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Theory of anisotropic spin relaxation in quantum dots O. Olendski and T. V. Shahbazyan Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA 共Received 14 December 2006; published 23 January 2007兲 We study theoretically phonon-assisted spin relaxation of an electron confined in an elliptical quantum dot 共QD兲 subjected to a tilted magnetic field. In the presence of both Rashba and Dresselhaus spin-orbit terms, the relaxation rate is anisotropic with respect to the in-plane field orientation. This anisotropy originates from the interference, at nonzero tilt angle, between the two spin-orbit terms. We show that, in a narrow range of magnetic field orientations, the relaxation rate exhibits anomalous sensitivity to variations of the QD parameters. In this range, the relative change in the relaxation rate with in-plane field orientation is determined solely by the spin-orbit coupling strengths and by the dot geometry. This allows simultaneous determination of both Rashba and Dresselhaus coupling parameters and the dot ellipticity from analysis of the angular dependence of the relaxation rate. DOI: 10.1103/PhysRevB.75.041306

PACS number共s兲: 73.21.La, 72.25.Rb, 71.70.Ej

Spin relaxation in semiconductor quantum dots 共QDs兲 has recently attracted intense interest because of the possible use of the electron spin as a qubit.1 Quantization of orbital states in a QD due to the zero-dimensional 共0D兲 confinement leads to the suppression of traditional spin-relaxation mechanisms 共e.g., D’yakonov-Perel兲 that are dominant in continuous systems. Indeed, recent experiments on GaAs QDs in a magnetic field B have revealed extremely long spin-relaxation times 共up to T1 = 170 ms at B ⯝ 1.75 T兲.2–8 For moderate and high fields 共B ⬎ 1 T兲, spin relaxation in QDs is dominated by phonon-assisted electronic transitions between Zeeman-split levels due to spin-orbit 共SO兲 coupling.9–16 There are two distinct types of SO couplings, one originating from bulk inversion asymmetry 共Dresselhaus coupling兲 and the other from structural inversion asymmetry along the growth direction 共Rashba coupling兲, that cause the admixture of orbital states with opposite spins.17 In the case of circular QDs in a perpendicular magnetic field, the Rashba and Dresselhaus terms mix different pairs of levels, and can, in principle, be distinguished if one such pair provides the dominant relaxation channel. This is the case, for example, when adjacent orbital levels, coupled via the Rashba term, are brought into resonance with changing magnetic field.12 However, in more realistic situations, deformations of the QD shape strongly alter the electronic spectrum18 and, in general, the effects of the two SO contributions are not separable.19 An important distinction between the Rashba and Dresselhaus terms is their different symmetry properties. The ˆ = ␣ 共 − 兲, former, described by the Hamiltonian H R R x y y x possesses an in-plane rotational symmetry, while the latter, ˆ = ␣ 共 − 兲, does not.17 Here = −i + eA, A beH D D x x y y ing the vector potential, is the Pauli matrix vector, and ␣R 共␣D兲 is the Rashba 共Dresselhaus兲 coupling constant 共we set ˆ leads to an ប = 1兲. This lack of rotational invariance for H D in-plane momentum azimuthal anisotropy in the presence of both SO terms that was recently reported in transport experiments in quantum wells.20,21 In a magnetic field, the anisotropy arises due to the interference between Rashba and Dresselhaus terms in the matrix elements22 in the presence of an in-plane field component. In QDs, this anisotropy reveals itself as a modulation of the spin-relaxation rate for different orientations of the in-plane field.11,14–16 1098-0121/2007/75共4兲/041306共4兲

Here we study the spin relaxation between Zeeman-split levels in elliptical QDs in a tilted magnetic field. We demonstrate that the interplay between SO interactions and QD geometry leads to dramatic changes in the relaxation rate in a certain range of field orientations for which Rashba and Dresselhaus contributions undergo destructive interference. Furthermore, in the vicinity of level anticrossings 共see Fig. 1兲, the SO contribution to the relaxation rate factors out from the phonon one. This allows simultaneous determination of the parameters for both SO interactions and QD geometry from the azimuthal anisotropy of the differential 共with respect to angle兲 relaxation rate. We start with the electronic spectrum in an elliptical QD in the presence of SO interactions subjected to a tilted magnetic field B = B⬜ + B储 = B共xˆ sin cos + yˆ sin sin + zˆ cos 兲, where and are the tilt and azimuthal angles, respectively. The system is described by the Hamiltonian ˆ +H ˆ , where H ˆ = 2 / 2m + V is the Hamiltonian ˆ =H ˆ +H H 0 SO Z 0 c of a 2D electron confined by the parabolic potential Vc = m共2x x2 + 2y y 2兲 / 2, x and y being the frequencies of the ˆ =H ˆ +H ˆ is the SO term, and QD in-plane potential, H SO R D 1 ˆ = g* · B is the Zeeman term 共m, g*, and stand H Z 2 B B for the electron effective mass, g factor, and Bohr magneton兲. For sufficiently strong 2D confinement, the orbital effect of the in-plane field can be neglected and ˆ depends only on the perpendicular field component via H 0 B A = 共 2⬜ 兲共−y , x , 0兲 共in symmetrical gauge兲. Using the transformation x = a1 P1 − a2 P2 , y = −b1m1Q1 + b2m2Q2 , x = a1Q1 − a2Q2 , y = b1 P1 / m1 − b2 P2 / m2, P j and Q j 共j = 1 , 2兲 being canonical momenta and coordinates, the ˆ can be brought to the canonical form of two Hamiltonian H 0 uncoupled oscillators with frequencies23 1 1,2 = 关冑共x + y兲2 + 2c ± 冑共x − y兲2 + 2c 兴, 2

