Spin-orbit-induced hole spin relaxation in a quantum dot molecule: the effect of $s$-$p$ coupling

TL;DR

This paper investigates how spin-orbit coupling and $s$-$p$ shell interactions in quantum dot molecules influence hole spin relaxation, revealing enhanced relaxation near resonance conditions due to strong spin mixing transfer.

Contribution

It introduces an effective model capturing spin-orbit effects in $p$ shells and demonstrates how $s$-$p$ coupling affects hole spin relaxation in quantum dot molecules.

Findings

Strong spin mixing in $p$ shell transfers to $s$ shell, increasing relaxation.

Enhanced spin relaxation occurs near $s$-$p$ resonance when dots are misaligned.

Magnetic field tilt influences the relaxation behavior.

Abstract

We study the effect of the coupling between the hole $s$ shell of one quantum dot and the $p$ shell in the other dot forming a quantum dot molecule on the spin relaxation between the sublevels of the hole $s$ state. Using an effective model that captures the spin-orbit effects in the $p$ shell irrespective of their origin, we show that the strong spin mixing in the $p$ shell can be transferred to the $s$ shell of the other dot, leading to enhanced spin relaxation in a certain energy range around the $s$-$p$ resonance if the dots are misaligned and the magnetic field is tilted from the sample plane.

Figures

Tables

Table 1. Table 1: Parameters

Electric permittivity	${\epsilon }_s$	$13.2$
Piezoelectric constant	$d$	$0.16$ C/m2
Longitudinal sound speed	$c_l$	$5600$ m/s
Transverse sound speed	$c_t$	$2800$ m/s
Density	${\rho }_c$	$5360$ kg/m3
Hole wave function widths
in-plane	$l_{\perp }$	$5.0$ nm
z direction	$l_z$	$1.5$ nm
Dots stacking
vertical	$D_z$	$7$ nm
horizontal	$D_x$	$1$ nm
Lande g-factor
For s-shell
in-plane	$g_s^{\left(x\right)},g_s^{\left(y\right)}$	$-0.1$
z direction	$g_s^{\left(z\right)}$	$-5.51$
For p-shell
in-plane	$g_p^{\left(x\right)},g_p^{\left(y\right)}$	$0.05$
z direction	$g_p^{\left(z\right)}$	$2.62$
SO coupling	$V_{SO}$	$-9.79$ meV
Envelope angular
momentum coefficient	$W$	$-0.532$ meV/T
Inter-dot s-p coupling	$t_p$	$0.1$ meV
Intra-dot s-p energy separation	$\Delta E_2$	$-40$ meV
Inter-dot s-p energy separation	$\Delta E_1$	$-7$ meV

Equations

$$H=H_0+H_{\mathrm{int}},$$

$$H_0=\sum _{\alpha =1}^7{H}_0^{(\alpha )}.$$

$$H_0^{(1)}=\mathrm{diag}\left(\Delta E_1+e{F}_z{D}_z,\Delta E_2,0,0\right)\otimes {\mathbb{I}}_2$$

$$H_0^{(2)}$$

$$\times \left(\text{outerproduct}1,01,0+\text{outerproduct}2,02,0\right)\otimes {\sigma }_i$$

$$H_0^{(3)}$$

$$\times \left(\text{outerproduct}2,+12,+1+\text{outerproduct}2,-12,-1\right)\otimes {\sigma }_i$$

$$H_0^{(4)}=V_a\left(\text{outerproduct}2,+12,-1+\text{outerproduct}2,-12,+1\right)\otimes {\mathbb{I}}_2,$$

$$H_0^{(5)}=t_p\left(\text{outerproduct}1,02,+1+\text{outerproduct}1,02,-1+h.c.\right)\otimes {\mathbb{I}}_2$$

$$H_0^{(6)}=V_{SO}\left(\text{outerproduct}2,+12,+1-\text{outerproduct}2,-12,-1\right)\otimes {\sigma }_z$$

$$H_0^{(7)}=W{B}_z\left(\text{outerproduct}2,+12,+1-\text{outerproduct}2,-12,-1\right)\otimes {\mathbb{I}}_2$$

