Phonon-assisted carrier tunneling in coupled quantum dot systems with hyperfine-induced spin flip

TL;DR

This paper investigates phonon-assisted hyperfine spin flips during electron and hole tunneling in coupled quantum dots, revealing the dominance of hyperfine processes over spin-orbit relaxation under certain magnetic field conditions and proposing experimental tests for hyperfine coupling effects.

Contribution

It provides a detailed calculation of hyperfine-induced spin flip rates in quantum dot tunneling, highlighting the conditions where hyperfine interactions dominate over spin-orbit effects and suggesting experimental verification methods.

Findings

Hyperfine processes dominate over spin-orbit relaxation for electrons up to a few Tesla.

For holes, hyperfine dominance occurs at sub-Tesla magnetic fields due to valence band hyperfine coupling.

A minimum in spin-flip probability arises from the interplay of hyperfine and spin-orbit mechanisms.

Abstract

We calculate the rates of phonon-assisted hyperfine spin flips during electron and hole tunneling between quantum dots in a self-assembled quantum dot molecule. We show that the hyperfine process dominates over the spin-orbit-induced spin relaxation in magnetic fields up to a few Tesla for electrons, while for holes this cross-over takes place at field magnitudes of a fraction of Tesla, upon the assumption of a large $d$-shell admixture to the valence band state, resulting in a strong transverse hyperfine coupling. The interplay of the two spin-flip mechanisms leads to a minimum of the spin-flip probability, which is in principle experimentally measurable and can be used as a test for the presence of substantial transverse hyperfine couplings in the valence band.

Tables

Table 1. Table 1: Nuclear Schliemann_JPC03 and atomic parameters.

	69Ga	71Ga	113In	115In	75As
$I$	$3/2$	$3/2$	$9/2$	$9/2$	$3/2$
$\zeta$	1.344	1.708	1.227	1.230	0.959
$r$	0.604	0.396	0.0428	0.9572	1
${\xi }_s$	3.9		3.9		4.4
${\xi }_p$	3.3		3.3		3.7
${\xi }_d$	10.5		8.9		11.9
$M_p$	0.050		0.050		0.050
$M_d$	0.33		0.20		0.33
$M_{sd}$	0.048		0.034		0.049
${\left|{\alpha }_d\right|}^2$	0.20		0.50		0.05
${\left|a_{C/A}^{\mathrm{cb}}\right|}^2$	0.50				0.50
${\left|a_{C/A}^{\mathrm{vb}}\right|}^2$	0.35				0.65

Equations

$$S_n(\mathbf{r})=z_n^{(\mathrm{WL})}+h_n\exp \left(-\frac{(x^2+y^2{)}^2}{r_n^4}\right),$$

$$C(\mathbf{r})=$$

$$H_{\mathrm{hf}}=3E_{\mathrm{hf}}\sum _{\alpha }{\zeta }_{\alpha }\mathbf{A}(\mathbf{r}-{\mathbf{R}}_{\alpha })\cdot {\mathbf{I}}_{\alpha }/\hslash ,$$

$$E_{\mathrm{hf}}=\frac{2{\mu }_0}{3\pi }{\mu }_{\mathrm{B}}{\mu }_{\mathrm{N}}{a}_{\mathrm{B}}^{-3}=0.5253\mu \mathrm{eV},$$

$$\mathbf{A}(\mathbf{r})=\frac{a_{\mathrm{B}}^3}{4\hslash }\left[\frac{8\pi }{3}\delta (\mathbf{r})\mathbf{S}+\frac{\mathbf{L}}{r^3}+\frac{3(\hat{\mathbf{r}}\cdot \mathbf{S})\hat{\mathbf{r}}-\mathbf{S}}{r^3}\right].$$

$$H_{ph}=\sum _{\mathbf{k},\lambda }\hslash {\omega }_{\mathbf{k},\lambda }{b}_{\mathbf{k},\lambda }^{†}{b}_{\mathbf{k},\lambda }+\sum _{\mathbf{k},\lambda }\left\{\Phi (\mathbf{r}),e^{i\mathbf{k}\cdot \mathbf{r}}\right\}\left(b_{\mathbf{k},\lambda }+b_{-\mathbf{k},\lambda }^{†}\right),$$

$$|{\Psi }_{n\sigma ;\dots m_{\alpha }\dots }⟩$$

$$+\sum _{\alpha }{c}_{\alpha +}^{(n\sigma )}|n\overline{\sigma };\dots m_{\alpha }+1\dots ⟩$$

$$+\sum _{\alpha }{c}_{\alpha -}^{(n\sigma )}|n\overline{\sigma };\dots m_{\alpha }-1\dots ⟩,$$

