Symmetry induced hole-spin mixing in quantum dot molecules
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Symmetry induced hole-spin mixing in quantum dot molecules
Josep Planelles, Fernando Rajadell, Juan I. Climente
Departament de Qu´ımica F´ısica i Anal´ıtica, Universitat Jaume I, E-12080, Castell´o, Spain ∗ (Dated: April 12, 2018)We investigate theoretically the spin purity of single holes confined in vertically coupledGaAs/AlGaAs quantum dots (QDs) under longitudinal magnetic fields. A unique behavior isobserved for triangular QDs, by which the spin is largely pure when the hole is in one of the dots,but it becomes strongly mixed when an electric field is used to drive it into molecular resonance.The spin admixture is due to the valence band spin-orbit interaction, which is greatly enhanced in C h symmetry environments. The strong yet reversible electrical control of hole spin suggests thatmolecules with C -symmetry QDs, like those obtained with [111] growth, can outperform the usual C -symmetry QDs obtained with [001] growth for the development of scalable qubit architectures. PACS numbers: 73.21.La,73.40.Gk,71.70.Ej
Single spins confined in III-V semiconductor QDs arecurrently considered as potential qubits for solid-statequantum information processing, which combine fastoptical and electrical manipulation with prospects ofscalability.
In the last years, heavy hole spin qubitshave emerged as a robust and long-lived alternative toelectron spins, as they can be less sensitive to dephasingfrom nuclear spins.
Significant advances have been re-ported on hole spin initialization, control and readout bymeans of optical excitations , and different pro-posals for electrical control have been put forward.
Although most works so far have focused on QDsgrown along [001], it has been noted that the C pointsymmetry of such systems gives rise to a splitting ofbright exciton states which limits the fidelity of opticalhole spin preparation. No such splitting is howeverexpected in [111] grown QDs owing to their higher ( C )symmetry, which hence become an alternative worthexploring. Early studies on single (In)GaAs/AlGaAsQDs grown along [111] have revealed that hole stateshave weak heavy hole-light hole (HH-LH) coupling dueto the large aspect ratio, which is a prerequesite to ob-tain pure hole spins and minimize the impact of hyperfineinteraction with the lattice nuclei. In turn, magneto-photoluminescence spectra have reported characteristicdifferences from [001] grown QDs which were ascribed tothe influence of the C symmetry on the hole states. In this paper, we move forward and study hole statesconfined in quantum dot molecules (QDM) formed by apair of vertically stacked QDs grown along [111]. QDMspresent several advantadges over single QDs for qubitdevelopment, including readout independency from ini-tialization and measurement protocols , higher fidelityof spin preparation and enhanced wavelength tun-ability with external electric fields, which greatly im-proves prospects of scalability. We consider [111] grownGaAs/AlGaAs QDMs with triangular shape, similar tothose reported in Refs. , adding longitudinal magneticand electric fields to control the Zeeman splitting andcharge localization. Our calculations show that the HHspin purity is high when the hole is confined in individ- ual QDs, but severe spin admixture takes place when theelectric field is used to form molecular orbitals. The spinadmixture follows from the formation of orbitals withapproximate C h point group symmetry, which enablesotherwise forbidden spin-orbit interactions (SOI). Thesymmetry-induced SOI does not mix nearby Zeeman sub-levels, but it couples bonding and antibonding molecularstates split by the tunneling energy. This is in sharp con-trast with usual [001] grown QDMs, with C h symmetry,where tunneling is normally a spin-preserving process. Since the activation of SOI mechanisms is gener-ally associated with a descent of the system symmetry(e.g. system and bulk inversion asymmetry , QDMmisalignment ), the enhancement of SOI for C QDsis apparently counterintuitive. Yet, we observe strongspin admixture between hole states over 1 meV apart,which is 2 . We providean explanation through group theory analysis of the themulti-band k · p Hamiltonian for holes, showing this is anexclusive property of C systems, and discuss the impli-cations of these findings for the development of hole spinqubit architectures.The Hamiltonian we use to describe hole states reads: H = H BF + H B + H strain + ( V ( r ) + e ( φ pz ( r ) − F z )) I . (1)Here H BF is the four-band Burt-Foreman Hamiltonian for [111] grown zinc-blende crystals, which considers HH-LH subband coupling as in the Luttinger model butincluding position-dependent effective masses. H B rep-resents the terms coming from a magnetic field appliedalong the growth ( z ) direction. H strain is the strainHamiltonian, V ( r ) the band-offset potential, e the holecharge, φ pz the piezoelectric potential, F an axial elec-tric field and I a rank-4 identity matrix. Further detailson the Hamiltonian can be found in the SupplementalMaterial. Hamiltonian (1) is solved numerically afterobtaining the strain tensors and piezoelectric fields us-ing the
Comsol package. The eigenstates are Luttingerspinors of the form: | n i = / X Jz = − / f nJ z ( r ) | J = 3 / , J z i , (2)where f nJ z ( r ) is the envelope function associated to | J =3 / , J z i , the periodic function with Bloch angular mo-mentum J z . J z = ± / J z = ± / h J z i = P J z J z h f nJ z | f nJ z i can be taken as a measure of the holespin purity, with h J z i ≈ ± / ⇑ ) or spin down ( ⇓ ) states.For our calculations we consider pyramidalGaAs/Al . Ga . As QDMs with triangular confine-ment, similar to those obtained by metallorganic vapordeposition.
