Abstract
We study the electricdipole transitions for a single electron in a double quantum dot located in a semiconductor nanowire. Enabled by spinorbit coupling (SOC), electricdipole spin resonance (EDSR) for such an electron can be generated via two mechanisms: the SOCinduced intradot pseudospin states mixing and the interdot spinflipped tunneling. The EDSR frequency and strength are determined by these mechanisms together. For both mechanisms the electricdipole transition rates are strongly dependent on the external magnetic field. Their competition can be revealed by increasing the magnetic field and/or the interdot distance for the double dot. To clarify whether the strong SOC significantly impact the electron state coherence, we also calculate relaxations from excited levels via phonon emission. We show that spinflip relaxations can be effectively suppressed by the phonon bottleneck effect even at relatively low magnetic fields because of the very large gfactor of strong SOC materials such as InSb.
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Introduction
Confined electron spins in semiconductor nanostructures are a viable option for implementing quantum computing and quantum information processing because of their long decoherence times^{1,2,3,4,5,6,7}, and quantum coherent manipulation of a single electron spin is an essential ingredient for such applications. Conventional approach for manipulating an electron spin uses magnetic dipole interaction to achieve electron spin resonance (ESR)^{8}. However, the very small electron spin magnetic moment dictates that a strong alternatingcurrent (AC) magnetic field is required to reach reasonable rate of spin rotation^{9,10}. In semiconductors, interestingly, spinorbit coupling (SOC) offers a viable alternative. Through SOC an AC electric field can also rotate an electron spin, leading to the socalled electricdipole spin resonance (EDSR)^{11,12,13,14,15,16,17,18}. Indeed, EDSR has proven to be an effective method for electron spin control in quantum dots^{19,20,21}.
Over the past decade semiconductor nanowire devices have attracted wide attention because of their onedimensionality, convenience of growth, and a variety of interesting physical properties^{19,20,21,22,23,24,25,26,27,28,29}. Experimentally, electron occupancy of quantum dots in a nanowire can be effectively controlled by regulating the local gate electrodes^{30,31,32,33}. Recently, nanowires with narrow bandgap, large SOC, and large gfactor have been of particular interest because they present intriguing opportunities for studying fast electrical control of spins^{19,20,21,25,26,27}, possible manipulation of entangled spins^{34,35}, and hybrid structures made of a superconductor and a largeSOC nanowire are a promising system to search for Majorana fermions^{28,29}.
A double quantum dot (DQD) is an interesting physical system that has attracted considerable attention over the past two decades^{2}. The tunnel coupling between two dots significantly alters the energy spectrum of the system as compared to a single dot, which allows fundamentally and technologically important phenomena such as Pauli spin blockade^{2,36}. Another example is the recent demonstration of strong spinphoton coupling in a double dot, where the DQD energy spectrum plays a crucial role in enhancing the spinphoton coupling strength^{14,37}.
In this paper, we investigate the electronic properties of a nanowire double quantum dot, with a particular focus on the interplay between SOC and the DQD potential on the electricdipole transitions of a single confined electron. We obtain the lowenergy spectrum of a single electron in the DQD using the linear combination of atomic orbital (LCAO) method^{38,39,40}. In our calculation the singledot singleelectron orbitals are obtained by accounting for the spinorbit coupling exactly while treating the external magnetic field as a perturbation^{41}. In the presence of an alternating electric field applied along the wire axis, EDSR can be generated by spin state hybridization from SOC. In a single or isolated QD, the state hybridization originates from the SOCinduced intradot orbital states mixing. In a DQD, on the other hand, interdot tunneling can also contribute to orbital mixings. Thus, in a DQD there are two mechanisms leading to the EDSR, and the dominant mechanism can be altered by changing system parameters. When orbital mixing is dominated by the interdot tunneling, we examine how the electricdipole transition rates depend on the magnitude and orientation of the applied magnetic field. The competition between contributions from the intradot and interdot orbital mixings can be revealed in the variations of the EDSR frequency with the magnetic field strength, at a large interdot distance. More specifically, we show that at lower applied magnetic field, spin flip assisted by interdot tunneling makes the dominant contribution to EDSR. With increasing the interdot distance and the associated suppression of tunneling, the main mechanism of EDSR in a DQD changes from the interdot spinflipped tunneling to the intradot orbital states mixing. Finally, we calculate the rates of phononassisted spin relaxation and show that the enhancement in relaxation would not significantly impact the quantum coherence quality factor of the electron spin. This study provides useful input for experimental studies of quantum coherent manipulations in a nanowire DQD.
Results
The model Hamiltonian
We consider a quasionedimensional double quantum dot with one confined electron, as shown in Fig. 1(a). The semiconductor materials for the nanowires we consider are those with large SOC, such as InAs and InSb^{42,43}, though our approach is sufficiently general so that our results should be applicable to material systems with weaker SOC as well. To better model a realistic nanowire DQD, we consider an asymmetric nanowire DQD, with system parameters taken from the experimental data of NadjPerge et al. in ref.^{2}.
As illustrated in Fig. 1(a), the nanowire axis is along the xdirection. Along the transverse directions, we have a strong harmonic potential along the ydirection and an asymmetric gradient potential along the zdirection (used to enhance the Rashba SOC^{43}). With DQD confinement potential much weaker than the y and z confinements, we treat our electron as quasionedimensional.
In the absence of an applied magnetic field, the Hamiltonian describing an electron in a quasionedimensional DQD along the xdirection is
where m_{ e } is the conductionband effective mass, and p_{ x } = −iħ∂/∂x. We choose to model the confinement potential along the x direction as an asymmetric doublewell potential \(V(x)=\frac{1}{2}{m}_{e}\,{\rm{\min }}\,\{{\omega }_{l}^{2}{(x+d)}^{2},{\omega }_{r}^{2}{(xd)}^{2}\}\), with 2d being the interdot distance and \({x}_{l/r}=\sqrt{\hslash /({m}_{e}{\omega }_{l/r})}\) being a characteristic length in the left/right dot [see Fig. 1(b)]. \({H}_{{\rm{so}}}^{x}\) corresponds to the effective SOC Hamiltonian along the axis direction of the nanowire DQD.
