Open Access

Pumped double quantum dot with spin-orbit coupling

Nanoscale Research Letters20116:212

DOI: 10.1186/1556-276X-6-212

Received: 13 August 2010

Accepted: 11 March 2011

Published: 11 March 2011

Abstract

We study driven by an external electric field quantum orbital and spin dynamics of electron in a one-dimensional double quantum dot with spin-orbit coupling. Two types of external perturbation are considered: a periodic field at the Zeeman frequency and a single half-period pulse. Spin-orbit coupling leads to a nontrivial evolution in the spin and orbital channels and to a strongly spin- dependent probability density distribution. Both the interdot tunneling and the driven motion contribute into the spin evolution. These results can be important for the design of the spin manipulation schemes in semiconductor nanostructures.

PACS numbers: 73.63.Kv,72.25.Dc,72.25.Pn

Introduction

Quantum dots, being one of the most intensively studied examples of natural and artificial nanostructures, attract attention due to the richness in the properties they demonstrate in the static and dynamic regimes [1]. A possible realization of qubits for quantum information processing can be done by using spins of electrons in semiconductor quantum dots [2]. Spin- orbit coupling makes the dynamics even in the basic systems such as the single-electron quantum dots extremely rich both in the orbital and spin channels. If the frequency of the electric field driving the orbital motion matches the Zeeman resonance for electron spin in a magnetic field, the spin-orbit coupling causes a spin flip. This effect was proposed in refs. [3, 4] to manipulate the spin states by electric means. The efficiency of this process is much greater than that of the conventional application of a periodic resonant magnetic field. The ability to cause coherently the spin flip in GaAs quantum dots was demonstrated in ref. [5] where the gate-produced electric field induced the spin Rabi oscillations. In ref. [6] periodic electric field caused the spin dynamics by inducing electron oscillations in a coordinate-dependent magnetic field. In addition, these results confirmed that the spin dephasing in GaAs quantum dots, arising due to the spin-orbit coupling [7, 8] is not sufficiently severe to prohibit a coherent spin manipulation.

The spin dynamics experiments [5, 6] necessarily use at least a double quantum dot to detect the driven spin state relative to the spin of the reference electron. Multiple quantum dots realizations become nowadays the subject of extensive investigation [9]. In double quantum dots an interesting charge dynamics occurs and requires theoretical understanding. In this article we address full driven by an external electric field spin and charge quantum dynamics in a one-dimensional double quantum dot [1013]. Despite the simplicity, these systems show a rich physics. In the wide quantum dots, where the tunneling is suppressed, and the motion is classical, the interdot transfer occurs only due to the over-the-barrier motion, and a chaos-like behavior is usually expected. The irregular driven behavior in the spin and charge dynamics in these systems was studied in ref. [14]. In the quantum double quantum dots, the tunneling between single quantum dots is crucial and the spin-orbit coupling makes the interdot tunneling spin-dependent [1517]. In quantum systems a finite set of energy eigenstates allows only for a strongly irregular rather than a real chaotic behavior. These orbital and spin dynamical irregularities are important for the understanding of the quantum processes in multiple quantum dots.

In this article we consider various regimes for a one-dimensional double quantum dot with spin-orbit coupling driven by an external electric field and analyze the probability and spin density dynamics in these systems.

Hamiltonian, time evolution, and observables

We use a quartic potential model to describe a one-dimensional double quantum dot [18],
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ1_HTML.gif
(1)
where the minima located at d and -d are separated by a barrier of height U 0, as shown in Figure 1. We assume that the interminima tunneling is sufficiently weak such that the ground state can be described with a high accuracy as even linear combination of the oscillator states with a certain "harmonic" frequency ω 0 located near the minima. The double quantum dot is located in a static magnetic field B z along the z-axis and is driven by an external electric field (t) parallel to the x-axis. The full Hamiltonian https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq1_HTML.gif , where the time-independent parts are given by
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Fig1_HTML.jpg
Figure 1

A schematic plot of the double-well potential described by Equation (1). Double green (red) lines correspond to the spin-split even (odd) tunneling-determined orbital states.

https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ3_HTML.gif
(3)
and the time-dependent perturbation is
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ4_HTML.gif
(4)
Here p x is the momentum operator, m is the electron effective mass, e is the electron charge, Δ z = |g|μ B B z (we assume below g < 0) is the Zeeman splitting, and σ i are the Pauli matrices. The electron Landé factor g determines the effect of B z , which in this geometry is reduced to the Zeeman spin splitting only. The bulk-originated Dresselhaus (β) and structure-related Rashba (α) parameters determine the strength of spin-orbit coupling and make the electron velocity defined as
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ5_HTML.gif
(5)

spin-dependent.