$$c_{\alpha \pm }^{(n\sigma )}$$

$$=\frac{3E_{\mathrm{hf}}{\zeta }_{\alpha }}{2\hslash {\omega }_{\sigma \overline{\sigma }}^{(nn)}}\sqrt{j(j+1)-m_{\alpha }(m_{\alpha }\pm 1)}⟨n\overline{\sigma }\left|A_{\mp }\right|n\sigma ⟩,$$

$${\Gamma }_{\sigma \to {\sigma }^{\prime }}^{\{m_{\alpha }\}\to \{m_{\alpha }^{\prime }\}}=$$

$$\frac{2\pi }{\hslash }\left|n_{\mathrm{B}}({\omega }_{\sigma {\sigma }^{\prime }}^{(12)})+1\right|\sum _{\mathbf{k},\lambda }\delta (\hslash {\omega }_{\mathbf{k},\lambda }-|{\omega }_{\sigma {\sigma }^{\prime }}^{(12)}|)$$

$$\times {\left|⟨{\Psi }_{2{\sigma }^{\prime };\dots m_{\alpha }^{\prime }\dots }|\left\{\Phi (\mathbf{r}),e^{i\mathbf{k}\cdot \mathbf{r}}\right\}|{\Psi }_{1\sigma ;\dots m_{\alpha }\dots }⟩\right|}^2.$$

$$F_{{\sigma }^{\prime }\sigma }(\mathbf{k})=⟨2{\sigma }^{\prime }|\left\{\Phi (\mathbf{r}),e^{i\mathbf{k}\cdot \mathbf{r}}\right\}|1\sigma ⟩$$

$$⟨{\Psi }_{2{\sigma }^{\prime };\dots m_{\alpha }^{\prime }\dots }|\left\{\Phi (\mathbf{r}),e^{i\mathbf{k}\cdot \mathbf{r}}\right\}|{\Psi }_{1\sigma ;\dots m_{\alpha }\dots }⟩=$$

$$F_{{\sigma }^{\prime }\sigma }(\mathbf{k})$$

$$+F_{{\sigma }^{\prime }\overline{\sigma }}(\mathbf{k})\sum _{\alpha ,\pm }{c}_{\alpha \pm }^{(1\sigma )}⟨\dots m_{\alpha }^{\prime }\dots |\dots m_{\alpha }\pm 1\dots ⟩$$

$$+F_{\overline{{\sigma }^{\prime }}\sigma }(\mathbf{k})\sum _{\alpha ,\pm }{c}_{\alpha \pm }^{(2{\sigma }^{\prime })\ast }⟨\dots m_{\alpha }^{\prime }\pm 1\dots |\dots m_{\alpha }\dots ⟩$$

$$+\dots .$$

$${\Gamma }_{\sigma \to \sigma }=2\pi R_{\sigma \sigma \sigma \sigma }(|{\omega }_{\sigma \sigma }^{(12)}|),$$

$$R_{{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4}(\omega )$$

$$\times \sum _{\mathbf{k},\lambda }{F}_{{\sigma }_1{\sigma }_2}(\mathbf{k})F_{{\sigma }_4{\sigma }_3}^{\ast }(\mathbf{k})\delta ({\omega }_{\mathbf{k},\lambda }-|\omega |).$$

$${\Gamma }_{\sigma \to \overline{\sigma }}^{(\mathrm{so})}=2\pi R_{\sigma \overline{\sigma }\overline{\sigma }\sigma }({\omega }_{\sigma \overline{\sigma }}^{(12)}),$$

$${\Gamma }_{\sigma \to \overline{\sigma }}^{(\mathrm{hf})}(\dots m_{\alpha }\dots )=\sum _{\alpha }{\Gamma }_{\sigma \to \overline{\sigma }}^{(\mathrm{hf}),\alpha }(\dots m_{\alpha }\dots ),$$

$${\Gamma }_{\sigma \to \overline{\sigma }}^{(\mathrm{hf}),\alpha }(\dots m_{\alpha }\dots )=2\pi \sum _{a=\sigma ,\overline{\sigma }}\sum _{b=\sigma ,\overline{\sigma }}{Q}_{ab}^{(\alpha )}{R}_{aabb}({\omega }_{\sigma \overline{\sigma }}^{(12)}),$$