共1兲

where c = eB⬜ / m is the cyclotron frequency and the coefficients a j and b j are given by a j = c j / D j, b j = 共2x − 2j 兲 / D j with D j = 关共2x − 2j 兲2 + 2x 2c 兴1/2. In the absence of SO coupling, the spectrum represents two ladders of equidistant levels 共for each spin projection兲 with energies En共0兲n ±

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= 共n1 + 1 / 2兲1 + 共n2 + 1 / 2兲2 ± Z, where Z = 21 g*BB is the Zeeman energy. ˆ causes an admixture of the ˆ +H ˆ =H The SO term H SO D R oscillator states 兩n1n2典± with different orbital 共n j兲 and spin 共⫾兲 quantum numbers. In a tilted field, the calculation of the SO matrix elements is carried out in two steps. First, the spin ˆ operators in H SO are rotated in spin space to align the spin-quantization axis along the total field B. Second, the ˆ are expressed via new canonical orbital operators in H SO variables P j, Q j. The expressions for the general matrix elements are rather cumbersome, and here we provide only those for the lowest adjacent levels with opposite spins, corresponding to the 2 ladder, ˆ

±具01兩HSO兩00典⫿

=

␣R

l2冑2 −

关共b2 cos ± a2兲cos + i共a2 cos ± b2兲sin 兴

i␣D

l2冑2

关共a2 cos ⫿ b2兲cos − i共b2 cos ⫿ a2兲sin 兴, 共2兲

where l j = 共m j兲 . These matrix elements explicitly depend on the tilt, , and azimuthal, , angles as well as on the QD geometry encoded in the coefficients a j , b j. In general, the magnitude of SO coupling is small compared to the level separation, ␣ / l j Ⰶ j, and, accordingly, the SO-induced level admixture is weak. However, this admixture becomes strong near the resonance, i.e., when the spacing between adjacent levels is of the order of the SO energy: 2 − Z ⬃ ␣R,D / l2 共see Fig. 1兲. This can be achieved, e.g., by varying the Zeeman energy with the tilt angle . The corresponding anticrossing ˆ 兩00典 円 is evaluated from Eq. 共2兲 as gap ⌬ = 2円 +具01兩H SO − −1/2

⌬2 =

␣R2 关共b2 cos + a2兲2 cos2 + 共b2 + a2 cos 兲2 sin2 兴 l22 +

␣D2 关共b2 cos − a2兲2 sin2 + 共b2 − a2 cos 兲2 cos2 兴 l22

+

␣ R␣ D 2 2 2 共a2 + b2兲sin sin 2 . l22

共3兲

The gap magnitude is governed by the angle-dependent interference between Rashba and Dresselhaus terms. Importantly, in elliptical QDs, such interference depends on the dot geometry via the coefficients a j , b j. Thus, near the resonance, 2 − Z ⬃ ⌬, the energies of the lowest excited states, E2,3 = 2 ⫿ 21 冑共2 − Z兲2 + ⌬2, acquire a strong angular dependence. At the same time, the phononassisted transitions between these states 兩j典 共j = 2 , 3兲 and the ground state 兩1典 are enhanced due to the strong admixture of constituent orbital levels 共see Fig. 1兲. As a result, the spin relaxation becomes anisotropic with respect to the in-plane field orientation . As we show below, the relaxation exhibits anomalous sensitivity to the system parameters in a narrow range of where the SO terms interfere destructively. The transition rate between state 兩j典 and the ground state

FIG. 1. 共Color online兲 共a兲 Energy levels and 共b兲 relaxation rates ⌫12 and ⌫13 for circular 共dashed line兲 and elliptical 共solid line兲 QDs vs in-plane field component B储. Arrows in 共a兲 indicate relevant transitions, and circles indicate resonance regions.

兩1典 is given by ⌫1j = 2兺Q円具1兩U兩j典円2␦共E1 − E j + Q兲, where the sum runs over acoustic phonon modes with dispersion Q = cQ, c being the sound velocity, and 3D momenta Q = 共q , qz兲. The transition matrix element is a product of phonon and electron contributions, 具1兩U兩j典 = M 共Q兲具1兩eiQ·R兩j典, where the phonon part M 共Q兲 = ⌳共Q兲 + i⌶共Q兲 includes piezoelectric, ⌳, and deformation, ⌶, contributions.24 In the numerical calculations we include longitudinal and two transverse piezoelectric acoustical modes. The electron matrix element can, in turn, be decomposed into a product of transverse and in-plane contributions, 具1兩eiQ·R兩j典 = f z共qz兲f 1j共q兲. The transverse contribution f z共qz兲 is determined by the 2D confinement, assumed parabolic below, while the in-plane contribution f 1j共q兲 is evaluated as follows. Not too far from the resonance region, 兩共2 − Z兲 / 共2 + Z兲兩 Ⰶ 1, it is sufficient to restrict calculations to the lowest four levels of the 2 ladder. The states 兩2典 and 兩3典, corresponding to the anticrossing levels, are superpositions of un共j兲 共j兲 perturbed states 兩j典 = d00− 兩00典− + d01+ 兩01典+, while the ground state acquires a small admixture from the upper orbital of 共1兲 共1兲 opposite spin 兩1典 = d00+ 兩00典+ + d01− 兩01典−. The coefficients d are obtained as 共3兲 共2兲 = d00− = 共1 + e−2兲−1/2 , d01+ 共3兲 共2兲* = ei共1 + e2兲−1/2 , = − d01+ d00− 共1兲 = 共1 + 2兲−1/2, d00+