There are two kinds of spinorbit interactions in A_{III}B_{V} heterostructures^{44}. One is the Dresselhaus SOC due to bulk inversion asymmetry^{45}. The other is the Rashba SOC generated by structure inversion asymmetry^{46}. In general, the SOC strengths depend on system parameters and spatial distributions of the electron wave function. By averaging over the transverse directions y and z, we obtain an effective linear SOC Hamiltonian \({H}_{{\rm{so}}}^{x}\) along the x direction (see Methods)
with the effective SOC strength \(\alpha =\sqrt{{\alpha }_{{\rm{D}}}^{2}+{\alpha }_{{\rm{R}}}^{2}}\). Here the spin quantization axis is defined by the SOC to be along \(\hat{{\bf{a}}}=(\cos \,\varphi ,\,\sin \,\varphi ,\mathrm{0)}\), with \(\varphi =\arctan ({\alpha }_{R}/{\alpha }_{D})\), so that \({\sigma }^{a}=\hat{{\bf{a}}}\cdot \sigma \), where \(\sigma =({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})\) are the Pauli matrices. α_{R} and α_{D} denote the effective strengths of the Rashba and Dresselhaus SOCs, respectively.
When an external magnetic field is applied in the direction \(\hat{{\bf{n}}}=(\cos \,\theta ,\,0,\,\sin \,\theta )\) with strength B, the singleelectron Hamiltonian becomes
with the vector potential A = (A_{ x }, A_{ y }, 0), where A_{ x } = −By sin θ, A_{ y } = −Bz cos θ, g_{ e } is the Landé factor, μ_{ B } is the Bohr magneton, and \({\sigma }^{n}=\hat{{\bf{n}}}\cdot \sigma \). With our assumption of an asymmetric double dot, it follows naturally that the specific value of the Landé factor g_{ e } in the left dot is different from that in the right dot, g_{ el } ≠ g_{ er }^{20}. Due to the strong confinements along the transverse directions, 〈y〉 ~ 0, the effects of the magnetic vector potential on the electron orbital dynamics is negligible (detailed calculations are given in Methods), so that the Hamiltonian for the DQD can be simplified as
with the Zeeman splitting Δ_{ Z } = −g_{ e }μ_{ B }B.
Traditionally SOC is treated as a perturbation in theoretical calculations for semiconductors. However, such a perturbative approach becomes problematic when SOC is strong, in materials such as InSb^{41}. For a comprehensive study of the effect of a strong SOC on the electricdipole transition in a nanowire DQD, in the following calculations we take the SOC term into consideration precisely while treating the Zeeman term perturbatively.
Energy spectrum of the DQD
The energy spectrum of the DQD is calculated by adopting the linear combination of atomic orbitals (LCAO) method. The localized electron wavefunctions are derived by solving the eigenstates of the individual quantum dots. The orthonormal bases used to project the DQD Hamiltonian are obtained by the Schmidt orthogonalization of the local wavefunctions.
Near each of the minima of the DQD potential well along the nanowire axis, V(x) can be approximated as parabolic, \({V}_{l/r}(x)=\frac{1}{2}{m}_{e}{\omega }_{l/r}^{2}{(x\pm d)}^{2}\). Including the SOC effect, the local Hamiltonian for each single quantum dot can be written as
which is isomorphic to the single dot Hamiltonian H_{0} in refs^{41} and^{47}.
The eigenstates of \({H^{\prime} }_{l/r}\) can be solved analytically. Let Φ_{ κnσ }〉 denote the eigenstates of \({H^{\prime} }_{\kappa }\), with orbital quantum number n = 0, 1, 2, 3, …, κ = l, r corresponding to the different quantum dots, and σ = ↑, ↓ denoting the electron spin states. Explicitly, Φ_{l/rn↑}〉 and Φ_{l/rn↑}〉 take the form
where ψ_{ κn }(x) represents an eigenstate of a harmonic oscillator with eigenvalue (n + 1/2)ħω_{ κ }, x_{so} is the effective SOC length x_{so} = ħ/(m_{ e }α), and ↑_{ a }〉 and ↓_{ a }〉 denote the eigenstates of σ^{a}: σ^{a}↑_{ a }〉 = ↑_{ a }〉 and σ^{a}↓_{ a }〉 = −↓_{ a }〉. Φ_{κn↑}〉 and Φ_{κn↓}〉 are degenerate (Kramers degeneracy), with the eigenvalue given by ε_{ κn } = (n + 1/2)ħω_{ κ } − (1/2)m_{ e }α^{2}. The energy levels of \({H^{\prime} }_{\kappa }\) are thus evenly spaced, with an energy splitting Δ_{ κS } = ħω_{ κ }.
In the presence of an applied magnetic field, the singledot Hamiltonian becomes
where Δ_{ κZ } corresponds to the Zeeman splitting in κ dot. The Zeeman term can be regarded as a perturbation if the ratio
i.e. the Zeeman splitting Δ_{ κZ } is much smaller than the orbital splitting Δ_{ κS }, dictating a relatively small magnetic field (see the estimate in ref.^{41}). Within firstorder perturbation theory, the two lowestenergy eigenstates of H_{ κ } are
where
Here \(\phi =\arccos \langle \hat{{\bf{a}}}\cdot \hat{{\bf{n}}}\rangle \) is the angle between unit vectors \(\hat{a}\) and \(\hat{n}\) (i.e., the angle between the effective field from SOC and the applied magnetic field), and \({\eta }_{\kappa }=\sqrt{{m}_{e}/(\hslash {\omega }_{\kappa })}\alpha \). It is a ratio between the effective dot size x_{ κ } and SOC length x_{so}, therefore is a measure of the SOC strength relative to the confinement energy. For a nanowire quantum dot, η_{ κ } is generally small, \({\eta }_{\kappa }\equiv {x}_{\kappa }/{x}_{{\rm{so}}}\ll 1\), even for materials with strong SOC. According to Eq. (9), an applied magnetic field generally leads to hybridization of different spinorbit states in \({{\rm{\Psi }}}_{\kappa }^{\pm }\rangle \), with the degree of orbital mixing proportional to \({\xi }_{\kappa }{\eta }_{\kappa }^{n}{e}^{{\eta }_{\kappa }^{2}}\).