$$Q_{\sigma \sigma }^{(\alpha )}$$

$$Q_{\overline{\sigma }\overline{\sigma }}^{(\alpha )}$$

$$Q_{\overline{\sigma }\sigma }^{(\alpha )}$$

$$+\left[j(j+1)-m_{\alpha }(m_{\alpha }-1)\right]⟨1\overline{\sigma }\left|A_{+}\right|1\sigma ⟩⟨2\sigma \left|A_{-}\right|2\overline{\sigma }⟩\}.$$

$${\Gamma }_{\sigma \to \overline{\sigma }}^{(\mathrm{hf}),\alpha }(\dots m_{\alpha }\dots )$$

$$=2\pi R({\omega }_{\sigma \overline{\sigma }}^{(12)})\frac{(3E_{\mathrm{hf}}{\zeta }_{\alpha }{)}^2}{4}\{\left[j(j+1)-m_{\alpha }(m_{\alpha }+1)\right]{\left|⟨2\sigma \left|A_{+}\right|2\overline{\sigma }⟩-⟨1\sigma \left|A_{+}\right|1\overline{\sigma }⟩\right|}^2$$

$$+\left[j(j+1)-m_{\alpha }(m_{\alpha }-1)\right]{\left|⟨2\sigma \left|A_{-}\right|2\overline{\sigma }⟩-⟨1\sigma \left|A_{-}\right|1\overline{\sigma }⟩\right|}^2\}.$$

$${\overline{\Gamma }}_{\sigma \to \overline{\sigma }}^{(\mathrm{hf})}=2\pi \sum _{a=\sigma ,\overline{\sigma }}\sum _{b=\sigma ,\overline{\sigma }}{\overline{Q}}_{ab}{R}_{aabb}({\omega }_{\sigma \overline{\sigma }}^{(12)}).$$

$${\overline{Q}}_{\sigma \sigma }$$

$$\times \left({\left|⟨2\sigma \left|A_{-}\right|2\overline{\sigma }⟩\right|}^2+{\left|⟨2\sigma \left|A_{+}\right|2\overline{\sigma }⟩\right|}^2\right),$$

$${\overline{Q}}_{\overline{\sigma }\overline{\sigma }}$$

$$\times \left({\left|⟨1\overline{\sigma }\left|A_{-}\right|1\sigma ⟩\right|}^2+{\left|⟨1\overline{\sigma }\left|A_{+}\right|1\sigma ⟩\right|}^2\right),$$

$${\overline{Q}}_{\overline{\sigma }\sigma }={\overline{Q}}_{\sigma \overline{\sigma }}^{\ast }$$

$$\times (⟨1\overline{\sigma }\left|A_{-}\right|1\sigma ⟩⟨2\sigma \left|A_{+}\right|2\overline{\sigma }⟩$$

Full text

I Introduction

Hyperfine coupling between carrier spins and nuclear magnetic moments in a crystal Glazov-book is one of the key factors determining the properties of semiconductor structures, like quantum dots (QDs), and their usability in new information processing devices. Once considered the main source of dephasing Schliemann_JPC03 ; Coish_PSSB09 ; Cywinski_APPA11 ; Urbaszek2013 ; Chekhovich2011c , it can now be controlled with growing precision and used as a manageable degree of freedom Smirnov2020 ; Chekhovich2020 ; Waeber2019 .

The hf coupling for electrons is dominated by the approximately isotropic contact interaction. Its transverse components can in principle lead to spin relaxation accompanied by a simultaneous change of one of the nuclear spins (a ‘flip-flop’ process). Due to a large mismatch between electronic and nuclear Zeeman energies, this process is restricted to very low magnetic fields Eble2009a ; Dou2012 ; Fras2012a or bright-dark exciton resonances Kurtze2012 . At magnetic fields exceeding a fraction of Tesla the hf processes become ineffective Braun_PRL05 ; Glasenapp2016 . Even though the energy gap can be closed by emitting a phonon Beyer2012a , it turns out that processes relying on the spin-orbit (SO) coupling dominate in this regime heiss07 ; bulaev05a due to their more favorable dependence on the magnetic field. A cross-over between the hf and SO regimes has been observed in gated GaAs QDs Camenzind2018 .

Theoretical description of hf-induced spin flips was developed in the context of electrons in gated QDs erlingsson02 ; Erlingsson_PRB04 : One considers corrections to the carrier state with a given spin, mediated by the hyperfine coupling with all the nuclei, treated as a perturbation. In this way, the original state gets an admixture of inverted spin, which allows transitions to a state with a nominally opposite spin via spin-conserving phonon couplings. For a transition within the Zeeman doublet, the combination of the $\propto B^{-2}$ scaling of the hf admixture (stemming from the carrier Zeeman energy, while the nuclear Zeeman splitting is negligible), frequency dependence of the phonon spectral density, and van Vleck cancellation leads to a $\propto B^{3}$ dependence on the magnetic field.