共1兲 d01− = *共1 + 2兲−1/2 ,

共4兲

ˆ 兩00典 共 + 兲−1 ⬃ 共␣ / l 兲共 + 兲−1 is where = −具01兩H SO + 2 Z R,D 2 2 Z the ratio of SO and orbital energies, while ˆ 兩00典 兲 is the phase of the SO matrix element. = arg共 +具01兩H SO − The parameter  characterizes the proximity to the resonance: sinh  = 共2 − Z兲 / ⌬ is the detuning in units of the anticrossing gap. In deriving Eqs. 共4兲, we neglected higherorder terms in the SO coupling suppressed by the additional

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FIG. 2. 共Color online兲 Angular dependence of the relaxation rate ⌫12 for 共a兲 circular and 共b兲 elliptical QDs for B⬜ = 6.0 T and several values of B储.

factor 兩兩 Ⰶ 1. In the resonance region  ⬃ 1, the coefficients d共2兲 ⬃ d共3兲 ⬃ 1, i.e., the levels 2 and 3 are well hybridized. Away from the resonance, ⌬ Ⰶ 兩2 − Z兩 Ⰶ 共2 + Z兲, corresponding to 兩兩 Ⰷ 1, the eigenstates 2 and 3 are mainly determined by the unperturbed states 兩00典− and 兩01典+ 共and vice versa兲 to the left 共right兲 from the resonance region. In terms of the coefficients d, the 2D matrix element takes 共1兲 共j兲* 共1兲 共j兲* the form f 1j共q兲 = d01− d00− f 0001共q兲 + d00+ d01+ f 0100共q兲, with f m⬘n⬘mn = 具m⬘n⬘兩eiq·r兩mn典. The relevant elements are f 0001 * = f 0100 = 共il2q / 冑2兲e−共a1l1+a2l2兲q /4共−a2 cos + ib2 sin 兲, = arg共q兲. We then obtain 22

f 1j = −

il2q

冑2 e

22

2

−共a21l21+a22l22兲q2/4

−i i 关F共j兲 + F共j兲 1 e 2 e 兴,

with

共5兲

where 1 共j兲 共1兲* 共j兲 共1兲* F共j兲 1 = 关共a2 + b2兲d01+d00+ + 共a2 − b2兲d00−d01− 兴, 2 1 共j兲 共1兲* 共j兲 共1兲* F共j兲 2 = 关共a2 − b2兲d01+d00+ + 共a2 + b2兲d00−d01− 兴. 2

共6兲

Substituting Eq. 共5兲 into the full transition matrix element 具1兩U兩j典 = M 共Q兲f z共qz兲f 1j共q兲, the scattering rate ⌫1j is obtained by summing up over phonon modes in a standard manner.9–13 The result is the product ⌫1j = G1j共兲W j,

j = 2,3,

共7兲

共j兲 2 2 where the geometric factor G1j = 兩F共j兲 1 兩 + 兩F2 兩 is determined only by the SO-induced admixture of electronic states, encoded in the coefficients d, while the phonon factor W j is the probability of phonon-assisted transitions between levels with energies E j and E1. Note that W j are nearly SO independent; in the resonance region, the SO contribution to W j is ⬃2 Ⰶ 1 and can be neglected. Thus, near the resonance, the dependence of scattering rate ⌫ on the SO parameters and, accordingly, on the azimuthal angle , comes only through the geometric factor G1j. Remarkably, this factor can be extracted directly from the experimental data via the differential relaxation rate normalized to its value at some angle 共e.g., = 0兲:

⌬⌫ j ⌫1j共兲 − ⌫1j共0兲 G1j共兲 − G1j共0兲 = = , ⌫ ⌫1j共0兲 G1j共0兲

共8兲

where the right-hand side is independent of the phonon contribution.

Below we present our numerical results for spin relaxation rates in a GaAs QD. Calculations were performed for a parabolic confining potential with x = 2 meV and y = 1.33 meV, corresponding to the ellipticity x / y = 3 / 2. For comparison, results for the circular QD 共x = y = 2 meV兲 are presented too. We choose the parabolic transverse confinement with lz = 4 nm. The values of the SO constants, if not specified otherwise, in GaAs were taken as ␣R = 5 meV Å and ␣D = 16.25 meV Å, while the phonon parameters were taken from Ref. 24. In Fig. 1 we plot the lowest energy levels and relaxation rates for both circular and elliptical QDs as a function of the in-plane field B储 at fixed B⬜ = 6.0 T and = 0. For chosen parameters, the level anticrossings at 2 = Z, indicated by circled regions, for elliptical QDs are achieved at lower B储 due to a weaker confinement along the y axis 关see Fig. 1共a兲兴. The relaxation rates ⌫12 and ⌫13 are plotted in Fig. 1共b兲. The sharp increase in ⌫12 共⌫13兲 is caused by a stronger SOinduced admixture of the 兩00典− and 兩01典+ states as one approaches the resonance from the left 共right兲. To the right of the resonance 共Z ⬎ 2兲, ⌫12 is dominated by the orbital transition between states 兩00典+ and 兩01典+; so is ⌫13 to the left of the resonance 共Z ⬍ 2兲. The flat B储 dependence in these regions is because in a narrow 2D layer the orbital wave functions depend only on B⬜. Apart from the magnitude of ⌫, the overall behavior is similar for circular and elliptical QDs. In Fig. 2, we plot the relaxation rate ⌫12 as a function of in-plane field orientation at several values of its magnitude B储 in the resonance region. A strong azimuthal anisotropy is apparent: at certain angles, ⌫12 reaches minima that turn into maxima as B储 sweeps through the resonance. For a circular QD, the extrema of ⌫12 occur at = ± / 4 regardless of the values of ␣R and ␣D 关see Fig. 2共a兲兴. This anisotropy originates from the angular dependence of the anticrossing gap ⌬共兲. Indeed, in the resonance region, 兩2 − Z兩 / ⌬ ⬃ 1, the SO contribution to ⌫12 takes the simple form

