Simulations of electric-dipole spin resonance for spin-orbit-coupled quantum dots in Overhauser field: fractional resonances and selection rules
We consider spin rotations in single- and two-electron quantum dots that are driven by external AC electric field with two mechanisms that couple the electron spatial motion and the spin degree of freedom: the spin-orbit interaction and a random fluctuation of the Overhauser field due to nuclear spin bath. We perform a systematic numerical simulation of the driven system using a finite difference approach with an exact account taken for the electron-electron correlation. The simulation demonstrates that the electron oscillation in fluctuating nuclear field is translated into an effective magnetic field during the electron wave packet motion. The effective magnetic field drives the spin transitions according to the electric-dipole spin resonance mechanism. We find distinct signatures of selection rules for direct and higher-order transitions in terms of the spin-orbital symmetries of the wave functions. The selection rules are violated by the random fluctuation of the Overhauser field.
Control and coherent manipulation of electron spin in quantum dotshanson () are extensively studied in the context of information storage and processing.lossdivi () The most basic operation on the single electron spin is its rotation. One of applicable procedures for spin rotation is the electron spin resonance (ESR). The ESR appears in constant magnetic field which splits the spin-up and spin-down energy levels. In presence of an additional AC magnetic field with frequency tuned to the Zeeman energy splitting, the electron undergoes the Rabi oscillation between the opposite spin states.ESR () It has been proposed for quantum wells EDSRQW () and quantum dots EDSR () that the oscillating external magnetic field can be replaced by oscillating electric field according to electric-dipole spin resonance (EDSR) mechanism.EDSRo () In presence of the spin-orbit (SO) coupling the electron motion induced by the electric field is translated into an effective magnetic field that depends on the electron momentum. The effective SO magnetic field induces Rabi oscillations and the corresponding spin rotations in a similar manner as the external field. The EDSR was observed in experiments that probe lifting of the Pauli blockade of the current across double quantum dots.EDSRm (); Sch (); Nadj () It has also been demonstrated that the role of SO coupling that translates the electron motion into appearance of an effective magnetic field can be played by a gradient of external magnetic field. Space () Successful experiments, in which EDSR is achieved due to the spatial fluctuation of the GaAs nuclear spin field created by the hyperfine interaction (HF) were also performed.EDSRnuc ()
The purpose of the present paper is the simulation of the EDSR that results from the random fluctuation of the HF field and determination of its signatures as compared to the process governed by SO interaction. We consider single and two-electron systems confined in quantum dots and transitions that are induced by oscillating electric field. We perform numerical simulations for systems with strong lateral confinement using a finite-difference approach that takes an exact account for the electron-electron correlation. We discuss the characteristic features of the resonant transitions due to the HF field and/or SO interaction. In particular the results indicate existence of selection rules for transitions governed by SO interaction. We discuss the symmetries behind the selection rules also for fractional resonances that have been observed in experiments with both SO coupling Sch (); Nadj () and HF field EDSRnuc () and were recently explainednowak () as second-order processes similar to multi-photon transitions in quantum optics.optics () We demonstrate that the selection rules are violated by the Overhauser field fluctuations. This finding is a relevant information for dynamical nuclear polarizationot (); johnson2005 (); inni () which reduces the randomness of the Overhauser field for which restoration of the selection rules should be observed.
The EDSR requires application of an external magnetic field to induce the Zeeman splitting. At non-zero the direct electron-nuclear spin flip-flopsjohnson2005 () become suppressed.hanson () Then, the field of nuclear spins separates from the electron wave function. The characteristic time for the fluctuation of the nuclear spin field due to the dipole-dipole and HF interactions is of the order of 10-100 s which is much longer hanson () than the time scales for the evolution of the electron spin. For the above reasons we will treat the HF field as a static Overhauser magnetic field.
We consider a quasi one-dimensional quantum dot with a strong lateral confinement – see Fig. 1 with external magnetic field oriented along the axis of the wire . In this configuration the orbital effects of the external magnetic field can be neglected, and the single-electron Hamiltonian takes the form,
where is the confinement potential, is the potential of the AC electric field which will drive the spin transitions, , describes the interaction of the electron spin with the Overhauser field
and introduces the linear Rashba spin-orbit interaction due to the electric field oriented perpendicular to the length of the quantum dot
We apply GaAs material parameters with the effective electron mass , and the Lande factor . We assume kV/cm which for the bulk SO coupling constant nm (after Ref. silva, ) gives meV nm. For this value of , SO coupling produces comparable effects to the Overhauser field fluctuation (see below).