In the case of holes the physics of hf interactions is more complex and some questions seem to remain open. Overall, the hole hf coupling is due to dipole interactions, which renders it much weaker than the contact interaction of electrons Fischer2008 ; Testelin2009a ; Fallahi2010 . Moreover, for a purely $p$-type valence band, the transverse components of the hole hf coupling can only result from weak band mixing effects Fischer2008 ; Eble2009a ; Testelin2009a , which would strongly limit hole spin relaxation. Indeed, a coherent population trapping experiment under transverse nuclear spin polarization in a low-noise device Prechtel2016 has led to the conclusion that the transverse hyperfine coupling is negligible. On the other hand, selective measurements of the Overhauser field for particular elements and isotopes in the crystal Chekhovich2013 yield results that can be explained by a substantial admixture of atomic $d$-shell states to the valence band, in line with earlier theoretical calculations Boguslawski1994 . According to theoretical models, such a $d$-shell admixture would give rise to a substantial transverse contribution to the hole hyperfine coupling Machnikowski2019 .

In quantum dot molecules (QDMs), built of two coupled QDs, even in the $s$ shell, spin relaxation can take place not only between states within one Zeeman doublet but can also accompany charge relaxation (dissipative tunneling) between the QDs. Understanding the role of hyperfine interactions in such a carrier tunneling process may be important for possible spin injection schemes as well as for spin readout protocols involving carrier transfer induced by gating pulses. On the other hand, hf flip-flops combined with spatio-temporal dynamics of the carrier may be used to imprint a particular state in the nuclear system. Finally, theoretically predicted characteristics of the spin relaxation, when confronted with the experiment, may verify the assumptions of the model and thus offer information on the nature of the hf coupling itself.

A reliable description of processes involving tunneling in self-assembled QD systems requires reasonable knowledge of wave functions, which is achievable with the $\bm{k}\!\cdot\!\bm{p}$ method in the envelope function approximation Gawarecki2012 . The $\bm{k}\!\cdot\!\bm{p}$ method allows one also to describe all kinds of phonon-assisted processes, including those involving SO-induced spin flips Mielnik-Pyszczorski2018a . Recently, we have also combined the $\bm{k}\!\cdot\!\bm{p}$ model with the hyperfine Hamiltonian and provided a description of hyperfine couplings based on multi-band wave functions Machnikowski2019 .

In this paper we calculate the rates of phonon-assisted hyperfine spin flip-flops during electron and hole relaxation between the two branches of the $s$ shell in a QDM (corresponding to states localized in two different QDs if the system is away from the resonance). We compare the result with the SO-induced phonon-assisted spin-flip tunneling and show that the hyperfine process dominates for fields below a few Tesla for electrons (depending on the axial electric field), while for holes it becomes important only for fields roughly below 0.1 T. The interplay of hf and SO mechanisms of spin relaxation leads to a non-monotonic dependence of the total spin flip probability, which may be used as a test for considerable transverse hyperfine couplings.

The paper is organized as follows. In Sec. II we desccribe the model of the system under study. Sec. III presents the theory for the phonon-assisted tunneling of a carrier with a simultaneous spin flip-flop between the carrier and a nuclear spin. In Sec. IV we present the results. Sec.

II Model

We consider two coupled, vertically stacked self-assembled InGaAs quantum dots. The dots are placed on the wetting layers of $a_{G}=0.565325$ nm thickness and $\mathrm{In_{0.4}\mathrm{Ga}_{0.6}As}$ composition. The shape of each dot is defined by the restriction of its upper limit by the surface Ardelt2016a

[TABLE]

where $n=\{1,2\}$ refers to the bottom and the top dot respectively, $z^{\mathrm{(WL)}}_{n}$ denotes the top of the wetting layer, $h_{n}$ is the dot height, and $r_{n}$ defines the lateral extension. We take $h_{1}=4.8$ nm, $h_{2}=5.4$ nm, $r_{1}=15.8$ nm, and $r_{2}=17$ nm. Both dots have a trumpet-shape composition jovanov12 , where the position-dependent In content is given by

[TABLE]

where $C_{\mathrm{t}}=0.7$, $C_{\mathrm{b}}=0.4$ are related to the composition in the top/bottom region of a given QD, $z_{\mathrm{c}}=1.1$ nm and $r_{\mathrm{c}}=2.3$ nm. To simulate material intermixing, the dots are processed by a Gaussian blur with the standard deviation of $0.6$ nm. The computations were carried out on $(200\times 200\times 200)a_{G}$ and $(100\times 100\times 70)a_{G}$ computational boxes for calculation of strain and electron states, respectively.