$$H_{\mathrm{int}}=\sum _{j,j^{\prime }}\text{outerproduct}j{j}^{\prime }\sum _{\mathbf{k},\lambda }{F}_{j{j}^{\prime }}^{(\lambda )}\left(\mathbf{k}\right)\left(b_{\mathbf{k}}+b_{-\mathbf{k}}^{†}\right)e^{i\mathbf{kr}},$$

$$F_{j{j}^{\prime }}^{(\lambda )}(\mathbf{k})=-i\sqrt{\frac{\hslash }{2\rho V{c}_{\lambda }k}}\frac{d_{p}e}{{\epsilon }_0{\epsilon }_r}{M}_{\lambda }(\hat{k}){\mathcal{F}}_{j{j}^{\prime }}(k,\hat{k}),$$

$${\mathcal{F}}_{j{j}^{\prime }}\left(\mathbf{k}\right)=\int {\mathrm{d}}^{3}r{\Psi }_j^{\ast }(\mathbf{r})e^{i\mathbf{kr}}{\Psi }_{j^{\prime }}(\mathbf{r}),$$

$${\mathcal{F}}_{j{j}^{\prime }}(k,\hat{k})=f_{j{j}^{\prime }}\left(\frac{1}{2}k\sin \theta \right)$$

$$\times \exp [-\frac{1}{2}{k}^2(l_{\perp }^2{\sin }^2\theta +l_z^2{\cos }^2\theta )]\exp \left[i(m_j-m_{j^{\prime }})\phi \right]$$

$$\displaystyle\times\left\{\begin{array}[]{l l}\exp\left[ikD_{x}\sin\theta\cos\varphi+ikD_{z}\cos\theta\right]&\textrm{for QD1}\\ 1&\textrm{for QD2},\\ \end{array}\right.$$

$$\left\{f_{jj^{\prime}}\right\}(q)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&-iq&-iq\\ 0&-iq&1-q^{2}&-q^{2}\\ 0&-iq&-q^{2}&1-q^{2}\\ \end{array}\right)\otimes\mathbb{I}_{2}$$

$$M_{\lambda }(\hat{k})=2\left[k_x{k}_y{\left({\hat{e}}_{\lambda ,\mathbf{k}}\right)}_z+k_y{k}_z{\left({\hat{e}}_{\lambda ,\mathbf{k}}\right)}_x+k_z{k}_x{\left({\hat{e}}_{\lambda ,\mathbf{k}}\right)}_y\right],$$

$$|\alpha ⟩=\sum _j{c}_{\alpha j}|j⟩$$

$$R_{j{j}^{\prime }n{n}^{\prime }}(\omega )=$$

$$\sum _{\lambda =l,t_1,t_2}\int d{\Omega }_{\hat{k}}\frac{1}{c_{\lambda }^3}{M}_{\lambda }^2(\hat{k})F_{j{j}^{\prime }}\left(\frac{\omega }{c_{\lambda }},\hat{k}\right)F_{n{n}^{\prime }}^{\ast }\left(\frac{\omega }{c_{\lambda }},\hat{k}\right),$$

$${\gamma }_{\alpha \to {\alpha }^{\prime }}=\sum _{j,j^{\prime },n{n}^{\prime }}{c}_{\alpha j^{\prime }}^{\ast }{c}_{{\alpha }^{\prime }j}{c}_{\alpha n^{\prime }}{c}_{{\alpha }^{\prime }n}^{\ast }{R}_{j{j}^{\prime }n{n}^{\prime }}({\omega }_{if}).$$