We assume that the confinement potential is separable into a lateral and longitudinal components: . We consider a quantum dot with a strongly an elongated geometry (see Fig. 1). For the length of the dot which largely exceeds the lateral extent of the wave function all the low-energy phenomena appear in the ground-state of the lateral quantization.pt () We assume that the quantum dot is defined in the direction by an infinite quantum well of length 158.2 nm: (see Fig. 1) and that the lateral confinement potential is strong enough to induce localization around the axis of the cylinder within a region of radius 10 nm. We assume that the lateral wave function can be described by a Gaussian for which an analytical form of the effective electron-electron interaction potential is known.pt () Namely, we take the single-electron lateral wave function of form
For the electron pair with a double quantum dot is considered in Sec. III. B we take , where is barrier applied centrally to the system of height 10 meV and width 13.56 nm.
For the electron-pair we consider the Hamiltonian including the electron-electron interaction
with the dielectric constant of . After integration of the Hamiltonian with the lateral wave function (4) one obtains
where the last term is the effective 1D interaction derived in Ref. [pt, ] as .
Numerical calculations are performed with a finite difference approach with discretized versions of the differential operators and a finite mesh constant . For a single electron we look for eigenstates and time evolution of the spinor components . For the two-electron wave function we solve the equations for the bi-spinor in form
The fermion symmetry with respect to the electron interchange implies that and . The spin unpolarized parts have no definite symmetry with respect to the exchange of the electron spatial coordinates separately, instead one has . The evolution of the system in time is given by the Schrödinger equation , which is solved with the explicit Askar-Cakmak schemeacs ()
ii.2 Effective magnetic field due to nuclear spins
In GaAs each nucleus of the crystal lattice carry an uncompensated spin that interacts with the electron spin via the Fermi contact hyperfine interaction,hanson () , where is the position of k-th nucleus, and are the nuclear and electron spin operators, and is the coupling constant which is proportional to the magnetic moment of nucleus and the probability density of finding the electron therein.
In nonzero when the entanglement between the electron wave function and the nuclear spins can be neglected, the electron interacts with the ensemble of spins through the Overhauser effective magnetic field , with . For fully polarized nuclei the maximal value of the Overhauser field is 5T. For the purpose of the numerical simulation we need the distribution of the Overhauser field along the quantum dot. The field is generated in the following manner. We consider all the nuclei present within the volume of the quantum dot. With each nucleus at position (eight nuclei per cubic unit cell of lattice constant nm) we attribute a local vector of an effective magnetic field of length 5T with orientation taken at random with the uniform distribution. For the purpose of the present study we do not distinguish between isotopes of Ga and As. The electron lateral wave function averages the magnetic field due to the nuclei.kh (); merk () We calculate the magnetic field in the cell of the finite difference mesh, between and points
The simulated field due to nuclear spins is plotted in Fig. 2. For nm, the field amplitude is of the order of 20 mT. The actual effective field perceived by the electron spin will be further reduced to a few mili TeslamT () by the spread of the wave function along the quantum dot (see below).
Iii Results and Discussion
iii.1 Single-electron electric-dipole spin resonance
Figure 3(a) shows the single-electron energy spectrum for a quantum dot of length 158.2 nm as a function of the magnetic field. Figures 3(b) and (c) present zoom at the ground-state energy level that is split by the Zeeman interaction. The spin-orbit coupling [Fig. 3(b)] lowers the energy levels and preserves the double (Kramers) degeneracy of the ground-state. When the Overhauser field is introduced [Fig. 3(c)], the energy levels are no longer degenerate at 0T. For the adopted parameters () the energy effects due to both spin-orbit coupling and hyperfine field are comparable and of the order of eV.
iii.1.1 EDSR due to SO coupling
Let us now consider the spin rotations due to spin-orbit coupling only (no hyperfine field). Figure 4 shows the real part of the wave functions of the two lowest energy levels for T. The ground-state [Fig. 4(a,b)] is nearly spin polarized. Its spin-down component [Fig. 4(b)] is of the odd spatial parity and corresponds to the first-excited energy level of the quantum dot without SO coupling. The SO coupling Hamiltonian commutes with the s-parity operator , where is the parity operator with respect to point inversion through the center of the dot. Therefore, the components of the spin-orbitals possess a definite but opposite spatial parities. The SO coupling appears through coupling of spatial energy levels of opposite parities.
