---
abstract: 'We calculate the spin relaxation rates in a parabolic InSb quantum dots due to the spin interaction with acoustical phonons. We considered the deformation potential mechanism as the dominant electron-phonon coupling in the Pavlov-Firsov spin-phonon Hamiltonian. By studying suitable choices of magnetic field and lateral dot size, we determine regions where the spin relaxation rates can be practically suppressed. We analyze the behavior of the spin relaxation rates as a function of an external magnetic field and mean quantum dot radius. Effects of the spin admixture due to Dresselhaus contribution to spin-orbit interaction are also discussed.'
address:
- '$^1$ Instituto de Física, Universidade Federal de Uberlândia, 38400-902 Uberlândia MG, Brazil'
- '$^2$ Departamento de Física, Universidad de Buenos Aires, C1428EHA Buenos Aires, Argentina'
- '$^3$ Departamento de Física, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil'
author:
- 'A. M. Alcalde$^1$, C. L. Romano$^2$, L. Sanz$^1$ and G. E. Marques$^3$'
title: 'Phonon modulation of the spin-orbit interaction as a spin relaxation mechanism in InSb quantum dots'
---

Introduction
============

The ability to manipulate and control processes that involve transitions between spin states is, at the moment, of extreme importance due to the recent applications in polarized spin electronics and quantum computation. Spin dephasing is the most critical aspect that should be considered in the elaboration of proposals of quantum computation based in single spin states as qubits in quantum dots (QDs) [@Imamoglu99]. While for bulk and for 2D systems the spin relaxation processes has been studied in some detail, the problem for QD’s still require deeper and further discussions. Several processes that can induce spin relaxation in semiconductors have been identified and were studied. At the moment remains in discussion which, between these processes, is dominant in zero-dimensional systems. Some experimental results have shown good agreement with the theoretical predictions for 2D systems [@Lau01] but, in general, the identification of the processes through direct comparison with the experimental results may become a formidable task. This problem is more critical for QDs, since few experimental results exist and the theoretical discussion of the spin relaxation mechanisms is still an open subject. Extensive theoretical works in QD systems have studied the main phonon mediated spin-flip mechanisms, including admixture processes due to spin-orbit coupling [@Khaetskii01] and phonon coupling due to interface motion (ripple mechanism) [@Woods02]. Spin relaxation rates strongly dependent on the dot size, magnetic field strength, and temperature, as reported by several authors [@Khaetskii01; @Falko05]. It was shown that the quantum confinement produces, in general, a strong reduction of the QD relaxation rates.

In this work, we calculate the spin-flip transition rates, considering the phonon modulation by the spin-orbit interaction. For this purpose will use the spin-phonon interaction Hamiltonian proposed by Pavlov and Firsov [@Pavlov66; @Pavlov67]. In this model, the Hamiltonian describing the transitions with spin reversal, due to the scattering of electrons by phonons, can be written in a general form, $H_{ph}=V_\mathrm{ph} + \beta[\sigma \times \nabla V_\mathrm{ph}]\cdot (\mathbf{p}+e/c \mathbf{A})$, where $V_\mathrm{ph}$ is the phonon operator, $\sigma$ is the spin operator, $\mathbf{p}$ is the linear momentum operator and $\mathbf{A}$ is the vectorial potential related with the external magnetic field $\mathbf{B}$. This model has the advantage of being easily adapted to the study of other interaction mechanisms with phonons.

Theory
======

Based on the effective mass theory applied to the problem of the interaction of an electron with lattice vibrations, including the spin-orbit interaction and in presence of an external magnetic field, Pavlov and Firsov [@Pavlov66; @Pavlov67] have obtained the spin-phonon Hamiltonian that describes the transitions with spin reversal of the conduction band electrons due to scattering with longitudinal lattice vibrations as $$\begin{aligned}
H_{ph}&=& d(q)\left( \frac{\hbar}{\rho_M V v q}\right)^{1/2}
\left\{ e^{i \mathbf{q \cdot r}} b_\mathbf{q}
\left[\begin{array}{cc}
0 &\mathbf{\hat{n}}^-\times \mathbf{\hat{e}_q}  \\
\mathbf{\hat{n}}^+\times \mathbf{\hat{e}_q} & 0
\end{array}
\right] %\right.\nonumber \\
%&&\left.
\left( \frac{\mathbf{p}}{\hbar} + \frac{e\mathbf{A}}{\hbar c} + \mathbf{q} \right)
+ \mathrm{h.c}\right\},
\label{spinphonon}\end{aligned}$$ where, $b_\mathbf{q} (b_\mathbf{q}^\dagger)$ are annihilation (creation) phonon operators, the magnetic vector potential $\mathbf{A}$ is obtained in the symmetric gauge considering an external magnetic field $\mathbf{B}$ oriented along the $z$ axis. $\mathbf{\hat{n}}^\pm= \mathbf{\hat{x}} \pm \mathbf{\hat{y}}$, where $\mathbf{\hat{x}}$, $\mathbf{\hat{y}}$ are unitary vectors along the $x$ and $y$ axis. $\mathbf{\hat{e}_q}$ is a unit vectors in the direction of the phonon polarization, $\mathbf{q}$ is the phonon wave vector, $\mathbf{p}$ is the momentum operator, $v$ is the average sound velocity, $\rho_M$ is the mass density, $V$ is the system volume and $d(q)$ is a coupling constant that depends on the electron-phonon coupling mechanism. Detailed expressions for the parameter $d(q)$ can be found in Ref. [@Pavlov67].