The orbital states localized in different quantum dots are not orthogonal in general. Nevertheless, from the four lowestenergy localized states \({{\rm{\Psi }}}_{\kappa }^{\pm }\rangle \) (κ = l, r) and using Schmidt orthogonalization method, we can construct local orthonormal basis states \({{\rm{\Psi }}}_{l\Uparrow }\rangle \), \({{\rm{\Psi }}}_{l\Downarrow }\rangle \), \({{\rm{\Psi }}}_{r\Uparrow }\rangle \), and \({{\rm{\Psi }}}_{r\Downarrow }\rangle \). Here \(\Uparrow \) and \(\Downarrow \) refer to the two pseudospin states, whose compositions have been modified by the applied magnetic field as compared to the zerofield Kramers degenerate pair. The analytical expressions for the bases are given in Methods.
Projecting the Hamiltonian H_{DQD} onto this orthonormal basis, the lowenergy part of the Hamiltonian H_{DQD} can be written as
with
Here t_{ σσ } (\(\sigma =\Uparrow ,\Downarrow \)) is spinconserved tunnel coupling, \({t}_{\sigma \bar{\sigma }}\) is spinflipped tunnel coupling and ε_{ κσ } is the corresponding singledot energy. These matrix elements can be obtained by dividing the original Hamiltonian H_{DQD} in Eq. (4) into two parts H_{DQD} = H_{ κ } + Δ_{ κ }V(x), with H_{ κ } either one of the singledot Hamiltonian, and Δ_{ κ }V(x) = V(x) − V_{ κ }(x) the double dot correction on H_{ κ }. Due to the orthogonality of Ψ_{ κσ }〉, the tunnelings can be calculated as t_{σσ′} = 〈Ψ_{ lσ }Δ_{ l }V(x)Ψ_{rσ′}〉, with its magnitude proportional to the interdot wave function overlap, \({t}_{\sigma \sigma ^{\prime} }\propto \exp ({d}^{2}/{\bar{x}}^{2})\) where \({\bar{x}}^{2}=({x}_{l}^{2}+{x}_{r}^{2}\mathrm{)/2}\).
The eigenstates of the nanowire DQD can be obtained numerically by the direct diagonalization of the Hamiltonian H_{DQD} in Eq. (11). We denote these states Φ_{ i }〉 (i = 1 − 4), with eigenvalues E_{1} ≤ E_{2} ≤ E_{3} ≤ E_{4}. In Fig. 2 we give an example energy spectrum of an InSb nanowire DQD, with the corresponding system parameters taken from the experimental data in ref.^{20}: ħω_{ l } = 5.0 meV, ħω_{ r } = 7.5 meV, g_{ el } = −32.2, g_{ er } = −29.7, \({x}_{{\rm{so}}}\simeq 200\,{\rm{nm}}\), and d = 40 nm. The effects of the anisotropic gfactors are neglected for simplicity. Except for the interdot distance and the magnetic field strength and orientation, the parameters of the typical InSb nanowire are used in the following calculations for convenience and consistency.
Equation (9) indicates that when \({\xi }_{\kappa }{\eta }_{\kappa }{e}^{{\eta }_{\kappa }^{2}}\ll {e}^{{d}^{2}/{\bar{x}}^{2}}\ll 1\) (κ = l, r), which is satisfied with the parameters used in Fig. 2, the intradot orbital states hybridization is negligible compared with the interdot states mixing. For weaker SOC or strongly coupled DQD, \({x}_{{\rm{so}}}\gg 2d\), the interdot spinflipped tunneling \({t}_{\sigma \bar{\sigma }}\) is much smaller than the spinconserved tunneling t_{ σσ }. Nevertheless, spinflipped tunneling leads to a high degree of pseudospin hybridization in states Φ_{2}〉 and Φ_{3}〉 around the anticrossing point B_{0}, as shown in Fig. 2.
In calculating the energy spectrum of the DQD, it is necessary to establish the validity of the perturbation expansion in Eq. (9) and the approximation to neglect orbital effects of the magnetic vector potential in our calculations. The perturbation expansion in Eq. (9) can be justified by the specific values of ξ_{ κ } (κ = l, r) at the upper limit of the magnetic field range we consider. With our chosen parameters, when B = 2 T, ξ_{ r } < ξ_{ l } < 0.75, which still (barely) satisfy the perturbation condition in Eq. (8). As for the orbital effect of the vector potential, we compare the effective magnetic length \({l}_{{\rm{B}}}=\sqrt{\hslash /{m}_{e}{\omega }_{{\rm{B}}}}\), where ω_{B} = eB_{t}/m_{ e } is the electron Larmor frequency, with the characteristic lengths along the transverse directions, with the specific values of y_{0} and z_{0} given in Methods. At a magnetic field B = 2.0 T, \({l}_{{\rm{B}}}\simeq 17.78\) nm, which is still larger than the characteristic lengths y_{0} and z_{0}. This relationship thus holds true for all the other (lower) fields in our considered parameter regime. Therefore, the approximations we have adopted here are valid in our calculations.
Electricdipole transitions
In the absence of SOC, electricdipole (ed) interaction induced transitions obey a strict spin selection rule. In the presence of the SOC, on the other hand, an electricdipole transition can involve spin flip, leading to EDSR^{11,12,13,14,15,16,17,18}. In a DQD with strong SOC, the pseudospin composition of the eigenstates vary with magnetic field and interdot distance/tunneling. Moreover, under certain circumstances, intradot spin mixing in the DQD can also affect EDSR. In this Section we investigate how EDSR transition rates depend on different system parameters.
When an AC electric field is applied in the x direction, the Hamiltonian describing the single electron in the DQD reads
with E and υ representing the amplitude and frequency of the electric field, respectively. The electricdipole interaction can be treated as a perturbation if \(2eEd\ll {{\rm{\Delta }}}_{ij}\equiv {E}_{i}{E}_{j}\), and the resonant electricdipole transition rate can be calculated as
where h is the Plank constant. Due to the spinless ed interaction, the compositions of the pseudospin states Φ_{ i }〉 and Φ_{ j }〉 are a crucial factor in determining the magnitude of Ω_{i↔j}. With the transitions involving state Φ_{4}〉 symmetric with respect to those involving state Φ_{1}〉, for simplicity we only consider the electricdipole transitions involving Φ_{1}〉 in the following calculations.