In order to quantitatively account for the hyperfine interactions in an inhomogeneous system like a QD, one needs to model the properties of wave functions on the mesoscopic scale of several or tens of nanometers and, simultaneously, describe the properties of Bloch functions near the nuclei, on a sub-nanometer scale. No single computational method is currently able to bridge these two scales, hence we use the hybrid approach developed in Ref. Machnikowski2019 , in which the mesoscopic scale is covered by an 8-band $\bm{k}\!\cdot\!\bm{p}$ theory in the virtual crystal approximation, while the atomic-scale properties are approximately accounted for by a simple model of effective rescaled hydrogen-like orbitals with averaging over specific alloying and isotope configurations.

Thus, on the mesoscopic scale, the static strain related to the lattice mismatch is accounted for within the continuous elasticity approach pryor98b . The strain-induced piezoelectric potential is calculated including the polarization up to the second order in strain tensor elements bester06b with parameters from Ref. caro15 . The wave functions for the four lowest electron and hole states (corresponding to the two QDs and two spin orientations) are obtained using the eight band $\bm{k}\!\cdot\!\bm{p}$ theory in the envelope function approximation. The Hamiltonian, the material parameters and implementation is given in much detail in Ref. Gawarecki2018 . We used also improvements of the model described in Ref. Krzykowski2020 .

The hyperfine Hamiltonian for the interaction of a carrier with nuclear spins is

[TABLE]

where $\alpha$ labels the ions, $\bm{R}_{\alpha}$ are their positions,

[TABLE]

$\mu_{\mathrm{B}}$ and $\mu_{\mathrm{N}}$ are Bohr and nuclear magnetons, respectively, $a_{\mathrm{B}}$ is the Bohr radius, $\mu_{0}$ is the vacuum permeability, $\bm{I}_{\alpha}$ is the nuclear spin, $\zeta_{\alpha}$ defines the nuclear magnetic moment for a given nucleus via $\bm{\mu}_{\alpha}=\zeta_{\alpha}\mu_{\mathrm{N}}\bm{I}_{\alpha}$, and

[TABLE]

We consider the two most common In isotopes, two Ga isotopes and one As isotope with the angular momenta $j=$9/2, 3/2, and 3/2, respectively.

The Bloch function is modeled as a sum of atomic orbitals corresponding to the outermost shells, centered around the two nuclei in the primitive cell. The atomic orbitals are taken as hydrogen-like wave functions $\psi_{nlm}(\bm{r})=\sqrt{n\xi_{nl}}\psi^{(\mathrm{H})}_{nlm}(n\xi_{nl}\bm{r})$, where $\psi^{(\mathrm{H})}_{nlm}(\bm{r})$ are the wave functions of the hydrogen atom and $\xi_{nl}$ is a scaling parameter. Since only one shell of a given symmetry is relevant for the topmost valence and lowest conduction bands, the principal quantum number $n$ will be omitted. The scaling parameters $\xi_{s}$ for the $s$-shell states are obtained from the experimentally determined values of the conduction band wave functions at the nucleus Chekhovich2017a ; Gueron1964 . The distribution of the wave functions between the anion and cation is chosen to be consistent with the known experimental Chekhovich2017a ; Gueron1964 and computational Boguslawski1994 data. In view of the lack of precise data, we choose $\xi_{p}=0.85\xi_{s}$ for each atom, following the general relation between the scaling parameters of Slater orbitals Clementi1963 ; Clementi1967 ; Benchamekh2015 (neither the Slater orbitals themselves nor their scaling commonly used in tight-banding computations can be used directly, as they are optimized against chemical bonding and band structures and fail to reproduce the correct behavior near the nucleus). The $d$-shell scaling parameter $\xi_{d}$ is estimated from the measured values of the hf coupling for holes Chekhovich2013 using the theoretically computed weights of $d$-shell admixtures in GaAs Boguslawski1994 . The resulting values of the parameters describing the hyperfine coupling are listed in Tab. 1, where we also give the quantities $M_{ll^{\prime}}=|\psi_{s}(0)|^{-2}\langle l|1/r^{3}|l^{\prime}\rangle$ that characterize the geometry of the wave functions for the dipole hyperfine interaction. The details of the model are described in Ref. Machnikowski2019 .

In view of the very small value of the nuclear Zeeman splitting, the probability for any nuclear configuration at thermal equilibrium is essentially the same at typical temperatures of experiments involving self-assembled QDs (a few Kelvin and above). Since the carrier Zeeman splitting is much larger than the nuclear one, the nuclear Zeeman energies can be neglected when considering the energy change in a carrier-nucleus spin flip-flop.