冉

G12共兲 = 共a22 + b22兲 1 −

冊

2 − Z . 关⌬2 + 共2 − Z兲2兴1/2

共9兲

For a circular QD, the expression 共3兲 for the gap simplifies 2 ⌬2 = 共2 / ⍀兲共D + R2 + 2DR sin 2兲 with D to11,22 2 = ␣D sin 共 / 2兲 / l2 and R = ␣R sin2共 / 2兲 / l2, so the extrema of ⌬ at = ± / 4 translate into the extrema of G1j. In contrast, in elliptical QDs, the angular dependence of ⌫ is determined by system parameters 关see Fig. 2共b兲兴. The additional asymmetry introduced by the QD ellipticity modifies the interference between Rashba and Dresselhaus terms and shifts the

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FIG. 3. 共Color online兲 Angular dependence of the differential relaxation rates for 共a兲 circular and 共b兲 elliptical QDs for several values of the coefficients ␣R at B⬜ = 6.0 T and B储 = 13.2 共a兲 and 7.7 T 共b兲.

extrema away from ± / 4. The extrema of ⌬, Eq. 共3兲, now depend on both the QD ellipticity and the SO parameters, and occur at tan 2e = − 2

␣R␣D a22 + b22 . ␣R2 − ␣D2 a22 − b22

共10兲

For the parameters of Fig. 2共b兲, e = 106°. Such interplay between QD geometry and SO interactions suggests a way to simultaneously determine both SO couplings and QD ellipticity from the measured angular dependence of the differential relaxation rate ⌬⌫ / ⌫, Eq. 共8兲. In the resonance region, the phonon contribution W j drops out, as does the prefactor in Eq. 共9兲, so ⌬⌫ / ⌫ is determined solely

Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 共1998兲 T. Fujisawa et al., Nature 共London兲 419, 278 共2002兲. 3 R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, Phys. Rev. Lett. 91, 196802 共2003兲. 4 J. M. Elzerman et al., Nature 共London兲 430, 431 共2004兲. 5 M. Kroutvar et al., Nature 共London兲 432, 81 共2004兲. 6 R. Hanson, L. H. Willems van Beveren, I. T. Vink, J. M. Elzerman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwenhoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94, 196802 共2005兲. 7 Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett. 96, 047202 共2006兲. 8 S. Amasha et al., cond-mat/0607110 共unpublished兲. 9 A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 共2000兲; 64, 125316 共2001兲. 10 L. M. Woods, T. L. Reinecke, and Y. Lyanda-Geller, Phys. Rev. B 66, 161318共R兲 共2002兲. 11 V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 共2004兲. 12 D. V. Bulaev and D. Loss, Phys. Rev. B 71, 205324 共2005兲. 13 C. F. Destefani and S. E. Ulloa, Phys. Rev. B 72, 115326 共2005兲. 14 V. I. Fal’ko, B. L. Altshuler, and O. Tsyplyatev, Phys. Rev. Lett. 1 D. 2

by the anticrossing gap ⌬共 , x , y兲. The angular dependence of ⌬⌫ / ⌫ is shown in Fig. 3 for several values of Rashba coupling ␣R which can be varied, e.g., with an external electric field.17 For a circular QD, change in ␣R does not affect the minima positions at = − / 4, as discussed above; however, the modulation depth varies strongly 关see Fig. 3共a兲兴. Note that the ␣R dependence is nonmonotonic; the deepest minimum occurs for the almost complete destructive interference of the SO terms at ␣R / ␣D ⬇ tan2共 / 2兲. Deviations from circular QD shape give rise to angular dispersion of spin relaxation in the parameter space. For elliptical QDs, variations of ␣R shift the minima positions of ⌬⌫ / ⌫ 关see Fig. 3共b兲兴. In fact, the sensitivity of ⌬⌫ / ⌫ to the system parameters is drastically enhanced for in the vicinity of the critical angles e, Eq. 共10兲, for which the destructive interference between the Rashba and Dresselhaus terms is strongest. Thus, a scan of the angular dependence of the experimental differential relaxation rate in this narrow domain would enable an unambiguous extraction of both SO and QD geometry parameters. In summary, we proposed a method for simultaneous extraction of both the spin-orbit constants as well as the quantum dot shape from the angular anisotropy of the differential spin-relaxation rate in a tilted field. The underlying mechanism is based upon the enhanced sensitivity of phononassisted spin-flip transitions to the system parameters in the vicinity of level anticrossings. This sensitivity arises from destructive interference between the SO terms in a narrow domain of in-plane field orientations in the presence of asymmetric confinement. Note that such interplay between SO interactions and QD geometry cannot be captured by simplified descriptions of elliptical QDs using circular QDs with effective parameters.2 This work was supported by NSF Grant No. DMR0606509 and by DoD contract No. W912HZ-06-C-0057.