Magnetic field dependence
In Sec. 2 we have shown that there are two mechanisms leading to different spin states hybridization in the eigenstates of DQD: the SOCinduced intradot states mixing and the interdot spinflipped tunneling. Because all the mechanisms show strong dependences on the external magnetic field, both the transition rates Ω_{2↔1} and Ω_{3↔1} will definitely change when varying the magnetic field. As is clearly illustrated in Fig. 3, the variations of Ω_{2↔1} and Ω_{3↔1} with the magnetic field strength B and orientation φ are shown.
For an InSb nanowire DQD with \({\xi }_{\kappa }{\eta }_{\kappa }{e}^{{\eta }_{\kappa }^{2}}\ll {e}^{{d}^{2}/{\bar{x}}^{2}}\ll 1\), the interdot tunneling dominates the orbital mixing in the eigenstates of DQD, and the effect of the intradot orbital states mixing can be negligible (the effect of the intradot orbital states mixing is investigated later in the next subsection). In a weak magnetic field, \(B\ll {B}_{0}\), the major pseudospin components of the state Φ_{1}〉 are the same as that of Φ_{3}〉 and different from that of Φ_{2}〉. It follows naturally that \({{\rm{\Omega }}}_{3\leftrightarrow 1}\gg {{\rm{\Omega }}}_{2\leftrightarrow 1}\). For a fixed φ (the angle between the applied magnetic field and the SOCinduced effective field), increasing the magnetic field strength enhances the degree of the interdot pseudospin hybridization in Φ_{2}〉 and Φ_{3}〉, which in turn leads to the rising (falling) of Ω_{2↔1} (Ω_{3↔1}), as shown in Fig. 3(c).
Φ_{1}〉 ↔ Φ_{2}〉 corresponds to the electricdipole spin transition for B < B_{0}, with Ω_{2↔1} representing the EDSR frequency when the AC electric field is on resonance with Δ_{12}. As demonstrated in refs^{16,17}, the magnitude of the EDSR frequency depends on the effective SOC strength, which can be controlled by changing the magnetic field direction. In Fig. 3(d), for a fixed magnetic field strength, the magnitude of the EDSR frequency Ω_{2↔1} as a function of the field orientation φ is shown. In particular, when the magnetic field is perpendicular to the SOC field direction, the effect of the SOCinduced mixing reaches its maximum, and the EDSR frequency reaches its peak value. Similarly, Ω_{3↔1} also has a strong φ dependence.
As B increases beyond B_{0}, the major pseudospin components of Φ_{2}〉 and Φ_{3}〉 are swapped. At this point, Φ_{1}〉 ↔ Φ_{3}〉 is the spinflip transition, with Ω_{3↔1} the corresponding EDSR frequency. When increasing magnetic field, the larger energy splitting between Φ_{2}〉 and Φ_{3}〉 weakens the interdot pseudospin hybridization in these levels. As a result, the EDSR frequency Ω_{3↔1} decreases, and the orbital transition rate Ω_{2↔1} saturates, as shown in Fig. 3(c).
The effect of the intradot spin mixing
In a DQD the interdot state mixing decreases exponentially with the increase of the interdot distance. When the interdot distance increases to a certain extend, the intradot orbit states mixing \({\xi }_{\kappa }{\eta }_{\kappa }{e}^{{\eta }_{\kappa }^{2}}\) becomes comparable to the interdot overlaps \({e}^{{d}^{2}/{\bar{x}}^{2}}\) in our considered range of magnetic field, so that intradot orbital mixing becomes an important factor in determining the overall spinflip transition rates. Here we examine the competition between the interdot and intradot mechanisms for spin flip transitions.
In a low magnetic field, the effect of the intradot orbital states mixing on the EDSR, compared with the interdot mechanism, is negligible as long as ξ_{ κ } is small. As the Zeeman splitting increases, the rising value of ξ_{ κ } enhances the strength of the intradot orbital states mixing, see Eq. (9). Meanwhile, the interdot pseudospin states mixing weakens with the increase of B for B > B_{0}. There thus exists a turning magnetic field B_{t}: for B < B_{t}, EDSR is dominated by the interdot state hybridization; for high fields the state mixing is dominated by the intradot mechanism. This change is also reflected in the variation of the EDSR frequency Ω_{3↔1} around the turning field B_{t}, as shown in the inset of Fig. 4. As the B field increases and approaches B_{t}, the magnitude of the EDSR frequency Ω_{3↔1} decreases with the growth of B as the interdot state mixing mechanism becomes less efficient, so it reverts that trend when B > B_{t} as the intradot mechanism becomes more effective.
The turning field B_{t} is a symbol for the competition between these two different mechanisms. Its magnitude mainly depends on the interdot distance. For the InSb nanowire DQD with φ = π/2, B_{t} as a function of d is shown in Fig. 4. The downward trend of B_{t} with the increase of d can be explained by the decline of the interdot state mixing, which requires a smaller magnetic field to counteract.
At large interdot distances, the magnitude of B_{ t } tends to be stable. This is because at a large interdot distance intradot orbital mixing dominates over interdot pseudospin hybridization, even for smaller magnetic field B ≤ B_{0}. Now B_{0} mainly depends on the orbital energy difference between the QDs Δ_{ o } ≡ ħω_{ r } − ħω_{ l }, and nearly independent of d, \({B}_{0}\simeq {{\rm{\Delta }}}_{o}/[({g}_{el}+{g}_{er}){\mu }_{B}]\). With our chosen QD parameters, we find \({B}_{0}\simeq 0.688\) T. Thus, once B increases beyond B_{0}, the electricdipole spin transition Φ_{1}〉 ↔ Φ_{3}〉 is dominated by the intradot orbital mixing, and the EDSR frequency increases with B.
The dependence on the interdot distance
The underlying dependence of the interdot barrier on the interdot distance means that spin tunneling, and singleelectron energy spectrum of the nanowire DQD in general, depend on d^{48}. In Fig. 5 we show the energy spectrum of the nanowire InSb DQD as a function of d, with B = 0.3 T and φ = π/2. Because of the asymmetry in the confinement potential along the wire axis, we limit ourselves to consider the case with a nonzero finite interdot distance exclusively.