We use the highly numerically accurate approach to describe the dynamics with the sum of Hamiltonians in Equations (2)-(4). As the first step we diagonalize exactly the time-independent H 0 + H so in the truncated spinor basis ψ n (x) |σ〉 of the eigenstates of the quartic potential in magnetic field without spin-orbit coupling with corresponding eigenvalues E . As a result, we obtain the basis set |ψ n 〉 where bold n incorporates the spin index. For the presentation, it is convenient to introduce the four-states subset: |ψ 1 〉 = ψ 1(x)|↑〉, |ψ 2 〉 = ψ 1(x)|↓〉, |ψ 3 〉 = ψ 2(x)|↑〉, |ψ 4 〉 = ψ 2(x)|↓〉, and to note that the spin-dependent bold index may not correspond to the state energy due to the Zeeman term in the Hamiltonian. The wavefunction ψ 1(x) (ψ 2(x)) is even (odd) with respect to the inversion of x. In the case of weak tunneling, assumed here, these functions can be presented in the form: https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq2_HTML.gif , where ψ L(x) and ψ R(x) are localized in the left and in the right dot, respectively.

As the second step we build in the full basis the matrix of time-dependent https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq3_HTML.gif and study the full dynamics with the wavefunctions:
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ6_HTML.gif
(6)
The expansion coefficients ξ n (t) are then calculated as:
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ7_HTML.gif
(7)
Where https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq4_HTML.gif . The spin-dependence of the matrix element of coordinate responsible for the spin dynamics is determined with
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ8_HTML.gif
(8)

and the spin-dependent velocity in Equation (5).

With the knowledge of the time-dependent wavefunctions (6) one can calculate the evolution of probability ρ(x, t) and spin S i (x, t)-density
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ10_HTML.gif
(10)
Since we are interested in the interdot transitions, with these distributions we find the gross quantities, e.g., for the right quantum dot:
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ12_HTML.gif
(12)

where ω R(t) is the probability to find electron and https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq5_HTML.gif is the analog of expectation value of the spin component.

Calculations and results

As the electron wavefunction at t = 0 we take linear combinations of two out of four low-energy states. The initial state in the form https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq6_HTML.gif is localized in the left quantum dot, corresponding to the parameters https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq7_HTML.gif .

Two types of electric field were considered as the external perturbation. The first one is the exactly periodic perturbation for all t > 0:
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Equ13_HTML.gif
(13)

Where T z (B z ) = 2πħ z is the Zeeman period. The second type is a half-period pulse, same as in Equation (13), but acting at the time interval 0 < t < T z (B z )/2 only. The spectral width of the pulse covers both the spin and the tunneling splitting of the ground state, thus, driving the spin and orbital dynamics simultaneously. Since Tz(Bz)ω 1, that is the corresponding frequencies are much less than those for the transitions between the orbital levels corresponding to a single dot, the higher-energy states follow the perturbation adiabatically. The field strength 0 is characterized by parameter f such that |e| 0f × U 0/2d. Here we concentrate on the regime of a relatively weak coupling (f = 1)

where the shape of the quartic potential remains almost intact in time, and the interdot tunneling is still crucially important. For the magnetic field we consider two different regimes Δ z = ΔE g /2 and Δ z = 2ΔE g to illustrate the role of the Zeeman field for the entire dynamics.

We consider a nanostructure with https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq8_HTML.gif nm and U 0 = 10 meV. The four lowest spin-degenerate energy levels are E 1 = 3.938 meV, E 2 = 4.030 meV, E3 = 9.782 meV, E 4 = 11.590 meV counted from the bottom of a single quantum dot with the tunneling splitting ΔE g = E 2 - E 1 = 0.092meV, and the corresponding timescale 2π ħE g = 45ps. The spin-orbit coupling is described by parameters α = 1.0 · 10-9 eVcm and β = 0.3 · 10-9 eVcm. The field parameter f = 0.125, corresponding to 0 = 177 V/cm. We use the truncated basis of 20 states with the energies up to 42 meV.