It has been assumed that the confinement along the $z$ axis is much stronger than the lateral confinement. Thus, the lateral motion is decoupled from the one along $z$ and the envelope functions separate $\psi(\mathbf{r})=f(x,y)\phi(z)$. The $z$-dependent part of $\psi(\mathbf{r})$ is an eigenfunction of a symmetric quantum well of width $L$. In lens-shaped quasi-two dimensional self assembled QDs, the bound states of both electrons and valence-band holes can be understood by assuming a lateral spatial confinement modeled by a parabolic potential with rotational symmetry in the $x-y$ plane [@Hawrylak99], $V(\rho)=\frac{1}{2}m\omega_0^2\rho^2$, where $\hbar\omega_0$ is the characteristic confinement energy, and $\rho$ is the radial coordinate. By using the one-band effective mass approximation and considering an external magnetic field $B$ applied normal to plane of the QD, the electron lateral wave function can be written as $$f_{n,l,\sigma}= C_{n,l}\frac{\rho^{|l|}}{a^{|l|+1}} e^{-\frac{\rho^2}{2a^2}} e^{il\varphi}
L_n^{|l|}\left(\rho^2 / a^2 \right) \chi(\sigma),
\label{funciondeonda}$$ where $C_{n,l}=\sqrt{n!/[{\pi(n + |l|)!}] }$, $L_n^{|l|}$ is the Laguerre polynomial, $n$ ($l$) is the principal (azimuthal) quantum number, and $\chi(\sigma)$ is the spin wave function for the spin variable $\sigma$. The corresponding eigenenergies are $E_{n,l,\sigma }=(2n+|l|+1)\hbar \Omega
+(l/2)\hbar \omega _{c}+(\sigma /2)g\mu _{B}B,$ where $\Omega =(\omega
_{0}^{2}+\omega _{c}^{2}/4)^{1/2}$, $\mu _{B}$ is the Bohr magneton, $%
a=(\hbar /m\Omega )^{1/2}$ is the effective length and $\omega _{c}=eB/m$. In our model, we also consider the effects of the Dresselhaus contribution that provides additional admixture between spin states. For 2D systems, the linear Dresselhaus Hamiltonian can be written as $$H_D=\frac{\beta}{\hbar}\left(\sigma_xp_x -\sigma_yp_y\right),$$ where $p_i = -i\hbar \nabla_i + (e/c)A_i$ and $\beta$ is the Dresselhaus coupling parameter for this confinement. If the confinement potential in the $z$-direction is considered highly symmetrical, then $\nabla V_z \sim 0$ and the Rashba contribution can be safely ignored.

The spin relaxation rates ($W$) between the electronic states: $(n,l,\uparrow (\downarrow)) \rightarrow (n^\prime, l^\prime, \uparrow (\downarrow))$, with emission of one acoustic phonon, are calculated from the Fermi golden rule. In the Hamiltonian (\[spinphonon\]), we only consider the deformation potential (DP) electron-phonon coupling, this is due to the large $g$-factor in narrow gap InSb ($|g| \sim 51$), the dominant electron-phonon coupling for spin relaxation is the DP mechanism [@Alcalde04]. The piezoelectric (PE) coupling governs the spin relaxation processes in wide or intermediate gap semiconductors. In the transition matrix elements calculation, we not only consider the linear term $i \mathbf{q}\cdot \mathbf{r}$ in the expansion of $\exp(i\mathbf{q} \cdot \mathbf{r})$ [@Khaetskii01], but the integral representation of Bessel function is used in the evaluation of electron-phonon overlap integrals. The linear approximation of $\exp(i\mathbf{q} \cdot \mathbf{r})$ may be valid for spin inversion transitions in the spin polarized ground-states of GaAs based QDs where, due to the small value of the electron $g$-factor, only long wavelength phonons are involved.