At a large interdot distance, the interdot tunneling (proportional to \({e}^{{d}^{2}/{\bar{x}}^{2}}\)) is much smaller than the orbital energy difference Δ_{ o } between the dots, so that the effect of the interdot states mixing on the energy spectrum is negligible. The energy spectrum of the DQD is essentially the sum of the energy spectrums of the single QDs in this case, with \({{\rm{\Phi }}}_{1}\rangle \simeq {{\rm{\Psi }}}_{l}^{+}\rangle \), \({{\rm{\Phi }}}_{2}\rangle \simeq {{\rm{\Psi }}}_{l}^{}\rangle \), \({{\rm{\Phi }}}_{3}\rangle \simeq {{\rm{\Psi }}}_{r}^{+}\rangle \) and \({{\rm{\Phi }}}_{4}\rangle \simeq {{\rm{\Psi }}}_{r}^{}\rangle \). When the interdot distance decreases, the interdot tunnel coupling increases exponentially, and the energy spectrum of the DQD changes correspondingly. When d ~ d_{0}, the energy scale of the interdot tunnel coupling becomes comparable to the orbital energy difference between the QDs, so that the eigenstates of the DQD are delocalized pseudospin states. If the interdot distance further decreases, the two dots start to merge. The “interdot tunneling” will be of the same magnitude as the orbital excitation energy in the individual QDs. At this limit, the character of the electronic states shifts back from molecularlike to atomiclike again like the case of large interdot distance^{48}, although the composition of the orbital states are dramatically different. The energy splitting between Φ_{2}〉 and Φ_{3}〉 is now dominated by the singleparticle singledot excitation energy (as compared to tunnel splitting in the case of a double dot), which results in a sharp rise in this energy gap, as shown in Fig. 5.
Since the compositions of the DQD eigenstates vary with the interdot distance, particularly near d_{0}, the electricdipole transition rates change quite dramatically as well. In the inset of Fig. 5 we plot the transition rates Ω_{2↔1} and Ω_{3↔1} as a function of d. At a large interdot distance \(d\gg {d}_{0}\), the DQD eigenstates can be approximated as the eigenstates of the individual QDs, as explained above. Thus, Φ_{2}〉 ↔ Φ_{1}〉 is an intradot spinflip transition while Φ_{3}〉 ↔ Φ_{1}〉 is an interdot transition. Because of the vanishingly small interdot state mixing, the magnitude of the interdot transition rate will be smaller than that of the intradot spinflip transition rate, Ω_{3↔1} < Ω_{2↔1}. As the interdot distance decreases, the rapidly rising interdot coupling means both transition rates increase quickly as the states become mixed. When d ~ d_{0}, the eigenstates of the DQD are delocalized, and the electricdipole spin transition Φ_{2}〉 ↔ Φ_{1}〉 is dominated by the interdot pseudospin tunneling. As d decreases further, the magnitudes of Ω_{2↔1} and Ω_{3↔1} become stable because the DQD merges into a single QD. The electricdipole transition in a single nanowire QD was investigated in ref.^{41}. In this limit, Ω_{3↔1} approaches \((\sqrt{2}\mathrm{/2})eE\bar{x}/h\), while Ω_{2↔1} can be approximated by \({\xi }_{l}{\eta }_{l}\,\exp ({\eta }_{l}^{2})eE{x}_{l}/h\). Thus, when \(d\ll {d}_{0}\) the main mechanism of the EDSR turns back to the intradot orbital states mixing again. In short, in the parameter range we have considered, the electricdipole transition rates depend sensitively on the interdot tunneling/distance.
Phononinduced relaxation between the energy levels
Electronphonon (eph) interaction, together with spinorbit coupling, is the main cause of spin relaxation in a quantum dot^{49,50,51,52,53,54,55,56,57,58,59,60,61}. Accurately determining the relaxation rates is thus a necessary condition for quantitatively assessing the fidelity of the electricdipole transitions. Recall that B_{0} is the field at which Φ_{2}〉 and Φ_{3}〉 cross in the absence of SOC. Based on the major pseudospin components of the eigenstates involved in relaxation, Γ_{2→1} corresponds to phononinduced spin relaxation for \(B\ll {B}_{0}\), while Γ_{3→1} is the phononinduced spin relaxation rate when \(B\gg {B}_{0}\).
For relaxation between energetically close levels, we only consider the eph interaction with acoustic phonons and ignore the optical phonons. For acoustic phonons, there are two types of eph interaction: the piezoelectric and deformation potential interactions^{62}. Including the eph interaction, the complete Hamiltonian describing the DQD reads
where the eph interaction is given by^{55,56}
For the deformation potential interaction, \({V}_{{\bf{q}},\lambda }^{{\rm{df}}}={D}_{e}{\delta }_{\lambda ,l}\); and for the piezoelectric interaction,\({V}_{{\bf{q}},\lambda }^{{\rm{pe}}}=2e{h}_{14}\) \(({q}_{x}{q}_{y}{\hat{e}}_{{\bf{q}},z}^{\lambda }+{q}_{y}{q}_{z}{\hat{e}}_{{\bf{q}},x}^{\lambda }+{q}_{z}{q}_{x}{\hat{e}}_{{\bf{q}},y}^{\lambda })/{q}^{3}\). Here q = (q_{ x }, q_{ y }, q_{ z }) is the phonon wave vector, with q representing its magnitude, r = (x, y, z) denotes the electron position, and λ is the polarization of the phonon, with \(\hat{e}\) and c_{ λ } being the polarization vector and sound velocity of the phonon mode. The phonon annihilation (creation) operator is denoted by b (b^{†}). ρ and V are the mass density and the volume of the sample, respectively.
