Triplet-Singlet Spin Relaxation in Quantum Dots with Spin-Orbit Coupling

We estimate the triplet-singlet relaxation rate due to spin-orbit coupling assisted by phonon emission in weakly-confined quantum dots. Our results for two and four electrons show that the different triplet-singlet relaxation trends observed in recent experiments under magnetic fields can be understood within a unified theoretical description, as the result of the competition between spin-orbit coupling and phonon emission efficiency. Moreover, we show that both effects are greatly affected by the strength of the confinement and the external magnetic field, which may give access to very long-lived triplet states as well as to selective population of the triplet Zeeman sublevels.

We estimate the triplet-singlet relaxation rate due to spin-orbit coupling assisted by phonon emission in weakly-confined quantum dots. Our results for two and four electrons show that the different triplet-singlet relaxation trends observed in recent experiments under magnetic fields can be understood within a unified theoretical description, as the result of the competition between spin-orbit coupling and phonon emission efficiency. Moreover, we show that both effects are greatly affected by the strength of the confinement and the external magnetic field, which may give access to very long-lived triplet states as well as to selective population of the triplet Zeeman sublevels. Semiconductor quantum dots (QDs) are called to play a central role in the emerging field of spintronics, because their zero-dimensional confinement constitutes an optimal environment to manipulate the spin of bound electrons. [1] This has stimulated their use as spinfilters [2], as well as several attempts to realize solid state implementations of spin-based qubits. [3] Understanding the spin relaxation in these structures is of utmost interest for their eventual use in practical devices. This has triggered a large number of experimental works in the last few years, where two main classes of spin transitions have been investigated, namely the spin-flip between single-electron Zeeman sublevels [3] and the tripletsinglet (TS) transition in QDs with an even number of electrons. [4,5,6,7] Remarkably, while the former systems have received much theoretical attention [3,8], the understanding of the latter is still rather limited. In particular, the TS relaxation due to spin-orbit (SO) coupling -which is often the dominant spin relaxation mechanism in semiconductor QDs -has only briefly been addressed in two-electron QDs [9,10], and many relevant features observed in experiments remain uncomprehended. This is the case of the role of an external magnetic field: experimental measuraments away from the TS anticrossings suggest that the influence of axial fields on the spin relaxation is fairly weak [4,5]. This is in strong contrast with the single-electron case, where a power-dependence of the relaxation rate on the field has been demonstrated [3], and it might seem surprising because the field reduces the energy splitting between the singlet and the triplet, what should enhance SO coupling. Besides, in the vicinity of the TS anticrossing, both decreased [5] and increased [6] relaxation rates have been reported. In this context, a unified picture describing the effect of an axial magnetic field on the TS spin relaxation rate is on demand.
In this Letter, we study the TS spin relaxation due to SO coupling in circular QDs with weak lateral confinement. Acoustic phonon emission, assisted by SO interaction, has been shown to be the dominant relaxation mechanism in this kind of QDs when cotunneling and nuclei-mediated relaxation are reduced. In fact, cotunneling is an extrinsic scattering process and can be controlled by means of reduced tunneling rates [5], while nuclei-mediated relaxation dramatically decreases with external magnetic fields. [3] We show that the current experimental evidence [4,5,6] can be reconciled within a unified picture, where the field dependence of the relaxation rate is determined by the interplay between spinorbit coupling and phonon emission efficiency. Furthermore, we show that such interplay can be tailored in order to obtain improved spin lifetimes.
