Spin relaxation and g-factor manipulation in quantum dots

NANOSTRUCTURES

Spin relaxation and g-factor manipulation in quantum dots

Carlos F. Destefani^*; Sergio E. Ulloa

Department of Physics and Astronomy and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701, USA

ABSTRACT

Phonon-induced spin relaxation rates and electron g-factor tuning of quantum dots are studied as function of in-plane and perpendicular magnetic fields for different dot sizes. We consider Rashba and Dresselhaus spin-orbit mixing in wide and narrow-gap semiconductors, and show how Zeeman sublevels can relax via piezoelectric (GaAs) and deformation (InSb) potential coupling to acoustic phonons. We find that strong confinement may induce minima in the rates at particular values of the magnetic field (due to a magnetic field-induced cancellation of the spin-orbit effects), where spin relaxation times can reach seconds. We also report on g-factor anisotropy. We obtain good agreement with available experimental values.

Keywords: g-factor; Phonon-induced spin relaxation; Quantum dots

Since the proposal of a qubit based on the electron spin of quantum dots (QDs) [1], much work has been done to understand the processes that may cause their relaxation, since long coherence times are required. One of those processes is related to the phonon-induced spin-flip rates of Zeeman sublevels in QDs in magnetic fields, where the spin purity of the levels is broken by the spin-orbit (SO) interaction. A recent experiment [2] has shown a spin relaxation time » 0.55 ms at an in-plane field of 10 T in a GaAs QD defined in a 2DEG. In general, SO effects have been considered via perturbation theory [3], although exact treatments have also been presented [4, 5]. The perturbative approach, which includes only a few states, has been called into question by the demonstration that a larger basis is needed in order to achieve convergence even for the lowest QD states when the QD vertical width is narrow [4], as a complex interplay between different energy scales can be present [6]. Insights on the purity of the spin degree of freedom of electrons in QDs can also be extracted from measurements of their effective g-factor, e.g., by means of capacitance [7] and energy [8] spectroscopies.

In this work we study spin-flip rates and g-factor tuning in QDs under SO influence. Our goal is to compare wide and narrow-gap materials under in-plane and perpendicular magnetic fields, at different QD sizes.

The QD is defined by an in-plane parabolic confinement, V(r) = /2, where m (w₀ = E₀/) is the electronic effective mass (confinement frequency); the QD lateral length is l₀ = . The vertical confinement V(z) is strong enough so that only the state in the first quantum well subband is relevant, and its function is j_z(z) = sin(pz/z₀) if a hard wall is assumed, z₀ being the QD vertical well thickness. In a magnetic field B, the unperturbed Hamiltonian, H₀ = ²k²/2m + V(r) + H_Z, has the well-known Fock-Darwin (FD) solution, where H_Z = g₀µ_BB · s/2 is the Zeeman term, g₀ is the bulk g-factor, and k is the kinetic momentum that includes the magnetic vector potential. We include all SO terms in 2D zincblende QDs, namely Rashba and Dresselhaus interactions. The former is due to surface inversion asymmetry (SIA) induced by the 2D confinement, while the latter is caused by the bulk inversion asymmetry (BIA) present in zincblende structures. The SIA Hamiltonian is H_SIA = as · ÑV(r, z) × k, while the BIA is H_BIA = g[s_xk_x () + s_yk_y () + s_zk_z ()], with coupling constants a and g. The z-confinement yields the electric field dV/dz in the SIA Hamiltonian as well as the momentum average áñ = (p/z₀)² in the BIA terms. The full QD Hamiltonian is then H = H₀ + H_SIA + H_BIA, which is diagonalized in a basis containing 110 FD states. Details about the derivation of H are found elsewhere [5].