Full text

double quantum dot, spin relaxation, piezoelectric field, phonon bath

I Introduction

The ability to confine a single carrier in a semiconductor quantum dot (QD) not only opened an exciting field of studies but also paved the way to many possible applications. The idea of implementing quantum computing schemes on such a paradigmatic physical qubit Loss and DiVincenzo (1998) has driven physical interest for more than 20 years, leading to the development of sophisticated schemes for spin initialization Brash et al. (2015); Ardelt et al. (2015); Mar et al. (2016); Atatüre et al. (2006); Mar et al. (2014), control Press et al. (2008, 2010); De Greve et al. (2011); Godden et al. (2012a), and readout Vamivakas et al. (2010); Delteil et al. (2014). Coupling and entangling spins to phonons De Greve et al. (2012); Gao et al. (2012); Schaibley et al. (2013); Vora et al. (2015) makes it possible to transfer quantum states from spins to photons He et al. (2017), entangle distant spins Delteil et al. (2014), and build quantum repeaters McMahon and De Greve (2015), which provides building blocks for quantum networks.

The possible applications of confined spins motivate studies of spin dynamic and, in particular, spin relaxation in QDs. While phase coherence of the spin may be limited by charge noise Godden et al. (2012b); Jovanov et al. (2011); Prechtel et al. (2015) and fluctuations of the effective magnetic field due to nuclear spins Merkulov et al. (2002), spin relaxation at moderate magnetic fields is mostly due to spi-orbit (SO) coupling that enables phonon-mediated spin-flip transitions Heiss et al. (2007); Bulaev and Loss (2005). In contrast to unstrained or uniformly strained systems, where the SO effects are characterized by usual Rashba Bychkov and Rashba (1984) or Dresselhaus Dresselhaus (1955) couplings, in strained self-assembled QDs the situation is more complex and the SO effect that predominantly affects spin relaxation is induced by shear strain Mielnik-Pyszczorski et al. (2018); Gawarecki and Krzykowski (2018). While the measured hole spin relaxation times vary from experiment to experiment, the observed upper limit seems to reach hundreds of microseconds Dahbashi et al. (2012); Heiss et al. (2007); Climente et al. (2013); Segarra et al. (2015a); Fras et al. (2012). These hole spin life times are shorter than those for electrons because of the much stronger SO coupling in the valence band.

It has been shown that systems build of coupled QDs, referred to as quantum dot molecules (QDMs), offer extended feasibility of spin initialization, manipulation, and readout, due to the sensitivity of confined states to external electric field and the possibility of exploiting tunneling processes Vamivakas et al. (2010); Müller et al. (2012); Economou et al. (2012). In a QDM, spin-orbit effects are enhanced by symmetry breaking due to QD off-axial misalignment, which typically appears in these structures Doty et al. (2010). Hole spin flip processes in QDMs were studied within the $\bm{k}\!\cdot\!\bm{p}$ method for a simplified QD geometry without strain Segarra et al. (2015b) near the resonance between the $s$ shells in the two QDs. It was shown that the spin relaxation rate is considerably enhanced by QD misalignment and by the presence of Dresselhaus SO interaction, with the spin life times around a $\mu$s at the magnetic field of $2$ T, roughly constant across the resonance. In a subsequent study Stavrou (2018) the spin-flip rate was also studied for an electron, using a more realistic model including strain in a QDM built of two nominally identical QDs.

The SO coupling that underlies the hole spin relaxation manifests itself in the band structure of a semiconductor in a rather complicated way that is captured e.g. by various inter-band terms of the $\bm{k}\!\cdot\!\bm{p}$ model Winkler (2003); Planelles and Climente (2013). For a carrier confined in a QD the SO coupling is particularly pronounced in the hole $p$ shell, where it dominates over the Zeeman terms, leading to reordering of the $p$-shell hole states Gawarecki (2018). It turns out that the results of exact $\bm{k}\!\cdot\!\bm{p}$ calculations are reproduced by an effective model with a few fitting parameters determined by comparison to $\bm{k}\!\cdot\!\bm{p}$ results Doty et al. (2010); Gawarecki (2018). When supplemented with a simple model of wave functions and a standard description of carrier-phonon couplings, such a model yields a simple, yet reasonably accurate, description of charge and spin relaxation in a QD, accounting for all the SO effects, including those induced by strain Gawarecki and Krzykowski (2018). Similar effective models were used in the description of tunneling between QDs in a QDM Stavrou and Hu (2005).