Now, we apply a harmonic AC field of amplitude kV/cm – for which the potential drop along the dot is meV. We consider the ground-state as the initial condition and apply the resonant frequency for the transition to the first excited energy level with inverted spin, , which corresponds to AC oscillation period of about 0.6 ns. We see in Fig. 5 that after 60 ns (nearly 100 periods of the external electric field) the electron spin gets inverted and the electron occupies the first excited state.
Figure 6(a) shows the maximal probability of the spin inversion obtained during 500 ns of the simulation as a function of the frequency of the AC field. We observe a pronounced resonance near 7.5 eV, that corresponds to the simulation of Fig. 5. We also notice a sign of fractional resonance near 2.5 eV. Note, that the half-resonancenowak () due to second-order transition that could be expected near 3.8 eV is missing. The reason for the missing half-resonance is the symmetry of the wave function for the initial and final states. The oscillator strength of the direct (first-order) transition from initial state to the final state which results from the harmonic perturbation of form , is determined by the Fermi golden rule with the matrix element . For the wave functions of Fig. 4 the components of the sum are non-zero due to the spin-orbit coupling that introduces non-zero overlap between the components of wave functions for the same spin orientation. Moreover, since the initial and final states have opposite parities for each of the components, and is the odd parity function, the integrands are even functions with respect to point inversion through the center of the dot. In general, the first-order transition are allowed between states of opposite s-parities.
According to the time-dependent perturbation theory,shankar () the second-order transitions are allowed if one the products is non-zero, where is any of the intermediate eigenstates of the unperturbed Hamiltonian. Now, since all the eigenstates of the unperturbed Hamiltonian possess a definite s-parity, and in our case and states are of the opposite s-parity, none of these products can be non-zero. The second-order transitions are allowed only between the states of the same s-parity (see below for EDSR in the two-electron system). The third order transitions that involve two intermediate states are again allowed between states of opposite s-parity, hence the 1/3 peak observed in Fig. 6(a).
iii.1.2 EDSR due to the hyperfine field
We now consider the hyperfine field only and apply the harmonic perturbation to the potential (same resonant frequency and amplitude as in Fig. 5). Charge density at opposite phases of the AC field is given in Fig. 7(a,b). Figure 7(c) shows that the center of the electron packet oscillates with an amplitude of about 10 nm. As the packet oscillates, the electron spin perceives a different effective magnetic field that is averaged by the moving charge density – see Fig. 7(c) for the components of the effective field. As a consequence, the electron spin is inverted after about 200 periods of the oscillating field.
The probability of the spin inversion as a function of are plotted in Fig. 6(b). In presence of the nuclear field, the s-parity of eigenstates is no longer defined, and we obtain the direct, second and third order transitions to the excited state. A reduced width of the direct transition can be noticed with respect to the SO case.
iii.1.3 EDSR: combined nuclear field and SO coupling
Figure 8 shows the wave functions with both SO coupling and the nuclear field (solid lines). For comparison, the results without nuclear field are also plotted (dashed lines). The nuclear field shifts the zero of the minority spin components, which are no longer of a strict odd spatial parity. The impact of the hyperfine field on the majority wave function is not visible at this scale. In spite of the fact, that the SO coupling seems to dominate, the transition probability as function of the frequency of Fig. 6(c) resembles rather the case of the nuclear field, due to the presence of the half-resonance. Nevertheless, the width of the direct transition is similar to the one observed for pure SO coupling.
iii.2 Two-electron results
The experiments EDSRm (); EDSRnuc () probe the consequences of EDSR in lifting the Pauli blockade of the current in double quantum dots. The blockade occurs when electrons in both the dots acquire the same spin. When the spin of one of electrons is inverted the current restarts to flow. For the rest of this paper we consider a double quantum dot [see the inset to Fig. 9(a)]. We increased the amplitude of the AC field twice to 0.1 kV/cm – to maintain the same potential drop within a single dot as compared to the results of the previous section. Figure 9 shows the energy spectrum for the electron pair. The Coulomb blockade experiments on lifting the Pauli blockade are performed for the spin-polarized triplet as the initial state.