We calculate spin relaxation rates between the two lowest Zeeman sublevels caused by piezoelectric and deformation acoustic phonons via Fermi's Golden Rule: G_fi = 2p/å_j,Q|g_fi(q)|² |Z(q_z)|² |M_j(Q)|² (n_Q + 1)d(DE + c_jQ) , where the sum is over the emitted phonon modes j (j = LA, TA1, TA2) with momentum Q = (q, q_z). The term Z(q_z) = áj_z||j_zñ (g_fi(q) = áf|e^iq.r|iñ) is the form factor perpendicular (parallel) to the 2D-plane (position is R = (r, z)), while n_Q is the phonon distribution with energy c_jQ; energies DE = e_f - e_i and states |iñ, |fñ are obtained via diagonalization of the total H, so that the SO mixing is fully taken into account. The element M_j(Q) = L_j(Q) + iX_j(Q) includes both piezoelectric L_j and deformation X_j potentials; in zincblende structures they become X_LA(Q) = X₀A_LA (only LA is present for X_j), L_LA(Q) = 3L₀A_LA sin(2q)q²q_z/2Q^7/2, L_TA1(Q) = L₀A_TA cos(2q)qq_z/Q^5/2, and L_TA2(Q) = L₀A_TAsin(2q) (2/q² - 1) q³/2Q^7/2 (both TA1 and TA2 modes are compacted as a single TA mode for L_j), where A_j = and L₀ = 4peh₁₄/k. The bulk phonon constants are X₀ and eh₁₄, c_j are the sound velocities (c_TA ¹ c_LA), k is the dielectric constant, and N₀ is the electron density. The triple space integral ([r, f_r, z]) allows an analytical form, the same as two ([f_q, q_z]) of the momentum integrations, leaving a numerical integral only in q. The only z₀-dependence in this remaining integral in G_fi reads F_j(z₀) = (d_jz₀ - (d_jz₀)³ /p²)^-2sin²(d_jz₀), where d_j = /2; q runs from 0 to DE/c_j, while F_j(z₀) is multiplied by polynomials and exponentials in q in the total G_fi. No approximation is needed in our derivation of G_fi, so that the 3D nature of the phonon is fully taken into account.

Regarding the effective g-factor, we consider two possible definitions involving the two lowest Zeeman sublevels for in-plane (B_||) and perpendicular (B_^) magnetic field, namely, /g₀ = DE/(g₀µ_BB_^,||) or /g₀ = á Ds_zñ /2 (á Ds_zñ is the spin expectation value difference of those levels under B_^,||). Although the first definition is used operationally in experiments where DE is measured, the latter is intuitively reasonable since g is a quantity intrinsically related to the spin value of those levels. For no SO interaction, both definitions yield g_^ = g_|| = g₀ (no anisotropy).

Figure 1 [9] shows the phonon-induced spin-flip rates as function of in-plane (panel A) and perpendicular (panel B) magnetic field, for different values of E₀, for GaAs QDs [10]. In panel A, smaller QDs present smaller spin-flip rates (longer relaxation times). This happens because as E₀ increases, the orbital levels become more separated and then the SO coupling becomes relatively less important. A strong dependence of the rates with B_|| and E₀ is clear. For GaAs QDs under in-plane fields, we find that the TA piezoelectric coupling dominates at low fields (< 14 T for E₀ = 5 meV), while at high fields the deformation potential takes over. The experimental spin-flip time at B_|| = 10 T for a 1.1 meV QD is 0.55 ms [2], while from panel A we find 1.5 µs. This time increases by one or two orders of magnitude in a larger z-well (» 0.05 ms at z₀ = 100 Å, not shown), which appears to be a better match for the experimental conditions. A 5 µs value was found perturbatively (z₀ = 50 Å, a=0) [11]. This confirms that the perturbative approach [3] yields the same rates as the exact calculation for GaAs in in-plane fields.

In panel B, the most striking feature under B_^ is the appearance of minima in the rates. The E₀-dependent minima (12 < B_^< 16 T) are due to a vanishing DE (see Fig. 4F). At such values of B_^, a sublevel crossing produces a sudden spin-flip. To understand the minima at low fields (6 < B_^< 8 T) we have to be mindful of the sine argument in F_j(z₀), which may induce a minimum in the rate at a particular value of the field, due to the interplay of the energy scales in the problem. We find that the three low-field minima occur at B_^-values where a magnetic field-induced cancellation of the SO influence is produced on the respective DE values (see Fig. 4F): below (above) such E₀-dependent values of B_^ - where DE recovers its pure Zeeman value of g₀µ_BB - the SO coupling increases (decreases) the sublevel splitting as compared to the QD without SO. Notice that the two smallest confinements do not show the low-field minima. This is due to the lowest sublevels acquiring the same spin (between 3 and 7 T at 0.7 meV, not shown), so that the rate decreases monotonically at that field-range. Like in the B_|| case, the dominating phonon mechanism is the TA piezoelectric coupling, but now at any field. Our results agree with available calculations [4] at small fields (B_^< 1 T), so that we can confirm that the perturbative approach is not adequate when dealing with SO effects in QDs in a perpendicular field (even in GaAs) if the vertical confinement is strong enough [12]. Notice that rates at small fields (< 2 T) have the same behavior under B_|| and B_^: the smaller E₀ the larger the rate; however, they are much larger under B_^.