Using the Fermi golden rule, the phononinduced relaxation rate between the eigenstates Φ_{ i }〉 and Φ_{ j }〉 (i > j) can be calculated as
where Δ_{ ij } is the energy difference between the eigenstates Φ_{ i }〉 and Φ_{ j }〉, n_{th} is the thermal occupation of the phonon mode with ħω_{ λ } = Δ_{ ij } ≡ E_{ i } − E_{ j }. At low temperatures when \({k}_{{\rm{B}}}T\ll {{\rm{\Delta }}}_{ij}\), n_{th} ≈ 0. The matrix element M_{ ij } depends on the spatial distribution of the electron wave functions in three dimensions (see Methods). Taking the phonon mode density into consideration, the phononinduced relaxation rate can be rewritten as
with the integral region satisfying the energy conservation condition ħc_{ λ }q = Δ_{ ij }.
The electronphonon interaction Hamiltonian in Eq. (16) is spin independent. As such the spin composition of Φ_{ i }〉 and Φ_{ j }〉 plays a key role in determining Γ_{i→j}: a mostly spinconserved relaxation would be much faster than a relaxation involving spin flip. Since the degree of spin mixing depends on the interplay between the external field and the spinorbit coupling, the relaxations between different eigenstates can generally be regulated by varying the magnetic field strength and direction^{54,55,56,57,58,59,60,61}. Here we focus on the dependence of the relaxation rates on the magnetic field strength for a fixed magnetic field direction. The numerical results are shown in Fig. 6, where we plot Γ_{2→1} and Γ_{3→1} as functions of the magnetic field strength, with the system parameters taking the values as in Fig. 2.
In a low magnetic field with \(B\ll {B}_{0}\), the major pseudospin components of Φ_{ i }〉 (i = 1 − 4) dictate that Φ_{3}〉 → Φ_{1}〉 is a charge transition, while Φ_{2}〉 → Φ_{1}〉 is a spinflip transition, thus \({{\rm{\Gamma }}}_{3\to 1}\gg {{\rm{\Gamma }}}_{2\to 1}\). As B increases, spinflipped tunneling results in the hybridization between \({{\rm{\Phi }}}_{2}^{0}\rangle \) and \({{\rm{\Phi }}}_{3}^{0}\rangle \), so that Γ_{3→1} decreases while Γ_{2→1} increases. The slight oscillations in both relaxation rates are most likely due to the matching between DQD charge density and phonon wave vector (recall that the eph interaction Hamiltonian contains a e^{iq·r} factor). Furthermore, with the large gfactor for InSb, Zeeman splitting reaches 1 meV when the magnetic field is only a fraction of 1 Tesla. The corresponding phonon wave length is in the order of 10 nm, already below the quantum dot size, so that phonon bottleneck effect starts to become apparent for spinflip relaxation^{63}.
When B = B_{0} the spin states of Φ_{2}〉 and Φ_{3}〉 are equally mixed. The energy gap between Φ_{2}〉 and Φ_{3}〉 means that the relaxation rates are generally not identical at B_{0}. The rates are determined by a competition mostly between phonon density of states consideration and the phonon bottleneck effect: the former favors the largerenergy 3 → 1 transition, while the latter favors the smallerenergy 2 → 1 transition.
When the magnetic field strength exceeds B_{0}, Φ_{2}〉 (Φ_{3}〉) becomes the pseudospin up (down) state. As illustrated in Fig. 2, for B > B_{0} the energy splitting Δ_{21} (proportional to the tunnel coupling in the DQD) tends to be stable with the growth of B, while Δ_{31}, now the Zeeman splitting, keeps increasing. Thus Γ_{2→1} approaches a constant value when \(B\gg {B}_{0}\). Γ_{3→1}, on the other hand, keeps decreasing due to the reduction in spin mixing and the increasing influence of the phonon bottleneck effect.
The relaxation calculation here is done using bulk phonons. In a suspended nanowire, confined phonons on the nanowire should be used, and we expect the relaxation rates to be further suppressed because of the much smaller phonon density of states and stronger anisotropy due to the nanowire geometry^{23,64}.
Discussion and Conclusion
In this paper, we study the electronic properties of a nanowire DQD within the frame of effective mass approximation (EMA). For a thin nanowire, the energy scale of the electronic dynamics along the axis direction is much smaller than the energy scale of the excitations in the transverse directions. As such in our consideration the confined electron always stays in the transverse ground state^{18,22,23,28}.
Our calculations are based on a truncated double harmonic potential. Within EMA, the DQD confinement potential is usually approximated by a quartic function, a biquadratic function, or a Gaussian function^{7,38,39,40,48}. All these model potentials give rise to results consistent with the experimental results at a qualitative level^{7,40,55,58}. Therefore, the simplicity associated with the truncated double harmonic well (biquadratic function) model potential becomes the deciding factor for our choice. The relatively concise expressions within this model allows us to get to the basic physics more easily.
The system parameters used in our calculations are taken from the experimental data in ref.^{20}. The lowenergy spectrum of a single electron in the DQD is obtained using the LCAO method. In the calculation, the SOC is taken into account precisely, while the applied magnetic field is treated as a perturbation.
Our calculations show that in a DQD, there exist two different mechanisms that lead to EDSR: the intradot pseudospin state mixing and the interdot spinflipped tunneling. The EDSR frequency is determined by the combined effect of these two mechanisms, in which the dominant role can be varied by changing the system parameters. When the EDSR is dominated by the interdot spinflipped tunneling, we show that the electricdipole transition rates depend sensitively on the magnitude and orientation of the applied field. The intradot orbital mixing becomes more important when we reduce the tunnel coupling, so the two dots become independent, or when we increase tunneling to the degree when the double dot merges into a single dot. In the intermediate regime the interdot spin mixing is more effective. For a fixed tunnel coupling/interdot distance, the electricdipole driven transition rates experience a dip as the magnetic field increases, when the DQD transitions from the interdotmixing dominated lowfield region to the intradotmixing dominated highfield region.
Finally, we have calculated phononinduced relaxation rates among the DQD energy levels. The very large gfactors for strong SOC materials, such as InSb that we consider, mean that phonon bottleneck effect kicks in at much lower magnetic field for spinflip transitions compared to materials such as GaAs. Overall, our results on lowenergy spectrum, controllable electricdipole transitions, and relaxations should provide useful input for experimental studies of quantum coherent manipulations in a nanowire DQD.