In this paper we study hole-spin relaxation between Zeeman sublevels of the ground ($s$-shell) state in one of the QDs forming a QDM, resulting from tunnel coupling to the $p$-shell states of the other QD. We use the effective model for SO couplings in the valence band extended by tunneling terms along with the standard model of carrier-phonon coupling. We show that the strong SO coupling in the hole $p$ shell can be transferred via tunnel coupling to the $s$-shell of the other QD in an axially misaligned QDM leading to considerably enhanced spin relaxation in a tilted magnetic field. The hole-spin flip rate becomes particularly large in the vicinity of the inter-dot $s$-$p$ resonance but the effect persists also away from the resonance, where the states are nearly completely localized in one of the QDs and nearly ideally spin-polarized.

The organization of this paper is as follows. In Sec. II we describe the system and introduce the effective Hamiltonian. In Sec. III we present the results for the relaxation rates as a function of electric and magnetic field. Sec. IV concludes the paper.

II Model

II.1 Description of the system

The structure consists of two self-assembled lens-shaped QDs grown in the [001] direction. The dots are stacked vertically with the inter-dot distance $D_{z}$. The cylindrical symmetry of the system is broken by a small relative in-plane displacement $D_{x}$ (see Fig. 1).

The wave function of the hole confined in a QD is the solution of a 2-dimensional quantum oscillator i.e. Fock-Darwin state Fock (1928); Darwin (1931), with the principal quantum number $n$ and the angular quantum number $m$ representing the envelope angular momentum. The wave functions are localized in the plane on a distance $l_{\bot}$ and in the growth direction on a distance $l_{z}$. We take into account the $s$-shell states ($n=0$, $m=0$) in the top QD, whereas in the bottom QD both the $s$-shell and the $p$-shell states ($n=0$, $m=0$ and $n=1$, $m=\pm 1$) are included. We take into account the two values of the band angular momentum (“spin”) of a heavy hole, represented by $s=\uparrow,\downarrow$. The basis states are then $\{\ket{l,m,s}\}=$ $\{\ket{1,0\uparrow}$, $\ket{1,0\downarrow}$, $\ket{2,0\uparrow}$, $\ket{2,0\downarrow}$, $\ket{2,+1\uparrow}$, $\ket{2,+1\downarrow}$, $\ket{2,-1\uparrow}$, $\ket{2,-1\downarrow}\}$, where the first number $l=1,2$ refers to QD1 and QD2, respectively. The off-axis misalignment leads to inter-dot $s$-$p$ coupling because of the symmetry breaking Gawarecki and Machnikowski (2012a). The system is placed in an arbitrarily oriented magnetic field and interacts with phonons.

The system is described by an effective Hamiltonian similar to that introduced in Ref. Gawarecki (2018) but generalized to the QDM structure,

[TABLE]

where $H_{0}$ is the effective Hamiltonian of the holes and $H_{\mathrm{int}}$ stands for hole interaction with the phonon bath.

The hole Hamiltonian $H_{0}$ contains various components accounting for the QD energy levels, external fields, tunnelling and SO coupling Gawarecki (2018),

[TABLE]

These components of $H_{0}$ can be written in the form of a tensor product corresponding to the decomposition $\ket{l,m}\otimes\ket{s}$, where the first component refers to the spatial degrees of freedom and the second one to the spin. The first part,

[TABLE]

describes the bare energies $\Delta E_{1}$ and $\Delta E_{2}$ of the $s$-shell states in the top and bottom QD, respectively, as well as the effect of the electric field $F_{z}$ applied along the stacking direction. The energies are defined with respect to the $p$-shell energy in the QD2. Next,

[TABLE]

and

[TABLE]

account for Zeeman splittings in the $s$- and $p$-shells, respectively. Anisotropic Lánde factors for $s$- and $p$-shell states are denoted by $g_{s}$ and $g_{p}$, $\sigma_{i}$ stand for Pauli matrices, $B_{i}$ are the components of the magnetic field and $\mu_{B}$ is the Bohr magneton. The next term,

[TABLE]

describes the influence of an anisotropic QD elongation. Next,

[TABLE]

is the inter-dot $s$-$p$ coupling of strength $t_{p}$.The part

[TABLE]

describes the SO coupling in the $p$ shell. Finally,

[TABLE]

accounts for the magnetic effect of the envelope angular momentum.