Figure 2 [9] shows results for InSb QDs [13] in in-plane (panel A) and perpendicular (panel B) magnetic fields. In panel A, in contrast to GaAs, minima in the rates are also visible under B_|| for the highest E₀ values. At small fields (< 2 T), a monotonic rate drop is again observed as E₀ increases. As shown for the 20 meV QD, the LA deformation potential is the dominant phonon mechanism for InSb at any field-value (and E₀). It is worth mentioning that, contrary to GaAs, where a unique low B_^-field minimum is present for a given E₀, the number of rate minima in InSb increases with E₀. Notice that both LA modes have the same number of minima - occurring at the same B_||-values -, while a larger number of minima in the TA mode occurs at different field-values; this happens because c_LA > c_TA, so that the sine argument in F_j(z₀) is smaller in the LA mode. Notice that the oscillatory rate is well defined until the Zeeman sublevels acquire the same spin; e.g., for a 15 meV QD, that occurs around 14 T.

In panel B, all spin-flip rates are well defined only for B_^ < 8 T. Such a field is even smaller (around 2 T) for small E₀, as it occurs for the 3.0 meV QD; the rate minimum around 1 T for this E₀ indicates a spin-flip, accompanied by the vanishing of DE (see Fig. 3F). Larger values of E₀ do not present spin-flip. The relaxation rates in panel B do not show the monotonic behavior seen at small fields for the in-plane field direction. Like in the B_|| case, the LA deformation potential dominates at any field-value. We emphasize that the perturbative approach finds no use in InSb QDs because of the inherent higher SO coupling.

Figures 3 [14] (InSb, [13]) and 4 [14] (GaAs, [10]) present the effective g-factor for different l₀ sizes, with left (right) panel for in-plane (perpendicular) fields, while dotted lines show results without SO coupling. Panels A and B (C and D) use the definition of g in terms of áDs_zñ (DE). The respective sublevel energy splitting is shown in panels E and F. Notice that for InSb both definitions yield basically the same low B_||-field results. The drop in g_|| is faster in the curves because the two lowest states acquire the same spin at higher fields. In the low B_^-field results, however, the two g-factor definitions yield different values mainly at weaker confinements, although the drop in g_^ is also faster in the curves. Observe that for the smallest E₀ (largest SO effect) of 3.0 meV, a sign change is seen in around 1 T, which relates to an unusual crossing involving the ground and first excited states. In both field-directions, smaller E₀ yields smaller g-factor, which shows that SO coupling provides a channel to manipulate g in QDs under magnetic fields. Notice the clear anisotropy in g^E (panels C and D): the same QD confinement shows , since the mixing with higher orbitals is stronger for B_^. If the definition is considered (panels A and B), such anisotropy is not as remarkable at low fields.

Figure 4 shows that for GaAs both definitions in panels A and C give essentially the same g_||-values. Results are totally different in a perpendicular field. At 1.1 meV, it can be seen (panel B) that has inverted sign at low fields, becomes zero for B_^ between 4 and 8 T, then acquires inverted sign again, and suddenly flips back to its 'normal' behavior at 13.7 T; under higher fields, goes to zero since the two lowest level spins are aligned. Panel B also shows that larger E₀-values (smaller SO coupling) cancels the field range where is zero. Notice that the weaker the confinement the smaller the field where the sign change occurs.