Methods
Derivation of the effective Hamiltonian
The nanowire DQD Hamiltonian in Eq. (4) is derived within the effective mass approximation. We choose our coordinate system according to the geometry of the nanowire and the applied field. Specifically, we choose the xaxis along the axis of the nanowire, as illustrated in Fig. 1(a). When an external magnetic field is applied, we choose it to lie in the xzplane, so that the field can be expressed as B = B(cos θ, 0, sin θ). The complete Hamiltonian describing an electron in a nanowire DQD is
where the first term is the kinetic energy, with the kinetic momentum P = p + eA and the vector potential A = B(−y sin θ, −z cos θ, 0), U(r) is the confinement potential in three dimensions, H_{so}(r) represents the spinorbit interaction, and the last term denotes the Zeeman term, with g_{ e } and μ_{ B } being the locationdependent Landé gfactor and Bohr magneton, respectively. Here g_{ e } is locationdependent, with the specific value of the gfactor of the left QD being different from that of the right QD g_{ el } ≠ g_{ er }.
The DQD confinement potential for the electron is modelled by a asymmetric double well harmonic potential along the nanowire axis, \(V(x)=\frac{1}{2}{m}_{e}\,{\rm{\min }}\,\{{\omega }_{l}^{2}{(x+d)}^{2},{\omega }_{r}^{2}{(xd)}^{2}\}\), where ω_{ l } ≠ ω_{ l }, and 2d is the interdot distance. In the transverse direction we consider a strong harmonic potential along the y direction, \(V(y)=\mathrm{(1/2)}{m}_{e}{\omega }_{y}^{2}{y}^{2}\), and a large gradient potential along the z direction, V(z) = eE_{ z }z for z ≥ 0 and V(z) = ∞ for z < 0. Due to the strong transverse confinements, we assume that the electron is always in the ground state along the y and z, so that the transverse orbital dynamics is frozen:
where \({y}_{0}=\sqrt{\hslash /({m}_{e}{\omega }_{y})}\) and τ = (2eE_{ z }m_{ e }/ħ^{2})^{1/3}. The characteristic length scales of the wavefunction along the y and z directions can thus be quantified by y_{0} and \({z}_{0}\equiv {\int }_{0}^{\infty }\,\varphi (z)z\varphi (z)dz=\mathrm{1.5581/}\tau \), respectively.
The lowestorder effective Hamiltonian for an electron moving along the xaxis can be obtained by averaging over the y and z directions,
where \(\langle \xi \rangle =\int \,{\psi }^{\ast }(y){\varphi }^{\ast }(z)\xi \psi (y)\varphi (z)dydz\). The first term on the right side of Eq. (21) represents the effective kinetic Hamiltonian. Substituting the kinetic momentum expression into Eq. (21), the effective kinetic Hamiltonian can be expanded as
with B_{ x } = B cos θ and B_{ z } = B sin θ. Since ψ(y) is an even function, the second term on the right side of Eq. (22) vanishes. The last term on the right side of Eq. (22) can also be ignored because it is a constant term and only affects the zeropoint energy of the effective Hamiltonian. In short, the applied magnetic field does not have any orbital effect within this mean field approximation.
The inversion asymmetry in A_{III}B_{V} heterostructures results in Dresselhaus and Rashba spinorbit interactions^{43,44},
Here interaction strength γ_{ D } and γ_{ R } are determined by the band structure parameters^{43,44}. \({\tilde{P}}_{x}={P}_{x}({P}_{y}^{2}{P}_{z}^{2})+{\rm{H}}{\rm{.c}}.\), while \({\tilde{P}}_{y}\) and \({\tilde{P}}_{z}\) can be obtained by cyclic permutations. The effective SOC Hamiltonian along the x direction can thus be calculated as
According to Eq. (23), the effective Hamiltonian describing the linear Dresselhaus SOC along the nanowire axis is
where we have used the identity 〈p_{ y }〉 = 0. In the considered range of magnetic field with \(\xi \ll 1\), the contribution of (eB_{ x })^{2}〈z^{2}〉 to \({H}_{D}^{x}\) is negligible compared with the other two terms in the bracket on the right side of Eq. (25). Similarly, the effective Rashba SOC along the x direction can be written as
Using the specific confinements along lateral directions, we obtain \({\partial }_{y}V(y)={m}_{e}{\omega }_{y}^{2}y\) and ∂_{ z }V(z) = eE_{ z }. After averaging over y and z, the effective Rashba SOC Hamiltonian takes the form
The total effective SOC Hamiltonian along the nanowire axis is thus given by
with
Substituting Eqs (22) and (28) into Eq. (21), the effective Hamiltonian describing the DQD along the wire axis can be simplified as Eq. (4) in the main text,
with \(\alpha =\sqrt{{\alpha }_{R}^{2}+{\alpha }_{D}^{2}}\) and Δ_{ Z } = −g_{ e }μ_{ B }B.
In the numerical calculations in this paper, we assume ħω_{ y } = 80 meV and E_{ z } = 0.6 mV/Å. The SOC length in an InSb nanowire DQD is x_{so} = 200 nm, and the characteristic lengths along the transverse directions are given by y_{0} = 8.2 nm and z_{0} = 12.4 nm. Other material parameters are all chosen for a nominal InSb nanowire, including m_{ e } = 0.013m_{0}, ρ = 5.77 × 10^{−27} kg/Å^{3}, γ_{ D } = 228 eVÅ^{3}, γ_{ R } = 500 Å^{2}, D_{ e } = 7.0 eV, eh_{14} = 0.061 eV/Å, c_{ l } = 3.69 × 10^{13} Å/s, and c_{ t } = 2.3 × 10^{13} Å/s, which are used in the main text for numerical calculations.