Coupling to phonons is described in the usual way. The phonon subsystem and the carrier-phonon interaction are described by the general Hamiltonian

[TABLE]

where $\bm{k}$ and $\lambda$ denote the wave vector and polarization of a phonon mode, respectively, $b_{\bm{k},\lambda}$, $b_{-\bm{k},\lambda}^{{\dagger}}$ are the corresponding annihilation and creation operators, and $\Phi(\bm{r})$ is an $8\times 8$ matrix of operators in the coordinate representation, corresponding to the 8-band structure of the $\bm{k}\!\cdot\!\bm{p}$ theory and accounting for deformation-potential and piezoelectric couplings. The detailed description of the carrier-phonon Hamiltonian is given in Ref. Gawarecki2019a .

III Phonon-assisted spin-flip transitions

In this section we present the theory for the phonon-assisted tunneling of carriers between the ground state manifolds of the two QDs with a simultaneous spin flip-flop between the carrier and a nuclear spin. The 8-band $\bm{k}\!\cdot\!\bm{p}$ theory treats the electron and hole states on equal footing (the latter upon a standard transition from the electron picture of the completely filled valence band to the hole picture) and we present our theory for single-particle states in a general form, without specifying the kind of the carrier. In the following, the term “spin” is used to identify one of the two sub-bands of the conduction or heavy-hole valence band.

In order to find the rate for a spin flip-flop process we first calculate the hyperfine flip-flop correction to the system state. We denote the carrier state in the $n$th QD ($n=1,2$) with the nominal spin orientation $\sigma$ (resulting from the $\bm{k}\!\cdot\!\bm{p}$ diagonalization) by $|n\sigma\rangle$ and its energy by $E_{\sigma}^{(n)}$. The state of the nuclei is labeled by $|\ldots m_{\alpha}\ldots\rangle$, where $m_{\alpha}$ is the quantum number for the $z$ projection of the respective nuclear spin. The states unperturbed by the hyperfine coupling are of a product form $|n\sigma;\ldots m_{\alpha}\ldots\rangle=|n\sigma\rangle\otimes|\ldots m_{\alpha}\ldots\rangle$. In the lowest order of perturbation theory with respect to the hyperfine interaction the eigenstates of the system are then

[TABLE]

where $\overline{\sigma}$ denotes inverted spin and the nuclear configuration on the right-hand side is the same as the one on the left-hand side except for the one explicitly given modified spin. The coefficients of the perturbative correction are

[TABLE]

where $A_{\pm}=A_{x}\pm iA_{y}$ and $\omega_{\sigma\sigma^{\prime}}^{(nn^{\prime})}=(E_{\sigma}^{(n)}-E_{\sigma^{\prime}}^{(n^{\prime})})/\hbar$.

From the Fermi golden rule, the probability of a phonon-assisted transition from the state with spin $\sigma$ in QD1 to the state with spin $\sigma^{\prime}$ in QD2 with a change of the nuclear configuration from $\{m_{\alpha}\}$ to $\{m^{\prime}_{\alpha}\}$ is

[TABLE]

Substituting the perturbation expansion from Eq. (3), taking into account the obvious fact that the phonon interaction conserves nuclear spins, and denoting

[TABLE]

one finds

[TABLE]

Since the multi-band carrier wave functions are dominated by one leading component (determining the nominal “spin” of the state), the couplings $F_{\sigma^{\prime}\sigma}(\bm{k})$ are large for $\sigma=\sigma^{\prime}$ and much smaller otherwise, when they stem from the band mixing involving SO couplings. The hyperfine admixture amplitudes $c_{\alpha\pm}^{(n\sigma)}$ are small, as well. Therefore, in Eq. (5) we kept only the contributions in the leading order in the SO or hf couplings, neglecting those relying on both these weak couplings simultaneously. Furthermore, for a nominally spin-conserving process ($\sigma=\sigma^{\prime}$), the transition amplitude is by far dominated by the first contribution $F_{\sigma\sigma}(\bm{k})$, which determines the spin-conserving phonon-assisted tunneling rate

[TABLE]

where we define the spectral densities for the phonon bath as

[TABLE]

For a spin-flip process, there are two mechanisms that may, in principle, yield comparable contributions: the SO channel entering via the first term and the hyperfine channel accounted for by the two other terms on the right-hand side of Eq. (5). The total spin-flip transition rate is then a sum of the SO rate,

[TABLE]

and the rates for hyperfine transitions, summed up over final configurations of the nuclear bath, differing by one nuclear spin-flip from the initial one,

[TABLE]

where we explicitly noted the dependence on the initial configuration of the nuclear bath and

[TABLE]

with

[TABLE]

Note that $R_{\sigma\sigma\overline{\sigma}\overline{\sigma}}(\omega)=R_{\overline{\sigma}\overline{\sigma}\sigma\sigma}^{*}(\omega)$ so $\Gamma_{\sigma\to\overline{\sigma}}^{(\mathrm{hf}),\alpha}$ is real.