The vertically stacked QDs are separatedby a barrier of thickness d = 2 nm, although they areinterconnected by a thin Al . Ga . As vertical quan-tum wire which enhances tunnel coupling -see structurein Fig. 1(a)-. A weak magnetic field of B = 0 . F . At F = 11 kV/cm we seetwo Zeeman-split doublets. The upper one (states | i and | i ) corresponds to the main component of the holespinor in the top QD, which is slightly bigger than thebottom QD. In turn, the lower doublet (states | i and | i ) has the main component in the bottom dot. Sincethe increasing electric field favors the occupation of thebottom dot, a charge transfer anticrossing takes placeat F = 14 . F r ), wherebonding and antibonding molecular orbitals are formed.The behavior closely resembles that of [001] grownQDMs, except for one anomaly: the Zeemansplitting of both doublets is quenched near the resonantelectric field, see Fig. 1(a) inset.The Zeeman splitting suppression can be seen as a van-ishing effective g-factor. Unlike in previous reports, how-ever, the origin of this effect cannot be ascribed to thedifferent g-factor of the QD and barrier materials, asthe QDs and the vertical wire connecting them have sim-ilar composition. Because electrically tunable g-factorsare of interest for spin manipulation, we further inves-tigate into the origin of this phenomenon. Fig. 1(b) showsthe hole Bloch angular momentum expectation value ofthe four states under consideration. As can be seen, awayfrom the resonant field, h J z i gradually approaches ± / F r , however, h J z i ≈
0, which means that for molecularstates the spin becomes completely mixed.For comparison, since [111] grown QDs normally haveeither triangular or hexagonal shape, in Fig. 1(c-d) we study a QDM formed by hexagonal QDs. One cansee that no spin mixing takes place near F r in this case.Actually, the behavior is now the same as observed invertically aligned [001] grown QDMs with C h symmetry -40.7-40.5-40.3-40.1 E ne r g y ( m e V ) -1.5-1-0.500.511.5 13 14 15 16F (kV/cm)-43-42.5-42-41.5 E ne r g y ( m e V ) -1.5-1-0.500.511.511 13 15 17 < J z > F (kV/cm)|1|2|3|4 |1|2|3|4 (a)(b) (c)(d) < J z > Fr Fr
F (kV/cm) | Δ B | ( μ e V ) FIG. 1. (a) and (b): hole energy levels and Bloch angu-lar momentum expectation value of a triangular QDM grownalong [111], as a function of a vertical electric field. (c) and(d): same but for a hexagonal QDM. Note the strong spinmixing for triangular QDMs near the resonant field F r . Thestructures in (a) and (c) show the hole localization for eachdoublet. The inset in (a) shows the Zeeman splitting of theupper (solid line) and lower (dashed line) doublet. QDs, where tunneling is a spin preserving process. Itfollows that the strong spin mixing in Fig. 1(b) is not aconsequence of the [111] crystal orientation, but ratherof the triangular envelope confinement. In fact, we alsoobserve it for triangles grown along [001], see Supple-mentary Material. It does not result from the presenceor absence of strain either, as similar results are obtainedusing lattice mismatched materials such as InAs/GaAs. Likewise, it is not induced by the magnetic field, as H B is a diagonal term which does not couple different spinorcomponent. It does not take place in single QDs either.It is an exclusive property of QDMs with triangular con-finement.Further insight into the hole spin mixing mechanismis obtained in Fig. 2, which plots the weight of the fourspinor components corresponding to each of the states | i to | i of the triangular QDM. Two conclusions canbe extracted: (i) LHs play a minor role in all cases, themixing is essentially between HH components with or-thogonal spin projections ( J z = ± / | i and | i (panels (a) and (d)) seem to exhibit complemen-tary behavior, and so do | i and | i (panels (b) and (d)).This suggests that the spin mixing is due to independentinteractions between | i and | i on the one hand, and | i and | i on the other.To understand the origin of the HH spin mix-ing we resort to a point group theory analysis.Eq. (1) Hamiltonian can be simplified as H ≈ H [111] LK + H B + ( V ( r ) + e F z )) I , where we have disre-garded strain terms –which are weak for GaAs/AlAsheterostructures– and approximated H BF by the (con-