The hole-phonon interaction Hamiltonian has the form

[TABLE]

where $j$, $j^{\prime}$ run through all the states in the basis. Here the phonon annihilation and creation operators of wave vector $\bm{k}$ and polarization $\lambda$ are denoted by $b_{\bm{k}}$ and $b_{-\bm{k}}^{\dagger}$, and $F_{jj^{\prime}}^{(\lambda)}\left(\bm{k}\right)$ are hole-phonon coupling constants,

[TABLE]

In the above expression, the wave vector $\bm{k}$ is represented by its length $k$ and its direction represented in the spherical coordinates as $\hat{k}=\left(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta\right)$, $\rho$ is the crystal density, $c_{\lambda}$ is the sound velocity, $d_{p}$ stands for piezoelectric constant, $\varepsilon_{r}$ denotes relative electric permittivity, and $\mathcal{F}_{jj^{\prime}}$ is the form-factor

[TABLE]

where $\Psi_{j}(\bm{r})$ is the envelope wave function of the hole. In the QDM system the form-factors take the form

[TABLE]

in which $f_{jj^{\prime}}$ is the matrix

[TABLE]

and $m_{j}$ is the orbital angular momentum of $j$-th state. The geometric factor $M_{\lambda}$ in Eq. (3) is defined as

[TABLE]

where $\hat{e}_{\lambda,k}$ is the unit polarization vector for the mode ($\lambda,\bm{k}$)

Note that our model does not include the usual SO terms responsible for spin mixing and spin relaxation within the $s$ shell, so that it allows us to sngle out the effect of $s$-$p$ coupling and $p$-shell SO structure.

The values of the parameters can be found in Tab. 1. The value of the hole $s$-$p$ tunnel coupling, which is a crucial parameter in the presented calculations, is assumed to be one order of magnitude lower than the analogous value calculated for the electron Gawarecki et al. (2014). This seems to be a safe estimate in view of the $\bm{k}\!\cdot\!\bm{p}$ results that yield similar orders of magnitude for the electron Gawarecki et al. (2010) and hole Gawarecki and Machnikowski (2012b) $s$-shell tunnel couplings.

II.2 Spin relaxation rate

We find the spin-flip rate between the $s$-shell Zeeman sublevels of the ground hole states in the QD1 using the Fermi golden rule. We diagonalize effective Hamiltonian to find the eigenstates

[TABLE]

and the transition frequencies $\omega_{if}=\left(E_{i}-E_{f}\right)/\hbar$, where initial and final states are the nominally spin-up and spin-down $s$-shell states in the top QD. We define the spectral densities (in terms of the basis states)

[TABLE]

where $l,t_{1},t_{2}$ denote the longitudinal and transverse phonon branches. Using the Fermi golden rule and writing the system eigenstates as a superposition of the basis states according to Eq. (10), we obtain a formula for the spin relaxation rate, which is a combination of the spectral densities taken at the transition frequency

[TABLE]

III Results

The rate of phonon-induced spin relaxation in the QD1 caused by the coupling to the $p$-shell of the QD2 was obtained by numerical implementation of the model discussed in Section II. Fig. 2(a) shows the diagram of energy levels as a function of electric field for the range of field magnitudes around the resonances between the $s$ shell of QD1 and the $p$ shell of QD2. The results were obtained for the magnetic field of $5$ T, tilted by $15^{\circ}$ off the plane. The states localized in QD1, distinguishable by their evolution in the electric field, are marked with colors. The two branches of the $s$ states are the Zeeman sublevels with respect to the spin quantization axis determined by the field orientation. The two $p$-shell states are the lower half of the four $p$-shell states split by the SO interaction Gawarecki (2018), corresponding to anti-parallel orientation of orbital and band angular momenta (the upper two states lie above the axis range).