95, 076603 共2005兲. J. Könemann, R. J. Haug, D. K. Maude, V. I. Fal’ko, and B. L. Altshuler, Phys. Rev. Lett. 94, 226404 共2005兲. 16 P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602 共2006兲; Phys. Rev. B 74, 045320 共2006兲. 17 See, e.g., R. Winkler, Spin-Orbit Coupling Effects in TwoDimensional Electron and Hole Systems 共Springer, Berlin, 2003兲, and references therein. 18 D. G. Austing, S. Sasaki, S. Tarucha, S. M. Reimann, M. Koskinen, and M. Manninen, Phys. Rev. B 60, 11514 共1999兲. 19 M. Valín-Rodríguez, A. Puente, and L. Serra, Phys. Rev. B 69, 085306 共2004兲. 20 J. B. Miller, D. M. Zumbühl, C. M. Marcus, Y. B. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Phys. Rev. Lett. 90, 076807 共2003兲. 21 S. D. Ganichev et al., Phys. Rev. Lett. 92, 256601 共2004兲. 22 V. I. Fal’ko, Phys. Rev. B 46, 4320 共1992兲. 23 N. G. Galkin, V. A. Margulis, and A. V. Shorokhov, Phys. Rev. B 69, 113312 共2004兲. 24 V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Metals and Semiconductors 共North-Holland, Amsterdam, 1987兲. 15

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Theory of anisotropic spin relaxation in quantum dots O. Olendski and T. V. Shahbazyan Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA 共Received 14 December 2006; published 23 January 2007兲 We study theoretically phonon-assisted spin relaxation of an electron confined in an elliptical quantum dot 共QD兲 subjected to a tilted magnetic field. In the presence of both Rashba and Dresselhaus spin-orbit terms, the relaxation rate is anisotropic with respect to the in-plane field orientation. This anisotropy originates from the interference, at nonzero tilt angle, between the two spin-orbit terms. We show that, in a narrow range of magnetic field orientations, the relaxation rate exhibits anomalous sensitivity to variations of the QD parameters. In this range, the relative change in the relaxation rate with in-plane field orientation is determined solely by the spin-orbit coupling strengths and by the dot geometry. This allows simultaneous determination of both Rashba and Dresselhaus coupling parameters and the dot ellipticity from analysis of the angular dependence of the relaxation rate. DOI: 10.1103/PhysRevB.75.041306

PACS number共s兲: 73.21.La, 72.25.Rb, 71.70.Ej

Spin relaxation in semiconductor quantum dots 共QDs兲 has recently attracted intense interest because of the possible use of the electron spin as a qubit.1 Quantization of orbital states in a QD due to the zero-dimensional 共0D兲 confinement leads to the suppression of traditional spin-relaxation mechanisms 共e.g., D’yakonov-Perel兲 that are dominant in continuous systems. Indeed, recent experiments on GaAs QDs in a magnetic field B have revealed extremely long spin-relaxation times 共up to T1 = 170 ms at B ⯝ 1.75 T兲.2–8 For moderate and high fields 共B ⬎ 1 T兲, spin relaxation in QDs is dominated by phonon-assisted electronic transitions between Zeeman-split levels due to spin-orbit 共SO兲 coupling.9–16 There are two distinct types of SO couplings, one originating from bulk inversion asymmetry 共Dresselhaus coupling兲 and the other from structural inversion asymmetry along the growth direction 共Rashba coupling兲, that cause the admixture of orbital states with opposite spins.17 In the case of circular QDs in a perpendicular magnetic field, the Rashba and Dresselhaus terms mix different pairs of levels, and can, in principle, be distinguished if one such pair provides the dominant relaxation channel. This is the case, for example, when adjacent orbital levels, coupled via the Rashba term, are brought into resonance with changing magnetic field.12 However, in more realistic situations, deformations of the QD shape strongly alter the electronic spectrum18 and, in general, the effects of the two SO contributions are not separable.19 An important distinction between the Rashba and Dresselhaus terms is their different symmetry properties. The ˆ = ␣ 共 − 兲, former, described by the Hamiltonian H R R x y y x possesses an in-plane rotational symmetry, while the latter, ˆ = ␣ 共 − 兲, does not.17 Here = −i + eA, A beH D D x x y y ing the vector potential, is the Pauli matrix vector, and ␣R 共␣D兲 is the Rashba 共Dresselhaus兲 coupling constant 共we set ˆ leads to an ប = 1兲. This lack of rotational invariance for H D in-plane momentum azimuthal anisotropy in the presence of both SO terms that was recently reported in transport experiments in quantum wells.20,21 In a magnetic field, the anisotropy arises due to the interference between Rashba and Dresselhaus terms in the matrix elements22 in the presence of an in-plane field component. In QDs, this anisotropy reveals itself as a modulation of the spin-relaxation rate for different orientations of the in-plane field.11,14–16 1098-0121/2007/75共4兲/041306共4兲

Here we study the spin relaxation between Zeeman-split levels in elliptical QDs in a tilted magnetic field. We demonstrate that the interplay between SO interactions and QD geometry leads to dramatic changes in the relaxation rate in a certain range of field orientations for which Rashba and Dresselhaus contributions undergo destructive interference. Furthermore, in the vicinity of level anticrossings 共see Fig. 1兲, the SO contribution to the relaxation rate factors out from the phonon one. This allows simultaneous determination of the parameters for both SO interactions and QD geometry from the azimuthal anisotropy of the differential 共with respect to angle兲 relaxation rate. We start with the electronic spectrum in an elliptical QD in the presence of SO interactions subjected to a tilted magnetic field B = B⬜ + B储 = B共xˆ sin cos + yˆ sin sin + zˆ cos 兲, where and are the tilt and azimuthal angles, respectively. The system is described by the Hamiltonian ˆ +H ˆ , where H ˆ = 2 / 2m + V is the Hamiltonian ˆ =H ˆ +H H 0 SO Z 0 c of a 2D electron confined by the parabolic potential Vc = m共2x x2 + 2y y 2兲 / 2, x and y being the frequencies of the ˆ =H ˆ +H ˆ is the SO term, and QD in-plane potential, H SO R D 1 ˆ = g* · B is the Zeeman term 共m, g*, and stand H Z 2 B B for the electron effective mass, g factor, and Bohr magneton兲. For sufficiently strong 2D confinement, the orbital effect of the in-plane field can be neglected and ˆ depends only on the perpendicular field component via H 0 B A = 共 2⬜ 兲共−y , x , 0兲 共in symmetrical gauge兲. Using the transformation x = a1 P1 − a2 P2 , y = −b1m1Q1 + b2m2Q2 , x = a1Q1 − a2Q2 , y = b1 P1 / m1 − b2 P2 / m2, P j and Q j 共j = 1 , 2兲 being canonical momenta and coordinates, the ˆ can be brought to the canonical form of two Hamiltonian H 0 uncoupled oscillators with frequencies23 1 1,2 = 关冑共x + y兲2 + 2c ± 冑共x − y兲2 + 2c 兴, 2