10 11 12 13 14 15 16 17 18 19F (kV/cm)00.20.40.60.810 11 12 13 14 15 16 17 18F (kV/cm)00.20.40.60.81 (a) (b)(c) (d) f Jz n | f Jz n f Jz n | f Jz n J z =+3/2Jz=+1/2Jz=-1/2Jz=-3/2 FIG. 2. Weight of the different Luttinger spinor componentsof the hole states in Fig. 1(a), as a function of the electric field.Panels (a) to (d) correspond to states | i to | i , respectively.The states are almost exclusively HH ( J z = ± / stant mass) Luttinger-Kohn Hamiltonian: H [111] LK = − ˆ P + ˆ Q − ˆ S ˆ R − ˆ S † ˆ P − ˆ Q R ˆ R † P − ˆ Q ˆ S R † ˆ S † ˆ P + ˆ Q (3)withˆ P ± ˆ Q = ( γ ± γ )( k x + k y ) + ( γ ∓ γ ) k z ˆ R = − √ ( γ + 2 γ ) k − + √ √ ( γ − γ ) k + k z ˆ S = − √ √ ( γ − γ ) k + √ (2 γ + γ ) k − k z (4)where γ , γ , γ are the Luttinger parameters and k ± = k x ± ik y . One can then see that H has C point symmetryset by the confining potential V ( r ). Near the resonantelectric field, however, an additional approximate sym-metry must be considered. Even if the two QDs formingthe QDM are not identical, the electric field restores an effective parity symmetry, the bonding and antibondingHH molecular orbitals forming even and odd functionswith respect to a mirror plane in between the QDs. The corresponding point group is then C h . Note that for H [111] LK to hold exact C h symmetry, we need to impose theso-called axial approximation ( γ = γ ), which is actuallyvalid for many III-V materials, such as GaAs. Actually,we do not impose exact symmetry in the numerical cal-culations and nevertheless, the obtained results reveal, asexpected, a high degree of symmetry.The anticrossing of Fig. 1(a) can be rationalized con-sidering the symmetry of the hole spinors in the doublegroup ¯ C h . Within this group | i and | i have E − / symmetry, while | i and | i have E / symmetry. Thedifferent symmetry of | i and | i ( | i and | i ) explainsthe lack of interaction within the Zeeman doublets, in-spite of the quasi-degeneracy. It also becomes clear why Interdot distance ( b) FIG. 3. (a) Diagram of single band hole energy levels undera longitudinal magnetic field for a QDM with symmetry C h .The labels ( X p , J z ) indicate the symmetry of the levels. X represents the rotational symmetry of the envelope function( A , E ± ) while p = ′ or ′′ represents the even/odd parity and J z indicates the non-zero component of the spinor. Thickarrows denote the symmetry allowed level couplings for theground state. (b) Energy structure of anticrossing hole statesbefore and after switching on off-diagonal terms in H [111] LK . (c)Typical dissociation energy spectrum of holes in QDMs at B = 0 T, showing a bonding-antibonding ground state rever-sal at d c . (d) Calculated spin purity of states | i and | i fordifferent interdot distances at the resonant electric field. Spinmixing is strongest around d c . | i and | i ( | i and | i ) interact separately, as observedin Fig. 2.The above picture differs from the widely studied [001]grown QDMs with circular confinement, where the sym-metry of all four spinorial states involved in the molecularanticrossing is different , which results in the absenceof interaction and hence spin-preserving tunneling. Siz-able hole spin mixing has been observed only in QDMswith significant misalignment , because the symmetry isthen completely reduced ( ¯ C point group). Nevertheless,the largest spin anticrossing measured for such system is0 . This is 2 . . To explain the unusual strength of the spin-orbit cou-pling in triangular QDMs, we can examine the envelopesymmetry of H [111] LK (within axial approximation) and theensuing eigenfuctions. In the C h point group, the termsof Eq. (3) form basis of the following irreducible repre-sentations: Γ H LK = A ′ E ′′− E ′ + E ′′ + A ′ E ′ + E ′− A ′ E ′′− E ′− E ′′ + A ′ . (5)A remarkable consequence is that for any hole state | n i ,the envelope functions of the spin up and down HH com-ponents, f n / and f n − / in Eq. (2), must have the samesymmetry except for the even/odd parity (e.g. A ′ and A ′′ ). This is a key factor in determining the strengthof the spin admixture, as we show in the perturbativeanalysis below.Hamiltonian H can be split as a sum of diagonal andoff-diagonal terms, H = H + H ′ , the latter being respon-sible for band coupling. If we disregard H ′ , the levelsanticrossing at F r are those represented at the top ofFig. 3(a), namely two Zeeman doublets formed by HHswith opposite spin ( J z = ± /
2) but the same rotationalsymmetry ( A ) and parity ( ′ or ′′ ). Each doublet is splitby a Zeeman term ∆ B and separated from each otherby an amount 2 t , where t is the HH tunneling integral.Considering H ′ as a perturbative term, the mixing be-tween the spin up ground state | k (0) i = ( A ′ , +3 /
2) andany spin down HH state | i (0) i is given by: | k (2) i = X i = k X j = k h i (0) | H ′ | j (0) i E k − E i h j (0) | H ′ | k (0) i E k − E j | i (0) i (6)where | j (0) i is the j -th intermediate state and E j itscorresponding energy. Notice that the strength of cou-pling is inversely proportional to ∆ E hh = E k − E i , i.e.the energy difference between the spin up and down HHstates. As explained with detail in the SupplementalMaterial, the symmetry of H ′ operators, off-diagonalterms of Eq.(5), translates into selection rules whichmake the numerator of Eq. (6) vanish for all except thetwo paths plotted with thick arrows in Fig. 3(a). Bothpaths involve excited LHs as intermediate states , andthe spin down HH is | i i = ( A ′′ , − / E hh is small (few meV at most), and is in contrast with otherpoint symmetries, where selection rules lead to couplingwith higher excited HH states, so that ∆ E hh is muchlarger. For example, if we consider QDMs with circu-lar QDs (point group C ∞ h ), the HH components coupledby H ′ no longer have the same rotational symmetry, butthey differ by three quanta of azimuthal angular momen-tum M z . Consequently, there is no coupling betweenthe M z = 0 HHs forming the molecular anticrossing. Having a QDM structure is also essential, as then A ′ and A ′′ are roughly split by the tunneling energy 2 t , whichcan be made small enough for the SOI to be efficient. Bycontrast, in single QDs the strong vertical confinementwould lead to several meV splitting.The suppression of the Zeeman splitting observed inFig. 1(a) can be also understood from the perturbative analysis. As indicated in Fig. 3(b), the band couplingoccurs between HH states belonging to different dou-blets. Because ∆ E hh is smaller for the innermost states(2 t − ∆ B ) than for the outermost ones (2 t + ∆ B ), theinteraction is stronger, leading to an effectively reducedZeeman splitting, ∆ B .It is clear from the discussions above that tunnelingmust be an important parameter to control the strengthof the spin mixing. One might then expect that spinmixing is enhanced for long interdot distances d , whentunneling energy t is small. Fig. 3(d) shows the spinpurity of the ground state HH for QDMs with different d at resonant electric