Fig. 2(b) shows the transition rate between the $s$-shell Zeeman sublevels of QD1 (orange to green in Fig. 2(a)) calculated from Eq. (11). The rate peaks at the resonances with the $p$-shell states, where the life time of the upper state is reduced to about 100 ps. In order to interpret this, we present the spin polarization of the two states in Fig. 2(c) (with color coding referring to Fig. 2(a)) and their localization in Fig. 2(d). At the resonance, one of the Zeeman branches in QD1 mixes with a state with the opposite spin from the $p$ shell of QD2. This results in a reduction of the spin polarization of this state from $\pm 1$ to 0, as can be seen in Fig. 2(c) and delocalization of the state between the QDs, visible in Fig. 2(d). As a result, the spin-flip time becomes comparable to the charge relaxation time (tens of picoseconds).

This resonant enhancement is, however, constrained to a very narrow parameter range, since the resonance between states with opposite spins is very narrow (the corresponding anticrossing is not even visible in Fig. 2(a)). Away from the resonance the rate decreases, as expected for an effect that relies on the admixture of $p$ states. Still, in a range of a few meV from the resonance, the spin-flip rates remain relatively high and correspond to spin life times in the $\mu$s range, which are shorter than those observed in some experiments. This shows that in a certain range of energies around the resonance with the $p$ shell of the QD2 the hole spin flip in the ground state of the QD1 is dominated by the inter-QD $s$-$p$ coupling and $p$-shell SO effects.

In Fig 3 we show the system spectrum and relaxation rates as a function of magnetic field for a fixed electric field of $0.12$ mV/nm (marked by a vertical line in Fig. 2.). The rates in Fig. 3(b) are plotted for three different tilt angles (marked by vertical lines in Fig. 4) and the spectrum in Fig. 3(a) is shown for one angle $\pi/12$, corresponding to the green line in Fig 3(b). The rates show a peak at the resonance, the origin of which is the same as in the previous discussion. Away from the resonances, the rate grows as $B^{5}$.

In Fig. 4 the eigenstate diagram and the spin relaxation rate are presented as a function of the magnetic field orientation for different magnetic field magnitudes. The rates in Fig 4(b) are plotted for various magnitudes of the magnetic field and the spectrum in Fig 4(a) corresponds to $B=5$ T (yellow line in Fig 4(b)). Apart from the resonant enhancements, one can see considerable reduction of the rate at angles 0 and $\pi/2$ (exact Voigt and Faraday geometry, respectively). The former is due to the strong anisotropy of the hole g-factor and nearly vanishing $s$-shell Zeeman splitting in the Voigt geometry, which suppresses the phonon-assisted relaxation due to low phonon spectral density at low frequencies. In the Faraday geometry the $z$ projection of the spin is strictly conserved in our model, as follows directly from the form of the Hamiltonian.

IV Conclusions

We have studied phonon-assisted hole-spin relaxation rate between Zeeman sublevels in a QD forming part of a QDM. We have calculated the energy states using an effective Hamiltonian that accounts for the spin-orbit coupling of the other QD in the QDM and tunnel coupling between the $s$ and $p$ shells of the two dots. We have shown that the inter-QD $s$-$p$ coupling induced by QD misalignment combined with strong SO effects in the $p$ shell leads to enhanced spin relaxation in the vicinity of the $s$-$p$ resonance, with the exception of exact Voigt and Faraday geometries, when the rates becomes small or vanish, respectively. The typical spin relaxation times resulting from the $s$-$p$ coupling mechanism are on the order of $\mu$s even away from the resonance, when the states are nearly completely localized and spin-polarized. The mechanism studied here can therefore dominate over other known spin-flip processes in a certain energy interval around the $s$-$p$ resonance.

The authors acknowledge funding from the Polish National Science Centre (NCN) under Grant No. 2016/23/G/ST3/04324 (KK) and Grant No. 2014/13/B/ST3/04603 (PM).