Since the shape of the wave function for a given spatial state in a given QD very weakly depends on the spin orientation, all the spectral densities in Eq. (11) are almost identical upon an appropriate choice of the arbitrary phases. This allows one to write

[TABLE]

For a transition between the Zeeman states in a single QD the distinction between “1” and “2” disappears, the two matrix elements of $A_{\pm}$ become identical, and the transition rate is suppressed by destructive interference. In contrast, in the QDM system, as long as the states are spatially separated (away from the level-crossing resonance at which the ground states in the two QDs are aligned) each ion is effectively coupled to at most one carrier state (the one localized in the same QD as the ion) and only one of the two interfering amplitudes can be large.

For unpolarized nuclei the physically meaningful rate is obtained by averaging Eq. (10) over all the initial configurations of the nuclear spins and summing up over all nuclear spin flips,

[TABLE]

Since $\langle j(j+1)-m_{\alpha}(m_{\alpha}\pm 1)\rangle=2j(j+1)/3$ one finds

[TABLE]

IV Results

In this section we analyze and compare the electron and hole phonon-assisted tunneling rates with hyperfine- and SO-induced spin flips, as well as the spin-conserving tunneling rates. The rates are calculated using Eq. (13), corresponding to the thermal equilibrium of the nuclear bath at temperatures much higher than the nuclear Zeeman energies. The matrix elements in Eqs. (14) are evaluated using the $\bm{k}\!\cdot\!\bm{p}$ formalism for hyperfine interactions developed in our previous paper Machnikowski2019 , while the spectral densities are computed directly from the $\bm{k}\!\cdot\!\bm{p}$ wave functions according to the definitions in Eq. (4) and Eq. (8).

Fig. 2(a) shows the spin-conserving relaxation rates between the two lowest electron states as a function of the electric field $F$ applied in the growth direction ($z$) that relatively shifts the ground-state manifolds of the two QDs. The results are shown at two values of the magnetic field oriented along the growth direction (Faraday configuration). The central maximum corresponds to the tunneling resonance, when the two levels are aligned and form an anti-crossing. The oscillations are due to the interplay between the QD separation and the wave length of the emitted phonon as the energy splitting between the two levels is changed gawarecki10 . The difference between the relaxation rates at the two values of magnetic field is marginal.

In Fig. 2(b) we show the spin-flip tunneling rates for the electron in the presence of electric and magnetic fields as previously, comparing the rates calculated according to the theory presented above with those resulting from the SO couplings Gaweczyk2019 . At weak magnetic fields (here we choose $B=1$ T) the hyperfine channel dominates over the spin-orbit one. On the other hand, for moderate and strong values of magnetic field (here $B=8.0$ T), the spin-flip transitions caused by the SO couplings are stronger. This is due to the fact, that the rate related to the hf interaction decreases like $\propto 1/B^{2}$, while for the SO mechanisms it increases slowly in the considered range of the magnetic fields. In the case of the hyperfine-induced transitions, we observe a minimum at the field value corresponding to the resonance condition (crossing of the ground states of the two QDs). Under such conditions both wave functions are delocalized over the two QDs and a destructive interference of the two matrix elements of $A_{\pm}$ in Eq. (III) takes place.

The ratio of spin-flip to spin-conserving transition rates is shown in Fig. 2(c). This value is equal to the probability of spin flip during the incoherent phonon-assisted tunneling and therefore is a measure of spin preservation in this process. Typical values are on the order of $10^{-4}$ at 1 T, dominated by the hf coupling and scaling as $1/B^{2}$ at lower fields, while at higher fields the spin-flip process is dominated by the SO coupling and its probability is reduced to $10^{-6}$. For the hf-induced process, the pattern of oscillations in the spin-flip rate $\overline{\Gamma}_{\sigma\to\overline{\sigma}}^{(\mathrm{hf})}$ is similar to that in the spin-conserving rate $\Gamma_{\sigma\to\sigma}$ and the relative ratio is relatively flat. In contrast, for the SO process, $\Gamma_{\sigma\to\overline{\sigma}}^{(\mathrm{so})}$ has a different pattern of oscillations compared to $\Gamma_{\sigma\to\sigma}$, hence the ratio $\Gamma_{\sigma\to\overline{\sigma}}^{(\mathrm{so})}/\Gamma_{\sigma\to\overline{\sigma}}$ as a function of the electric field exhibits pronounced maxima and minima.