Results and discussion
======================

The calculations were performed for a parabolic InSb QD at $T\sim 0$ K. The material parameters for the InSb system are listed in Ref. [@Destefani04a]. We only have considered electron transitions between ground state electron Zeeman levels $(0,0,\uparrow) \rightarrow (0,0,\downarrow)$ and $(0,1,\downarrow) \rightarrow (0,1,\uparrow)$. The temperature dependence for one-phonon emission rate is determined from $W=W_{0}(n_{B}+1)$, where $%
n_{B}$ is the Bose-Einstein distribution function and $W_{0}$ is the rate at $T=0$ K. In the temperature regime $T \leq $10 K, we obtain $n_{B}+1\approx
1 $ and $W\approx W_{0}$. For temperatures larger than few Kelvin degrees, two-phonon processes should be considered as the dominant spin relaxation mechanism. These types of processes have not been considered in the present calculation.

![Spin relaxation rates, $W$, for a parabolic InSb QD considering the DP coupling mechanism. Panel a) shows $W$ as a function of the magnetic field $B$, for two different electronic transitions and several lateral dot radius $r_0$ = 50, 75, 100, 125, 150, 175, and 200 Å(same $r_0$ ordering for both transitions). b) Contour plot of the spin relaxation rate as a function of $B$ and $r_0$.[]{data-label="rates"}](figura2a.eps "fig:"){width="18pc"} ![Spin relaxation rates, $W$, for a parabolic InSb QD considering the DP coupling mechanism. Panel a) shows $W$ as a function of the magnetic field $B$, for two different electronic transitions and several lateral dot radius $r_0$ = 50, 75, 100, 125, 150, 175, and 200 Å(same $r_0$ ordering for both transitions). b) Contour plot of the spin relaxation rate as a function of $B$ and $r_0$.[]{data-label="rates"}](figura2b.eps "fig:"){width="18pc"}

In the Figs. \[rates\]a) and \[rates\]b) we show the spin relaxation rates due to DP electron-phonon mechanism, as a function of the external magnetic field $B$ and considering some typical values for the effective lateral QD size, $r_0=\sqrt{\hbar/m\omega_0}$. Some interesting facts about these results should be pointed out: i) The rates show a strong dependence with the magnetic field. This fact can be explained from the dependence of the rates with the transition energy $\Delta E$. In general, we obtain that $W \sim [g^\ast \mu_B B]^n=(\Delta E)^n$, $n$ being an integer number that depends on the electron-phonon coupling process and $g^\ast$ the effective $g$-factor. As can be seen in Fig. \[rates\] a), when the magnetic field increases, the rates also increase until reaching a maximum near $B \sim 0.5$ T. The position of this maximum it is defined from the transition energy conservation: $E_{n l \sigma^\prime}-E_{n^\prime l^\prime \sigma^\prime}=\hbar v q$. ii) The oscillatory behavior of the rates, observed for $B>0.7$T are mainly produced by the Dresselhaus spin admixture, which modifies the effective Landè $g^\ast$-factor. As is showed in Fig. \[rates\] a), the $g^\ast$-factor effects are particulary important for the ground-state Zeeman transition. For small magnetic fields, $g^\ast \rightarrow g_\mathrm{bulk}$ and we may neglect the spin admixture effects. Therefore, the spin relaxation shows no oscillations and becomes almost independent of $r_0$. This small QD size dependence is in agreement with the experimental observations of Gupta and Kikkawa [@Gupta99]. iii) The rates dependence with the lateral QD size $r_0$, are related to the interplay effects between the spatial and magnetic confinements. This competing effects are contained in the electron-phonon overlap integral, $I \propto \int f^\ast_{n^\prime,l^\prime,\sigma^\prime}(\rho) \exp(i\mathbf{q}\cdot\mathbf{r}) f_{n,l,\sigma}(\rho) d\mathbf{r}$. For large fields, the magnetic confinement causes a gradual decrease in the overlap integral as the $r_0$ increases. For small magnetic fields, the spatial confinement is dominant. Thus, when $r_0$ diminishes the wave functions become more localized and the overlap integral should increase. This effects explain the behavior of the spin transition $(0,1,\downarrow) \rightarrow (0,1,\uparrow)$ showed in Fig. \[rates\] a) (red lines). The Zeeman ground-state rates (black lines) are strongly dependent on $\Delta E$ and, for small $B$, the rates are weakly dependent on $I$. iv) The same rates calculated for GaAs (not showed here), are in general, one order of magnitude smaller than InSb rates. As we expected, the relaxation via PE coupling is more efficient than via the DP phonon processes.

In Fig. \[rates\] b) we have plotted the spin relaxation rates for the ground-state Zeeman transition as a function of $r_0$ and $B$. We clearly identify a region of strong spin coherence, defined by $B > 1$ T and $r_0 > 100$ Å. In this regime, the relaxation times are in the ns order and this is an important feature for spin qubit engineering. In the $B < 0.1 $T regime, the relaxation times are approximately of few $\mu$s. This spin frozen region are not robust against the temperature and will disappear whenever the thermal energy is larger than the spin transition energy.

This work has been supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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