In weakly confined QDs, correlation effects may strongly influence charge and spin excitations. [11] In order to calculate the relaxation time of excited fewelectron correlated states, we need to know both ground and excited states with comparable accuracy: our method of choice is the full configuration interaction (FCI). [12] The single-electron states are calculated within the effective mass approximation for a typical "vertical" GaAs/AlGaAs QD, with confinement potential V (r) = V z (z) + 1/2 m * ω 2 0 (x 2 + y 2 ), V z (z) representing the (finite) vertical confinement of a quantum well of thickness W ,hω 0 being the single-electron energy spacing of a lateral two-dimensional harmonic trap, and m * the effective mass. The lateral confinement is much weaker than the vertical one, and a magnetic field, B, is applied along z. Under these conditions, the low-lying single-electron states are well described by the Fock-Darwin spectrum and the lowest eigenstate of the quantum well. [13] The single-electron levels can be classified by their radial quantum number n = 0, 1 . . ., azimuthal angular momentum m = 0, ±1 . . ., and spin s z =↑, ↓. In turn, the few-electron states can be labelled by the total azimuthal angular momentum M , total spin S, its z-projection S z , and by the number N = 0, 1, . . . indexing the energy order.
We introduce the SO coupling via the linear Rashba and Dresselhaus terms, H R and H D respectively. For a quantum well grown along the [001] direction, these terms can be written as: [14] where α and β are the Rashba and Dresselhaus coefficients for the sample under study. π ± and s ± z are ladder operators, which change m and s z by one unit, respectively. Since the few-electron M and S z quantum numbers are given by the algebraic sum of their singleelectron counterparts, Rashba interaction mixes (M, S z ) states with (M ± 1, S z ∓ 1) ones, and Dresselhaus interaction mixes (M, S z ) states with (M ± 1, S z ± 1) ones.
In our calculation, the SO terms of Eq. (1) are diagonalized in a basis of few-electron states, which are computed as linear combinations of Slater determinants, according to the FCI method. [12,15] In the general case the Rashba and Dresselhaus terms break S and M symmetries. However, for GaAs QDs, SO coupling is but a small perturbation and the quantum numbers M , S and S z are approximately valid except in the vicinity of the anticrossing regions. [9,16] Thus, we will still use them for clarity of the discussion. We estimate the relaxation rate at zero temperature due to acoustic phonon emission. The electron-phonon interaction is taken into account as in Ref. 17. Hence, we consider not only deformation potential as in previous works (Ref. 9), but also piezoelectric field scattering. The piezoelectric field interaction is dominant when the phonon energy is small, [8,17] so that it provides the main contribution to the relaxation in the interesting regions of TS anticrossings. GaAs material parameters are taken in the calculations, [17] along with a Landé factor g = −0.44. We start our discussion with the two-electron case (see Fig. 1). We use a typical value of the Dresselhaus coefficient for a GaAs QD, β = 25 meV·Å, and a Rashba coefficient α = 5 meV·Å, which could be ascribed e.g. to a small accidental asymmetry of the quantum well. The low-lying singlet state and the excited triplet state with three Zeeman sublevels are shown in Fig. 1(a). With increasing magnetic field, the singlet anticrosses with all triplet sublevels. The anticrossing energy gap is very small (of the order of µeV), as expected for GaAs QDs [4], and it is particularly small for the S z = 0 triplet sublevel. This is because the Dresselhaus (Rashba) interaction mixes the singlet with the triplet S z = −1 (+1) sublevel, but does not mix states with ∆S z = 0, which, therefore, takes place only indirectly through higher-lying states. Figure 1(b) illustrates the expectation value of S z of the four lowest-lying levels around the TS anticrossing. One can see that SO interaction barely affects the spin quantum numbers except in a narrow magnetic field range around the anticrossings. [9] In Fig. 2 we analyze the relaxation rate from the three first excited to the ground state of two-electron QDs with different dimensions. Left (right) panels correspond to structures without (with) Rashba interaction. For the QD studied in the upper panels, one can see that the relaxation rate increases slowly with the magnetic field, and then it suddenly drops in the anticrossing region (B ∼ 2.25 − 3.25 T). [18] This behavior, whose physical mechanism will be explained in the following, is in qualitative agreement with various recent experiments in weakly-confined QDs. In particular the rather weak dependence with the field before the anticrossing agrees well with TS relaxation measuraments [4,5], the increased relaxation rate before the anticrossing has been reported in Ref. 6, and the reduced rate in the anticrossing region may be inferred from the long triplet lifetimes for eight-electron QDs with small ST energy splittings [5].