The results for a hole are presented in Fig. 3. Again, we show the electric field dependence of the spin-flip rate for the two spin relaxation mechanisms at two magnitudes of the magnetic field. For a hole, the spin-conserving phonon-assisted relaxation, shown in Fig. 3(a), is slower than for an electron due to weaker deformation potential coupling. In addition, when the states are localized in different QDs, the phonon-assisted tunneling is less efficient because of the higher hole mass, hence stronger localization.

For a hole, as can be seen in Fig. 3(b), the two spin-flip channels depend in different ways on the energy splitting controlled by the external electric field. As a result, at a given magnetic field, one or the other process may dominate. In the vicinity of the resonance, when the phonon-assisted relaxation or tunneling process is effective, the cross-over between the two mechanisms occurs at magnetic field amplitudes of a fraction of Tesla, much lower than in the case of an electron. This is due to the fact that the hyperfine-induced spin flip is much less probable than for an electron as a consequence of the much lower hf coupling for holes, while the SO-induced process is a few times more effective. Here the SO channel is nearly magnetic-field-independent, while the rate for the hf channel scales as $1/B^{2}$.

The spin-flip probability, given by the ratio of the spin-conserving to spin-flip rates [Fig. 3(c)], is dominated by the hf-induced process at very low fields, on the order of 0.01 T and below, with a remarkably high probability of the spin flip on the order of $1\%$ at $B=0.01$ T except for the closest vicinity of the resonance. At higher fields the importance of this process is reduced, leading to a much lower relative efficiency of the spin-flip process near the resonance, while far away from the resonance the increasing SO contribution yield overall probabilities on the order of $10^{-3}$.

In Fig. 4 we present the crossover between the SO and hf-dominated hole spin flip as a function of the magnetic field at the electric field magnitude of $F=-5$ kV/cm, near the maximum of the relaxation rate (this time using a linear scale). At this electric field value the cross–over takes place at $B\approx 0.2$ T (the exact value obviously depends on the electric field). While the hf channel manifests the $1/B^{2}$ dependence visible already on the previous figure, the SO contribution shows a very weak increase with the magnetic field (see the inset in Fig.  4), which was not noticeable earlier. As a result, the total relative spin-flip rate (spin-flip probability) is a non-monotonic function of the magnetic field over an experimentally accessible range of field magnitudes. A similar dependence can also be observed for an electron but with a much less pronounced minimum that is shifted to magnetic field magnitudes over 10 T (well above the SO-hf crossover) due to a very weak magnetic field dependence of the SO component for electrons.

V Conclusions

We have calculated the rates of phonon-assisted hyperfine and spin-orbit induced spin flips during electron and hole relaxation in a self-assembled QDM using a multi-band theory of hyperfine couplings based on the $\bm{k}\!\cdot\!\bm{p}$ model. We have predicted a cross-over between the two processes as dominant spin-flip mechanisms at magnetic fields of a few Tesla and on the order of $0.1$ T for electrons and holes, respectively, with the hf mechanism scaling as $1/B^{2}$ and dominating at lower fields over the nearly magnetic-field-independent SO channel. For the QDM structure considered here, the probability of spin flip during electron tunneling between the QDs can be large at low fields (about 1% for electrons and holes at magnetic fields of 0.1 T and 0.01 T, respectively) but decreases strongly with increasing field, reaching values around $10^{-6}$ and $10^{-4}$ for electrons and holes, respectively, when the SO coupling dominates the relaxation.

The interplay between the two channels with opposite magnetic-field dependence of the relative spin-flip rates leads to a non-monotonic dependence of the total relative spin-flip rate. The resulting minimum is particularly pronounced for holes. This prediction relies on the assumed strong transverse hf coupling for holes, in line with some of the recent experiments and with theoretical predictions based on the $d$-shell admixture to the valence band states. As the spin-flip efficiency is in principle measurable in optical experiments, this prediction might be used for testing the presence of the transverse hf couplings.

Acknowledgements.

The authors acknowledge support by the Polish National Science Centre under Grant No. 2014/13/B/ST3/04603 (PM, KG) and Grant No. 2016/23/G/ST3/04324 (PK). Calculations have been carried out using resources provided by Wroclaw Centre for Networking and Supercomputing (http://wcss.pl), Grant No. 203.