共1兲

where c = eB⬜ / m is the cyclotron frequency and the coefficients a j and b j are given by a j = c j / D j, b j = 共2x − 2j 兲 / D j with D j = 关共2x − 2j 兲2 + 2x 2c 兴1/2. In the absence of SO coupling, the spectrum represents two ladders of equidistant levels 共for each spin projection兲 with energies En共0兲n ±

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= 共n1 + 1 / 2兲1 + 共n2 + 1 / 2兲2 ± Z, where Z = 21 g*BB is the Zeeman energy. ˆ causes an admixture of the ˆ +H ˆ =H The SO term H SO D R oscillator states 兩n1n2典± with different orbital 共n j兲 and spin 共⫾兲 quantum numbers. In a tilted field, the calculation of the SO matrix elements is carried out in two steps. First, the spin ˆ operators in H SO are rotated in spin space to align the spin-quantization axis along the total field B. Second, the ˆ are expressed via new canonical orbital operators in H SO variables P j, Q j. The expressions for the general matrix elements are rather cumbersome, and here we provide only those for the lowest adjacent levels with opposite spins, corresponding to the 2 ladder, ˆ

±具01兩HSO兩00典⫿

=

␣R

l2冑2 −

关共b2 cos ± a2兲cos + i共a2 cos ± b2兲sin 兴

i␣D

l2冑2

关共a2 cos ⫿ b2兲cos − i共b2 cos ⫿ a2兲sin 兴, 共2兲

where l j = 共m j兲 . These matrix elements explicitly depend on the tilt, , and azimuthal, , angles as well as on the QD geometry encoded in the coefficients a j , b j. In general, the magnitude of SO coupling is small compared to the level separation, ␣ / l j Ⰶ j, and, accordingly, the SO-induced level admixture is weak. However, this admixture becomes strong near the resonance, i.e., when the spacing between adjacent levels is of the order of the SO energy: 2 − Z ⬃ ␣R,D / l2 共see Fig. 1兲. This can be achieved, e.g., by varying the Zeeman energy with the tilt angle . The corresponding anticrossing ˆ 兩00典 円 is evaluated from Eq. 共2兲 as gap ⌬ = 2円 +具01兩H SO − −1/2

⌬2 =

␣R2 关共b2 cos + a2兲2 cos2 + 共b2 + a2 cos 兲2 sin2 兴 l22 +

␣D2 关共b2 cos − a2兲2 sin2 + 共b2 − a2 cos 兲2 cos2 兴 l22

+

␣ R␣ D 2 2 2 共a2 + b2兲sin sin 2 . l22

共3兲

The gap magnitude is governed by the angle-dependent interference between Rashba and Dresselhaus terms. Importantly, in elliptical QDs, such interference depends on the dot geometry via the coefficients a j , b j. Thus, near the resonance, 2 − Z ⬃ ⌬, the energies of the lowest excited states, E2,3 = 2 ⫿ 21 冑共2 − Z兲2 + ⌬2, acquire a strong angular dependence. At the same time, the phononassisted transitions between these states 兩j典 共j = 2 , 3兲 and the ground state 兩1典 are enhanced due to the strong admixture of constituent orbital levels 共see Fig. 1兲. As a result, the spin relaxation becomes anisotropic with respect to the in-plane field orientation . As we show below, the relaxation exhibits anomalous sensitivity to the system parameters in a narrow range of where the SO terms interfere destructively. The transition rate between state 兩j典 and the ground state

FIG. 1. 共Color online兲 共a兲 Energy levels and 共b兲 relaxation rates ⌫12 and ⌫13 for circular 共dashed line兲 and elliptical 共solid line兲 QDs vs in-plane field component B储. Arrows in 共a兲 indicate relevant transitions, and circles indicate resonance regions.

兩1典 is given by ⌫1j = 2兺Q円具1兩U兩j典円2␦共E1 − E j + Q兲, where the sum runs over acoustic phonon modes with dispersion Q = cQ, c being the sound velocity, and 3D momenta Q = 共q , qz兲. The transition matrix element is a product of phonon and electron contributions, 具1兩U兩j典 = M 共Q兲具1兩eiQ·R兩j典, where the phonon part M 共Q兲 = ⌳共Q兲 + i⌶共Q兲 includes piezoelectric, ⌳, and deformation, ⌶, contributions.24 In the numerical calculations we include longitudinal and two transverse piezoelectric acoustical modes. The electron matrix element can, in turn, be decomposed into a product of transverse and in-plane contributions, 具1兩eiQ·R兩j典 = f z共qz兲f 1j共q兲. The transverse contribution f z共qz兲 is determined by the 2D confinement, assumed parabolic below, while the in-plane contribution f 1j共q兲 is evaluated as follows. Not too far from the resonance region, 兩共2 − Z兲 / 共2 + Z兲兩 Ⰶ 1, it is sufficient to restrict calculations to the lowest four levels of the 2 ladder. The states 兩2典 and 兩3典, corresponding to the anticrossing levels, are superpositions of un共j兲 共j兲 perturbed states 兩j典 = d00− 兩00典− + d01+ 兩01典+, while the ground state acquires a small admixture from the upper orbital of 共1兲 共1兲 opposite spin 兩1典 = d00+ 兩00典+ + d01− 兩01典−. The coefficients d are obtained as 共3兲 共2兲 = d00− = 共1 + e−2兲−1/2 , d01+ 共3兲 共2兲* = ei共1 + e2兲−1/2 , = − d01+ d00− 共1兲 = 共1 + 2兲−1/2, d00+

