occupancy in each dot play a decisive role in obtaining the spin-polarized pumping mechanism with unpolarized leads.

4. Conclusions Using the Markovian master equation approach, we have studied the dynamics of a double QD system with up to two electrons per dot in the presence of an AC field in the pumping configuration. For N
2, we find from the time evolution of the RDM, that ‘spin blockade’ of the current takes place due to the presence of the spin-triplet state. We verify that by taking a fully spin-polarized injection from the leads, this blocking is removed. Also, by including three- and fourparticle states, we realize a novel pumping mechanism of a spin-polarized current with unpolarized leads, via double occupancy states in each dot. By applying a pulsed AC field in this set-up, we find that Rabi oscillations of ↓ spins can be resolved through current measurements. Finally, we have shown that the pumping mechanism in our device is robust with respect to spin-relaxation and decoherence effects for parameters corresponding to typical experimental values.

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