Construction of the orthonormal basis
The analytic formulas for the orthonormal bases \({{\rm{\Psi }}}_{\kappa \Uparrow }\rangle \) and Ψ_{κ⇓}〉 (κ = l, r) are given. Using the perturbation theory, the two lowestenergy eigenstates of the local Hamiltonian H_{ κ } can be approximated as the equation (9) in the main text
with the parameters given in Eq. (10). As is indicated by Eq. (9), the Zeeman field leads to the mixing of different spinorbit states in \({{\rm{\Psi }}}_{\kappa }^{\pm }\rangle \), with the degree of orbital mixing proportional to \({\xi }_{\kappa }{\eta }_{\kappa }^{n}{e}^{{\eta }_{\kappa }^{2}}\). Here ξ_{ κ } denotes the ratio between the Zeeman splitting and the orbital splitting in κ dot, ξ_{ κ } ≡ Δ_{ κZ }/Δ_{ κS }, and is much less than one, which ensures the validity of the perturbation theory. η_{ κ } corresponds to the ratio between the effective dot size x_{ κ } and SOC length x_{so}. In a nanowire quantum dot, η_{ κ } is generally a small number \({\eta }_{\kappa }\equiv {x}_{\kappa }/{x}_{{\rm{so}}}\ll 1\), even for materials with strong SOC. Therefore, in order to facilitate the numerical calculations in the main text and account the effect of high orbital states, the summation in Eq. (31) is truncated, and only keep the n = 1 term. Thus, the corresponding normalized local wave functions can be written as
with \({\chi }_{\kappa }=(\sqrt{2}\mathrm{/2}){\xi }_{\kappa }{\eta }_{\kappa }{e}^{{\eta }_{\kappa }^{2}}\,\sin \,\phi \) and \({{\vartheta }}_{\kappa }=\mathrm{(1/2)}\,\arccos (\cos \,\phi /{f}_{\kappa })\).
On the basis of Eq. (32), we can construct the two orthonormal bases
where \({s}_{}=\langle {{\rm{\Psi }}}_{1l}^{}{{\rm{\Psi }}}_{1r}^{}\rangle \) and \({g}_{}=(1\sqrt{1{s}_{}{}^{2}})/{s}_{}\). In order to construct the other two orthonormal bases, first we introduce two auxiliary states
with \({s}_{1l}=\langle {{\rm{\Psi }}}_{l\Downarrow }^{}{{\rm{\Psi }}}_{1l}^{+}\rangle \), \({s}_{2l}=\langle {{\rm{\Psi }}}_{r\Downarrow }^{}{{\rm{\Psi }}}_{1l}^{+}\rangle \), \({s}_{1r}=\langle {{\rm{\Psi }}}_{l\Downarrow }^{}{{\rm{\Psi }}}_{1r}^{+}\rangle \), and \({s}_{2r}=\langle {{\rm{\Psi }}}_{r\Downarrow }^{}{{\rm{\Psi }}}_{1r}^{+}\rangle \). Finally, basing on the auxiliary states, the other two orthonormal bases can be calculated as
where \({s}_{+}=\langle {\hat{{\rm{\Psi }}}}_{l\Uparrow }{\hat{{\rm{\Psi }}}}_{r\Uparrow }\rangle \) and \({g}_{+}=(1\sqrt{1{s}_{+}{}^{2}})/{s}_{+}\).
Calculation of the phononinduced relaxation rates
For relaxation between energy levels of a nanowire DQD through a singlephonon process, we only consider the eph interaction with acoustic phonons and ignore the optical phonons. For acoustic phonons, there are two types of eph interaction: the piezoelectric and deformation potential eph interactions^{62}. The corresponding Hamiltonian is given by Eq. (16).
At low temperatures with \({k}_{{\rm{B}}}T\ll {{\rm{\Delta }}}_{ij}\), the phononinduced relaxation rate between states Φ_{ i }〉 and Φ_{ j }〉 (i > j) can be calculated via the Fermi golden rule:
where Δ_{ ij } denotes the energy difference between Φ_{ i }〉 and Φ_{ j }〉, Δ_{ ij } = E_{ i } − E_{ j }, and M_{ ij } represents the transition matrix element of e^{iq·r} in three dimensions. In our model calculation, the electron is in the ground state along the transverse directions. The transition element M_{ ij } thus takes the form of
with Π being the average of \({e}^{i({q}_{y}y+{q}_{z}z)}\) over the transverse directions, i.e., \({\rm{\Pi }}=\langle {e}^{i({q}_{y}y+{q}_{z}z)}\rangle \). During this calculation, the wavefunction along the z direction is truncated for the account of a finite length along the transverse direction. Using threedimensional phonon density of states, the relaxation rate can be written as
with the integral region satisfying the energy conservation condition ħc_{ λ }q = Δ_{ ij }. This result should be the most accurate when the nanowire is buried inside a substrate. For a suspended nanowire, the relaxation rate should be further suppressed because of the reduced density of state for phonons.
In a cylindrical coordinate system, the relaxation rate caused by deformation potential \({V}_{{\bf{q}},\lambda }^{{\rm{df}}}={D}_{e}{\delta }_{\lambda ,l}\) can be written as
with \({{\rm{\Delta }}}_{ij}^{zl}=\hslash {c}_{l}{q}_{z}\) and ϑ the azimuth angle. Similarly, the relaxation rate caused by the piezoelectric interaction is
with
where \({{\rm{\Delta }}}_{ij}^{z}=\hslash {c}_{\lambda }{q}_{z}\) and \(\varpi ={{\rm{\Delta }}}_{ij}^{z2}+({{\rm{\Delta }}}_{ij}^{2}{{\rm{\Delta }}}_{ij}^{z2})\,{\sin }^{2}\,{\vartheta }\). The overall phononinduced relaxation rate between states Φ_{ i }〉 and Φ_{ j }〉 is then
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Acknowledgements
This work is supported by the National Key Research and Development Program of China (grant No. 2016YFA0301200), the NSFC (grant No. 11774022) and the NSAF (grant No. U1530401). R.L. is supported by the NSFC (grant No. 11404020) and Postdoctoral Science Foundation of China (grant No. 2014M560039). X.H. acknowledges financial support by US ARO through grant W911NF1210609 and W911NF1710257 and thanks the CSRC for hospitality during the visit.
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Z.H.L. performed the derivations and numerical calculations under the guidance of J.Q.Y. and X.H. Also, R.L. participated in the discussions. All authors contributed to the interpretation of the work and the writing of the manuscript.
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Liu, ZH., Li, R., Hu, X. et al. Spinorbit coupling and electricdipole spin resonance in a nanowire double quantum dot. Sci Rep 8, 2302 (2018). https://doi.org/10.1038/s41598018207065
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DOI: https://doi.org/10.1038/s41598018207065
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