共1兲 d01− = *共1 + 2兲−1/2 ,

共4兲

ˆ 兩00典 共 + 兲−1 ⬃ 共␣ / l 兲共 + 兲−1 is where = −具01兩H SO + 2 Z R,D 2 2 Z the ratio of SO and orbital energies, while ˆ 兩00典 兲 is the phase of the SO matrix element. = arg共 +具01兩H SO − The parameter  characterizes the proximity to the resonance: sinh  = 共2 − Z兲 / ⌬ is the detuning in units of the anticrossing gap. In deriving Eqs. 共4兲, we neglected higherorder terms in the SO coupling suppressed by the additional

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FIG. 2. 共Color online兲 Angular dependence of the relaxation rate ⌫12 for 共a兲 circular and 共b兲 elliptical QDs for B⬜ = 6.0 T and several values of B储.

factor 兩兩 Ⰶ 1. In the resonance region  ⬃ 1, the coefficients d共2兲 ⬃ d共3兲 ⬃ 1, i.e., the levels 2 and 3 are well hybridized. Away from the resonance, ⌬ Ⰶ 兩2 − Z兩 Ⰶ 共2 + Z兲, corresponding to 兩兩 Ⰷ 1, the eigenstates 2 and 3 are mainly determined by the unperturbed states 兩00典− and 兩01典+ 共and vice versa兲 to the left 共right兲 from the resonance region. In terms of the coefficients d, the 2D matrix element takes 共1兲 共j兲* 共1兲 共j兲* the form f 1j共q兲 = d01− d00− f 0001共q兲 + d00+ d01+ f 0100共q兲, with f m⬘n⬘mn = 具m⬘n⬘兩eiq·r兩mn典. The relevant elements are f 0001 * = f 0100 = 共il2q / 冑2兲e−共a1l1+a2l2兲q /4共−a2 cos + ib2 sin 兲, = arg共q兲. We then obtain 22

f 1j = −

il2q

冑2 e

22

2

−共a21l21+a22l22兲q2/4

−i i 关F共j兲 + F共j兲 1 e 2 e 兴,

with

共5兲

where 1 共j兲 共1兲* 共j兲 共1兲* F共j兲 1 = 关共a2 + b2兲d01+d00+ + 共a2 − b2兲d00−d01− 兴, 2 1 共j兲 共1兲* 共j兲 共1兲* F共j兲 2 = 关共a2 − b2兲d01+d00+ + 共a2 + b2兲d00−d01− 兴. 2

共6兲

Substituting Eq. 共5兲 into the full transition matrix element 具1兩U兩j典 = M 共Q兲f z共qz兲f 1j共q兲, the scattering rate ⌫1j is obtained by summing up over phonon modes in a standard manner.9–13 The result is the product ⌫1j = G1j共兲W j,

j = 2,3,

共7兲

共j兲 2 2 where the geometric factor G1j = 兩F共j兲 1 兩 + 兩F2 兩 is determined only by the SO-induced admixture of electronic states, encoded in the coefficients d, while the phonon factor W j is the probability of phonon-assisted transitions between levels with energies E j and E1. Note that W j are nearly SO independent; in the resonance region, the SO contribution to W j is ⬃2 Ⰶ 1 and can be neglected. Thus, near the resonance, the dependence of scattering rate ⌫ on the SO parameters and, accordingly, on the azimuthal angle , comes only through the geometric factor G1j. Remarkably, this factor can be extracted directly from the experimental data via the differential relaxation rate normalized to its value at some angle 共e.g., = 0兲:

⌬⌫ j ⌫1j共兲 − ⌫1j共0兲 G1j共兲 − G1j共0兲 = = , ⌫ ⌫1j共0兲 G1j共0兲

共8兲

where the right-hand side is independent of the phonon contribution.

Below we present our numerical results for spin relaxation rates in a GaAs QD. Calculations were performed for a parabolic confining potential with x = 2 meV and y = 1.33 meV, corresponding to the ellipticity x / y = 3 / 2. For comparison, results for the circular QD 共x = y = 2 meV兲 are presented too. We choose the parabolic transverse confinement with lz = 4 nm. The values of the SO constants, if not specified otherwise, in GaAs were taken as ␣R = 5 meV Å and ␣D = 16.25 meV Å, while the phonon parameters were taken from Ref. 24. In Fig. 1 we plot the lowest energy levels and relaxation rates for both circular and elliptical QDs as a function of the in-plane field B储 at fixed B⬜ = 6.0 T and = 0. For chosen parameters, the level anticrossings at 2 = Z, indicated by circled regions, for elliptical QDs are achieved at lower B储 due to a weaker confinement along the y axis 关see Fig. 1共a兲兴. The relaxation rates ⌫12 and ⌫13 are plotted in Fig. 1共b兲. The sharp increase in ⌫12 共⌫13兲 is caused by a stronger SOinduced admixture of the 兩00典− and 兩01典+ states as one approaches the resonance from the left 共right兲. To the right of the resonance 共Z ⬎ 2兲, ⌫12 is dominated by the orbital transition between states 兩00典+ and 兩01典+; so is ⌫13 to the left of the resonance 共Z ⬍ 2兲. The flat B储 dependence in these regions is because in a narrow 2D layer the orbital wave functions depend only on B⬜. Apart from the magnitude of ⌫, the overall behavior is similar for circular and elliptical QDs. In Fig. 2, we plot the relaxation rate ⌫12 as a function of in-plane field orientation at several values of its magnitude B储 in the resonance region. A strong azimuthal anisotropy is apparent: at certain angles, ⌫12 reaches minima that turn into maxima as B储 sweeps through the resonance. For a circular QD, the extrema of ⌫12 occur at = ± / 4 regardless of the values of ␣R and ␣D 关see Fig. 2共a兲兴. This anisotropy originates from the angular dependence of the anticrossing gap ⌬共兲. Indeed, in the resonance region, 兩2 − Z兩 / ⌬ ⬃ 1, the SO contribution to ⌫12 takes the simple form