Figure 9(b) compares the spectrum without SO and without the hyperfine field (dashed lines), to the case of pure SO coupling. The avoided-crossing opened by SO interaction between the and energy levels is distinctly enhanced by the HF field [see Fig. 9(c)], which also lifts the degeneracy of the triplet states at T.
iii.2.1 Two-electron EDSR: pure SO coupling
We have performed time dependent simulations starting from the ground state – the triplet at T. The minimal spin that was obtained during 100 ns of the simulation is displayed in Fig. 10(a). Fig. 10(b) shows the maximal value of the square of projection of the wave function on the Hamiltonian eigenstates. We can see that the direct transition , that is most relevant for the lifting of the spin Pauli blockade,EDSRm () is missing in this plot. The reason for this is again the symmetry of the wave functions, Fig. 11 shows the real parts of the components of , , and wave functions. Each of the four components possesses a definite parity with respect to simultaneous inversion of both electron coordinates. In the basis of , , , , the two-electron s-parity operator has the form
As it can be noticed in Fig. 11, the only state of positive s-parity is . and (the last state was not included in Fig. 11), have negative s-parity. Hence, the matrix element for the direct transition from to both and final states vanishes. On the other hand the second-order transition to state with half resonance is observed near eV. No half resonance is observed for the transition to , in consistence with the discussion of selection rules given above for the single electron.
Figure 10(b) shows that an admixture of state is observed in the final state for the energy equal half of the energy splitting between and . Note, that in this case the energy spacing between and is exactly the same as between and . Fig. 12 shows the occupation probability as a function of time for the frequency of the external field tuned to resonance. We can see that first a finite probability of occupation appears, but it never approaches 1. Instead the probability occupancy of increases. Hence in this case we observe a two-step process, first pumping the electron from to state and next from to .
iii.2.2 Two-electron EDSR with nuclear field
Figure 13 shows the transitions for the HF field without the SO coupling. As compared to the previous case we notice: a distinct transition to the state (s-parities are no longer definite), and reduced probability of finding in the final state, due to the energy difference of the transitions between the triplets. Also, a half resonance for transitions to can be observed. The peaks for half resonances are much smaller than in the SO case. Note, that also the direct transitions occur significantly slower (cf. the height of the peak).
In Fig. 14 we show the scans obtained as a function of both AC frequency and the magnetic field for AC amplitude of the electric field kV/cm that was considered above (a) and for a stronger amplitude of kV /cm (b). In Fig. 14(a) at this scale we can see only the direct transitions to and , which exactly agree in energy with spectral separation of the eigenstates from the ground state (dashed lines). In Fig. 14(b) we notice also fractional transitions to the singlets. However, there is a detectable redshift of resonant frequencies for singlets. This shift is due to appearance of the double occupancy of the dot that is induced by stronger slope of the confinement potential. The double occupancy lowers the energy of the singlets with respect to the triplets, for which the double occupancy is forbidden by the Pauli exclusion.szafranexchange ()
The scan of the transitions for both SO and hyperfine field present is given in Fig. 15. As compared to the pure HF case, the height of peak is reduced with respect to . In contrast to the pure SO coupling case the half-transitions occur for both and .
Iv Summary and Conclusions
In summary, we have performed a systematic numerical simulation for the EDSR mechanisms in single electron (electron spin flip) and two-electron (transitions from the spin-up polarized triplet to excited states) driven by AC electric field and mediated by both spin-orbit coupling and the abrupt fluctuations of the static Overhauser field. We have demonstrated that the latter is translated into a smooth effective magnetic field felt by the electron spin due to the wave packet oscillation. The simulation indicates the presence of both integer and fractional resonances that correspond to first and higher-order transitions. The results for transitions in presence of the pure spin-orbit coupling bear distinct signatures for the selection rules which result from the spin-orbital symmetry of wave functions. In presence of the Overhauser field the selection rules no longer hold. In experiments the randomness of the Overhauser field can be reduced or eliminated by the dynamical nuclear polarization ot (); inni (); johnson2005 () or in the strong external magnetic field of a few Tesla. As the hyperfine field becomes ordered, the selection rules should be restored and the observed spectrum for the spin transitions should start to resemble the case of pure SO coupling.
Acknowledgements.This work was supported by the funds of Ministry of Science and Higher Education (MNiSW) for 2012–2013 under Project No. IP2011038671, and by PL-Grid In- frastructure. M.P.N. gratefully acknowledges the sup- port from the Foundation for Polish Science (FNP) un- der START and MPD programme co-financed by the EU European Regional Development Fund.
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