The general trends described above can be explained by the opposite effect of the magnetic field on the SO mixing and the phonon emission efficiency. On the one hand, as the singlet-triplet energy splitting decreases, SO interaction couples the states more efficiently, favouring spin relaxation. On the other hand, the phonon energy decreases, reducing the efficiency of the electron-phonon interaction. The latter effect, which follows from the different orbital quantum numbers of the initial and final electron states, occurs at a rate that is determined by the interplay between the acoustic phonon wavelength and the dimensions of the QD. [8,17,19] For the QD of the upper panel in Fig. 2, the effect of the magnetic field on the SO interaction and phonon emission is mostly of similar magnitude, which explains the weak changes of the relaxation rate. At the anticrossing point, in spite of the fact that the SO mixing is maximum [see Fig. 1(b)], the phonon energy is so small (few µeV) that the spin relaxation is strongly supressed. It is worth mentioning that this result is opposed to that predicted for the TS anticrossing in a lateral QD, where maximum relaxation rate is predicted at the anticrossing point. [9] The origin of this difference might lie on the fact that the electron-phonon interaction matrix elements of lateral QDs with strongly asymmetric (non-parabolic) confinement potential may be significant even for very small phonon energies. [21] As a result, SO interaction alone would dominate spin relaxation in such structures. As stated before, the dimensions of the QD are known to play a critical role to determine the phonon emission efficiency. [8,17,19] To illustrate this effect on the spin relaxation, in the bottom panels of Fig. 2 we report the rate of a QD with larger height and stronger lateral confinement than the previous one. As compared to the upper panels, one can see a visibly stronger dependence of the relaxation rate on the field. This is because for the present QD dimensions, the phonon emission efficiency turns out not to be balanced with the SO mixing effect. As a result, when the effect of SO coupling prevails the relaxation rate strongly increases with B. Still, in the anticrossing region, that is now shifted towards higher magnetic fields, the suppression of phonon emission again yields a relaxation minimum. Furthermore, a new feature emerges in this case, namely a dip at about B ∼ 0.5 T (shaded area in the panels) This dip, which we had previously predicted for charge relaxation in QDs, [17] comes from the geometrically-induced suppression of the phonon emission, occuring when the quantum well width is a multiple of the phonon wavelength z−projection. This feature may give access to very long-lived triplet states at finite values of the magnetic field [22], where phonon-induced relaxation is usually the dominant scattering mechanism. Moreover, this dip takes place at a B value where the initial and final electron states are well-resolved energetically, which renders this minimum more useful than the one coming from the TS anticrossing. In the illustrated case, the average relaxation rate of the three Zeeman sublevels in this dip corresponds to a triplet relaxation time of tenths of seconds, two orders of magnitude over the longest triplet lifetime reported to date. [6] The position and depth of this kind of relaxation minima depend on the QD height and the emitted phonon energy. Therefore, they are almost independent of the SO interaction in the structure, which in GaAs has a neglegible influence on the phonon energy.
Next, we focus on the effect of the separate Rashba and Dresselhaus contributions over the spin relaxation by comparing the left and right panels of Fig. 2 in the magnetic field region before the TS anticrossing. When only Dresselhaus terms are present (left panels), the singlet mixes directly only with the higher-lying (S z = −1) Zeeman sublevel of the triplet. As a result, relaxation from such Zeeman sublevel (dot-dashed line) is about two orders of magnitude faster than from the S z = 0, +1 sublevels, and it exhibits a stronger dependence on the field. When a small Rashba interaction is switched on (right panels), direct mixing of the singlet with the triplet S z = +1 sublevel is enabled. This accelerates the relaxation rate from this sublevel (dashed line) in one order of magnitude and introduces a stronger dependence on B. It is worth noting that the order-of-magnitude enhancement of the relaxation rate due to the Rashba interaction is present away from the anticrossing region, where the effect of the SO interaction on S z is barely visible [see Fig. 1(b)]. From the above discussion it follows that in a magnetic field both Rashba and Dresselhaus interactions play an important role in determining the TS spin relaxation rate, as opposed to the well-known single-electron case, where the relaxation is mostly due to Rashba coupling only. [3] Moreover, we see that the lifetimes of the triplet Zeeman sublevels may strongly differ depending on the relative Rashba and Dresselhaus contributions. This may be useful to selectively populate the triplet sublevels.