冉

G12共兲 = 共a22 + b22兲 1 −

冊

2 − Z . 关⌬2 + 共2 − Z兲2兴1/2

共9兲

For a circular QD, the expression 共3兲 for the gap simplifies 2 ⌬2 = 共2 / ⍀兲共D + R2 + 2DR sin 2兲 with D to11,22 2 = ␣D sin 共 / 2兲 / l2 and R = ␣R sin2共 / 2兲 / l2, so the extrema of ⌬ at = ± / 4 translate into the extrema of G1j. In contrast, in elliptical QDs, the angular dependence of ⌫ is determined by system parameters 关see Fig. 2共b兲兴. The additional asymmetry introduced by the QD ellipticity modifies the interference between Rashba and Dresselhaus terms and shifts the

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FIG. 3. 共Color online兲 Angular dependence of the differential relaxation rates for 共a兲 circular and 共b兲 elliptical QDs for several values of the coefficients ␣R at B⬜ = 6.0 T and B储 = 13.2 共a兲 and 7.7 T 共b兲.

extrema away from ± / 4. The extrema of ⌬, Eq. 共3兲, now depend on both the QD ellipticity and the SO parameters, and occur at tan 2e = − 2

␣R␣D a22 + b22 . ␣R2 − ␣D2 a22 − b22

共10兲

For the parameters of Fig. 2共b兲, e = 106°. Such interplay between QD geometry and SO interactions suggests a way to simultaneously determine both SO couplings and QD ellipticity from the measured angular dependence of the differential relaxation rate ⌬⌫ / ⌫, Eq. 共8兲. In the resonance region, the phonon contribution W j drops out, as does the prefactor in Eq. 共9兲, so ⌬⌫ / ⌫ is determined solely

Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 共1998兲 T. Fujisawa et al., Nature 共London兲 419, 278 共2002兲. 3 R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, Phys. Rev. Lett. 91, 196802 共2003兲. 4 J. M. Elzerman et al., Nature 共London兲 430, 431 共2004兲. 5 M. Kroutvar et al., Nature 共London兲 432, 81 共2004兲. 6 R. Hanson, L. H. Willems van Beveren, I. T. Vink, J. M. Elzerman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwenhoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94, 196802 共2005兲. 7 Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett. 96, 047202 共2006兲. 8 S. Amasha et al., cond-mat/0607110 共unpublished兲. 9 A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 共2000兲; 64, 125316 共2001兲. 10 L. M. Woods, T. L. Reinecke, and Y. Lyanda-Geller, Phys. Rev. B 66, 161318共R兲 共2002兲. 11 V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 共2004兲. 12 D. V. Bulaev and D. Loss, Phys. Rev. B 71, 205324 共2005兲. 13 C. F. Destefani and S. E. Ulloa, Phys. Rev. B 72, 115326 共2005兲. 14 V. I. Fal’ko, B. L. Altshuler, and O. Tsyplyatev, Phys. Rev. Lett. 1 D. 2

by the anticrossing gap ⌬共 , x , y兲. The angular dependence of ⌬⌫ / ⌫ is shown in Fig. 3 for several values of Rashba coupling ␣R which can be varied, e.g., with an external electric field.17 For a circular QD, change in ␣R does not affect the minima positions at = − / 4, as discussed above; however, the modulation depth varies strongly 关see Fig. 3共a兲兴. Note that the ␣R dependence is nonmonotonic; the deepest minimum occurs for the almost complete destructive interference of the SO terms at ␣R / ␣D ⬇ tan2共 / 2兲. Deviations from circular QD shape give rise to angular dispersion of spin relaxation in the parameter space. For elliptical QDs, variations of ␣R shift the minima positions of ⌬⌫ / ⌫ 关see Fig. 3共b兲兴. In fact, the sensitivity of ⌬⌫ / ⌫ to the system parameters is drastically enhanced for in the vicinity of the critical angles e, Eq. 共10兲, for which the destructive interference between the Rashba and Dresselhaus terms is strongest. Thus, a scan of the angular dependence of the experimental differential relaxation rate in this narrow domain would enable an unambiguous extraction of both SO and QD geometry parameters. In summary, we proposed a method for simultaneous extraction of both the spin-orbit constants as well as the quantum dot shape from the angular anisotropy of the differential spin-relaxation rate in a tilted field. The underlying mechanism is based upon the enhanced sensitivity of phononassisted spin-flip transitions to the system parameters in the vicinity of level anticrossings. This sensitivity arises from destructive interference between the SO terms in a narrow domain of in-plane field orientations in the presence of asymmetric confinement. Note that such interplay between SO interactions and QD geometry cannot be captured by simplified descriptions of elliptical QDs using circular QDs with effective parameters.2 This work was supported by NSF Grant No. DMR0606509 and by DoD contract No. W912HZ-06-C-0057.

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