We begin with the exactly periodic driving force, as illustrated in Figure 2 where |ξ n |2 for three states are presented. Since the motion is periodic, here we use the Floquet method [13, 19, 20] based on the exact calculation at the first period and then transformed into the integer number of periods. Figure 2 demonstrates the interplay between the tunneling and the spin-flip process. The results indicate that the exact matching of the driving frequency with the Zeeman splitting generates the spin flip which is clearly visible as the initial spin-up (ξ 1 and ξ 3 ) components are decreasing to zero and, at the same time, the opposite spin-down components (ξ 2 and ξ 4 ) reach their maxima (not shown in the upper panel). The spin-flip time is approximately 350T z (B z ) (or 31 ns) for the weak magnetic field (upper panel) and 24T z (B z ) (or 528 ps) for the strong field (lower panel). Such an increase in the Rabi frequency with increasing magnetic field is consistent with previous theoretical [3, 4] and experimental results [5].
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Fig2_HTML.jpg
Figure 2

Motion driven by the exactly periodic field. Upper panel: B z = 1.73T, Δ z = ΔE g /2 and T z (B z ) = 90 ps; lower panel: B z = 6.92T, Δ z = 2ΔE g , and T z (B z ) = 22 ps. The states for ξ n(t) are marked near the plots. The upper panel demonstrates a relatively slow dynamics on the top of the fast oscillations. The increase in the ξ 2(t) term corresponds to the possible spin-flip due to the external electric field.

As the second example we consider the probabilities ω R(t) and https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq9_HTML.gif for the pulse-driven motion, presented in Figure 3. As one can see in the figure, the initial stage is the preparation for the tunneling, which develops only after the pulse is finished. Electric field of the pulse induces the higher-frequency motion by involving higher-energy states, as can be seen in the oscillations at t ≤ T z (B z )/2, however, prohibits the tunneling. Such a behavior of the probability and spin density can be explained by taking into account the detailed structure of matrix elements x nm . Namely, due to the symmetry of the eigenfunctions in a symmetric double QW the largest amplitude can be found for the matrix element of https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_IEq10_HTML.gif -operator for the pairs of states with opposite space parity having the same dominating spin projection. Hence, the dynamics involving all four lowest levels first of all triggers the transitions inside these pairs which do not involve the spin flip and only after this the spin-flip processes can become significant. As a result, Figure 3 shows that the spin flip has only partial character while the free tunneling dominates as soon as the pulse is switched off. A detailed description of other processes of nonresonant driven dynamics in the case of a half-period perturbation can be found in ref. [21].
https://static-content.springer.com/image/art%3A10.1186%2F1556-276X-6-212/MediaObjects/11671_2010_Article_146_Fig3_HTML.jpg
Figure 3

Motion driven by the single-pulse field. Upper panel: B z = 1.73T, Δ z = ΔE g /2, and T z (B z ) = 90 ps; lower panel: B z = 6.92T, Δ z = 2ΔE g , and T z (B z ) = 22 ps Black line is the probability to find the electron in the right quantum dot. Red and blue dashed lines show corresponding spin components, as marked near the lines, determined both by the spin-orbit coupling and external magnetic field.

Conclusions

We have studied the full driven quantum spin and charge dynamics of single electron confined in one-dimensional double quantum dot with spin-orbit coupling. Equations of motion have been solved in a finite basis set numerically exactly for a pulsed field and by the Floquet technique for the periodic fields. We explored here the regime of relatively weak coupling to the external field, where a nontrivial dynamics already occurs. Our results are important for the understanding of the effects of spin-orbit coupling for nanostructures as we have demonstrated a possibility to achieve a controllable spin flip at various time scales and in various regimes by the electrical means only.

Declarations

Acknowledgements

D.V.K. is supported by the RNP Program of Ministry of Education and Science RF (Grants No. 2.1.1.2686, 2.1.1.3778, 2.2.2.2/4297, 2.1.1/2833), by the RFBR (Grant No. 09-02-1241-a), by the USCRDF (Grant No. BP4M01), by "Researchers and Teachers of Russia" FZP Program NK-589P, and by the President of RF Grant No. MK-1652.2009.2. E.Y.S. is supported by the University of Basque Country UPV/EHU grant GIU07/40, Basque Country Government grant IT-472-10, and MCI of Spain grant FIS2009-12773-C02-01. The authors are grateful to L.V. Gulyaev for assistance.