One finds totally different results for (panel D), which may even assume values 12 times larger than g₀ for the smallest E₀ at low fields; still at low fields, larger E₀ tend to reduce towards g₀. At high fields (inset G), goes to zero when the level crossing involving the ground state occurs. For every E₀ there is a magnetic field - indicated by the dashed lines connecting panels D and F - where goes from higher to smaller values than g₀ (see discussion for the low-field minima in Fig. 1B). Such result emphasizes the intricate competition between external magnetic field and intrinsic SO coupling in QDs [6]. In GaAs QDs, the anisotropic nature of the g-factor is much more pronounced, despite the small values of the SO constants. For an experimental comparison [8], at GaAs QD with E₀ = 1.1 meV and B_||=10 T, DE @ 200 µeV was reported, while from Fig. 4E one finds DE @ 180 µeV; in a linear fit, |g| = 0.29±0.01 was found and from Fig. 4A (4C) one has || = 0.30 (|| = 0.31).

Even though both QD materials show minima in the spin-flip rates - with B_^ for GaAs and with B_|| and B_^ for InSb - their origin is slightly different. Minima come from the nature of the z-confinement, and the field where they occur depend on l₀. The SO coupling mixes spins and alters splitting of sublevels in distinct ways according to field-direction and QD material, so that spin relaxation can be induced by piezoelectric (wide-gap) and deformation (narrow-gap) phonons. The external field opens channels (at the rate minima) where long spin relaxation times (@ 1 s) may be reached, so that the spin coherence required for quantum computing could be improved. The SO coupling is able to tune the anisotropic electron g-factor in QDs and even change its sign. There is a complex interplay between SO and magnetic energies, which creates distinct phases in the GaAs QD spectra under B_^, where the SO interaction increases or decreases the Zeeman sublevel splitting. These features would not have been accessed if a perturbative approach had been used, especially for QDs with large (small) lateral (vertical) size.

[1] D. Loss et al., Phys. Rev. A 57, 120 (1998).

[2] J. M. Elzerman et al., Nature 430, 431 (2004).

[3] A. V. Khaestkii et al., Phys. Rev. B 64, 125316 (2001); L. M. Woods et al., Phys. Rev. B 66, 161318(R) (2002).

[4] J. L. Cheng et al., Phys. Rev. B 69, 115318 (2004).

[5] C. F. Destefani et al., Phys. Rev. B 69, 125302 (2004); C. F. Destefani et al., Phys. Rev. B 70, 205315 (2004).

[6] M. Valín-Rodríguez, Phys. Rev. B 70, 033306 (2004).

[7] G. Medeiros-Ribeiro et al., Appl. Phys. Lett. 80, 4229 (2002).

[8] R. Hanson et al., Phys. Rev. Lett. 91, 196802 (2003).

[9] C. F. Destefani et al., Phys. Rev. B, 72, 115326 (2005).

[10] m/m₀ = 0.067, g₀ = -0.44, k = 12.4, a = 4.4Ų, g = 26eVų, X₀ = 7eV, eh₁₄ = 0.140eV/Å, N₀ = 5.32 × 10^-27Kg/ų, c_LA = 4.73 × 10¹³Å/s, c_TA = 3.35 × 10¹³Å/s, z₀ = 40Å, dV/dz = -0.5meV/Å.

[11] V. N. Golovach et al., Phys. Rev. Lett. 93, 016601 (2004).

[12] If a Gaussian well is considered, one has ák²_zñ = 1/z²₀, a BIA term a factor of 10 smaller than used in this work.

[13] m/m₀ = 0.014, g₀ = -51, k = 16.5, a = 500Ų, g = 160eVų, X₀ = 7eV, eh₁₄ = 0.061eV/Å, N₀ = 5.77 × 10^-27Kg/ų, c_LA = 3.40 × 10¹³Å/s, c_TA = 2.29 × 10¹³Å/s, z₀ = 40Å, dV/dz = -0.5meV/Å.

[14] C. F. Destefani et al., Phys. Rev. B 71, 161303(R), 2005.

Received on 4 April, 2005

*

present address: Physics Department, University of Ottawa, Ottawa, Ontario K1N6N5, Canada.

Publication Dates

• Publication in this collection
  10 July 2006
• Date of issue
  June 2006

History

• Received
  04 Apr 2005