We now investigate the TS spin relaxation in a fourelectron QD. The energy spectrum of the lowest-lying triplet and singlet states in a magnetic field, plotted in Fig. 3(a), is very different from that of the two-electron case, but it closely resembles the one found experimentally for eight-electron QDs in Ref.5 (except for the absence of eccentricity features in the zero field limit [23]).
Here, we investigate the spin relaxation rate in the region B ∼ 0.3 − 3 T, where the ground state is a singlet (M = −2) and the first excited state is a triplet with two possible values of the angular momentum, depending on the magnetic field: for B < 1 T the angular momentum is M = 0, and for B > 1 T it is M = −3. These states are well separated from higher-lying states, so they might be used as a two-level system for quantum computation purposes. We compare the lifetimes of both triplet states in Fig. 3(b), where the averaged lifetimes of the three Zeeman sublevels are plotted as a function of the singlet-triplet energy splitting, ∆ ST , with (dashed lines) and without (solid lines) Rashba interaction (Dresselhaus interaction is present in both cases). The qualitative behavior is similar for both states: the lifetime is roughly constant for large energy splittings (∆ ST > 0.25 meV), and it increases when the energy splitting is small (∆ ST < 0.25 meV). This behavior, which is in agreement with the experimental findings of Ref. 5, can be understood in the same terms of compensation between SO coupling and phonon emission efficiency as in the two-electron cases studied above. The strong dip of the M = −3 triplet at ∆ ST ∼ 0.15 meV is due to the anticrossing of the upper Zeeman sublevel with the M = −4 singlet at strong magnetic fields [at B ∼ 2.8 T in Fig. 3(a)], which strongly enhances spin relaxation. For smaller ∆ ST , though, the small phonon energy again leads to increased lifetimes. An important result shown in Fig. 3(b) is that the average lifetime of the triplets differs by over one order of magnitude dependening on their angular momentum, regardless of the (here fairly strong) Rashba interaction. This is because the M = 0 triplet differs from the M = −2 singlet ground state in two quanta of angular momentum, and therefore direct SO mixing is not possible. In con-trast, direct mixing is possible for the M = −3 triplet, and this makes the relaxation much faster. It then follows that, by using four-electron QDs instead of twoelectron ones, one can use an external magnetic field to select excited states whose spin transition is "forbidden" even in the presence of linear SO interaction. This result is consistent with recent measuraments, where different lifetimes were observed for triplet states with different orbital quantum numbers. [5] However, in the experiment the triplet lifetimes changed by a factor of two only. The main reason for this difference is probably the ellipticity of their QDs, which mixes states with different angular momenta and hence weakens the efficiency of the ∆M = ±1 selection rule.
In summary, we have estimated the electron TS spin relaxation rate due to SO coupling in weakly-confined cylindrical GaAs/AlGaAs QDs. Experimentally observed trends of TS relaxation in magnetic fields [4,5,6] are well understood in terms of the competing SO coupling and phonon emission efficiency. Significant differences have been found as compared to the well-known single-electron spin-flip case, including a critical role of the dot confinement to determine the phonon emission efficency. These differences arise from the different orbital initial and final states, and the (usually larger) transition energies. We predict very long triplet lifetimes using QD geometries that lead to suppressed phonon emission. Improved lifetimes can also be obtained in four-electron QDs by selecting triplet states which do not fulfill the ∆M = ±1 selection rule.
We are grateful to X. Cartoixa, M. Florescu and F. Troiani for discussions. We acknowledge support from the Italian Ministry for University and Scientific Research under FIRB RBIN04EY74, CINECA Calcolo parallelo 2006, and Marie Curie IEF project MEIF-CT-2006-023797.