Authors’ Affiliations

(1)
Department of Physics, University of Nizhny Novgorod
(2)
Department of Physical Chemistry, Universidad del País Vasco
(3)
IKERBASQUE Basque Foundation for Science

References

  1. Kohler S, Lehmann J, Hänggi P: Driven quantum transport on the nanoscale. Phys Rep 2005, 406: 379. 10.1016/j.physrep.2004.11.002View Article
  2. Burkard G, Loss D, DiVincenzo DP: Coupled quantum dots as quantum gates. Phys Rev B 1999, 59: 2070. 10.1103/PhysRevB.59.2070View Article
  3. Rashba EI, Efros AlL: Orbital mechanisms of electron-spin manipulation by an electric field. Phys Rev Lett 2003, 91: 126405. 10.1103/PhysRevLett.91.126405View Article
  4. Rashba EI, Efros AlL: Efficient electron spin manipulation in a quantum well by an in-plane electric field. Appl Phys Lett 2003, 83: 5295. 10.1063/1.1635987View Article
  5. Nowack KC, Koppens FHL, Nazarov YuV, Vandersypen LMK: Coherent control of a single electron spin with electric fields. Science 2007, 318: 1430. 10.1126/science.1148092View Article
  6. Pioro-Ladriere M, Obata T, Tokura Y, Shin Y-S, Kubo T, Yoshida K, Taniyama T, Tarucha S: Electrically driven single-electron spin resonance in a slanting Zeeman field. Nat Phys 2008, 4: 776. 10.1038/nphys1053View Article
  7. Semenov YG, Kim KW: Phonon-mediated electron-spin phase diffusion in a quantum dot. Phys Rev Lett 2004, 92: 026601. 10.1103/PhysRevLett.92.026601View Article
  8. Stano P, Fabian J: Theory of phonon-induced spin relaxation in laterally coupled quantum dots. Phys Rev Lett 2006, 96: 186602. 10.1103/PhysRevLett.96.186602View Article
  9. Busl M, Sanchez R, Platero G: Control of spin blockade by ac magnetic fields in triple quantum dots. Phys Rev B 2010, 81: 121306. 10.1103/PhysRevB.81.121306View Article
  10. Sánchez D, Serra L: Fano-Rashba effect in a quantum wire. Phys Rev B 2006, 74: 153313.View Article
  11. Lü C, Zülicke U, Wu MW: Hole spin relaxation in p-type GaAs quantum wires investigated by numerically solving fully microscopic kinetic spin Bloch equations. Phys Rev B 2008, 78: 165321.View Article
  12. Romano CL, Tamborenea PI, Ulloa SE: Spin relaxation rates in quasi-one-dimensional coupled quantum dots. Phys Rev B 2006, 74: 155433. 10.1103/PhysRevB.74.155433View Article
  13. Jiang JH, Wu MW: Spin relaxation in an InAs quantum dot in the presence of terahertz driving fields. Phys Rev B 2007, 75: 035307. 10.1103/PhysRevB.75.035307View Article
  14. Khomitsky DV, Sherman EYa: Nonlinear spin-charge dynamics in a driven double quantum dot. Phys Rev B 2009, 79: 245321. 10.1103/PhysRevB.79.245321View Article
  15. Lin WA, Ballentine LE: Quantum tunneling and chaos in a driven anharmonic oscillator. Phys Rev Lett 1990, 65: 2927. 10.1103/PhysRevLett.65.2927View Article
  16. Tameshtit A, Sipe JE: Orbital instability and the loss of quantum coherence. Phys Rev E 1995, 51: 1582. 10.1103/PhysRevE.51.1582View Article
  17. Murgida GE, Wisniacki DA, Tamborenea PI: Landau Zener transitions in a semiconductor quantum dot. J Mod Opt 2009, 56: 799. 10.1080/09500340802263109View Article
  18. Romano CL, Ulloa SE, Tamborenea PI: Level structure and spin-orbit effects in quasi-one-dimensional semiconductor nanostructures. Phys Rev B 2005, 71: 035336. 10.1103/PhysRevB.71.035336View Article
  19. Shirley JH: Solution of the Schrdinger equation with a Hamiltonian periodic in time. Phys Rev 1965, 138: B979. 10.1103/PhysRev.138.B979View Article
  20. Demikhovskii VYa, Izrailev FM, Malyshev FM: Manifestation of Arnold diffusion in quantum systems. Phys Rev Lett 2002, 88: 154101. 10.1103/PhysRevLett.88.154101View Article
  21. Khomitsky DV, Sherman EYa: Pulse-pumped double quantum dot with spin-orbit coupling. EPL 2010, 90: 27010. 10.1209/0295-5075/90/27010View Article

Copyright

© Khomitsky and Sherman; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.