field. Interestingly, the maximumspin mixing is found at intermediate distances, d ≈ As shown in theschematic of Fig. 3(c), there is a critical distance d c wherebonding and antibonding hole states are reversed. Atthis point, t has a relative minimum combined with largewave function delocalization, which enables the strongspin mixing. For d < d c , t increases rapidly, reducingthe interaction. For d > d c , t eventually decreases butso does the wave function delocalization. As a result wegradually retrieve the single QD limit, were spin mixingis weak.Electrical control of hole spins in QDMs has beenproposed as a key ingredient for scalable qubitarchitectures. So far, however, only [001] grown QDMshave been considered, where the main source of spinmixing was misaligment between the vertically stackeddots, which is a difficult parameter to regulate exper-imentally. The C h -symmetry-induced spin mixing de-scribed here arises as a more robust and manageablemechanism. It can also help increase the fidelity of spincontrol gate operations, as this requires the spin mixedstates to (i) be energetically well resolved, and (ii) beable to form indirect excitons with large optical dipolestrength. As for (i), we predict strong mixing betweenstates 1 meV away from each other, larger than anyprevious measurement. As for (ii), unlike in misalignedQDMs, the spin mixing we describe is strong at the reso-nant electric field, where direct and indirect excitons havecomparable optical strength. Another advantadge is thepossibility to use weak magnetic fields, which limits theinfluence of the g-factor inhomogeneity of different QDsin the qubit scaling.In summary, we have shown that triangular QDs canbe used to build QDMs with electrically controllable holespin. The hole spin is well defined inside the individ-ual QDs, but the formation of delocalized molecular or-bitals with C h symmetry enables SOI induced mixingwith unprecedented strength. 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Josep Planelles, Fernando Rajadell, Juan I. Climente
Departament de Qu´ımica F´ısica i Anal´ıtica,Universitat Jaume I, E-12080, Castell´o, Spain ∗ (Dated: April 12, 2018) PACS numbers: 73.21.La,73.40.Gk,71.70.Ej . THEORETICAL MODEL AND PARAMETERS The Hamiltonian we use to describe hole states in QDs grown along the [001] directionis (atomic units): H [001] = H [001] BF + H [001] B + H [001] strain + (cid:0) V ( r ) + e ( φ [001] pz ( r ) − F z ) (cid:1) I . (1)Here H [001] BF is the four-band Burt-Foreman Hamiltonian for zinc-blende crystals, whichconsiders HH-LH subband coupling and position-dependent effective masses: H [001] BF = ( 12 x,y,z X i k i L + M k i ) I − x,y,z X i k i L − M k i J i + i k x ( N − N ′ ) k y − k y ( N − N ′ ) k x ) J z − x,y,z X i 00 0 0 − (3)By inserting eq. 3 in eq. 2 we end up with the following matrix representation: H [001] BF = − ˆ P ′ ˆ S − − ˆ R S −† ˆ P ” − ˆ C − ˆ R − ˆ R † − ˆ C † ˆ P ” ∗ − ˆ S + † − ˆ R † − ˆ S + ˆ P ′∗ (4)2here, P ′ = ( k x ( L + M ) k x + k y ( L + M ) k y + k z (2 M ) k z ) + i ( k x ( N − N ′ ) k y − k y ( N − N ′ ) k x ) P ” = ( k x ( L + 5 M ) k x + k y ( L + 5 M ) k y + 2 k z (2 L + M ) k z ) + i ( k x ( N − N ′ ) k y − k y ( N − N ′ ) k x ) P ” ∗ = ( k x ( L + 5 M ) k x + k y ( L + 5 M ) k y + 2 k z (2 L + M ) k z ) − i ( k x ( N − N ′ ) k y − k y ( N − N ′ ) k x ) P ′∗ = ( k x ( L + M ) k x + k y ( L + M ) k y + k z (2 M ) k z ) − i ( k x ( N − N ′ ) k y − k y ( N − N ′ ) k x ) R = √ [ k x ( L − M ) k x − k y ( L − M ) k y − i ( k x ( N + N ′ ) k y + k y ( N + N ′ ) k x )] R † = √ [ k x ( L − M ) k x − k y ( L − M ) k y + i ( k y ( N + N ′ ) k x + k x ( N + N ′ ) k y )] S − = − √ [ k − N k z + k z N ′ k − ] S †− = − √ [ k z N k + + k + N ′ k z ] S + = − √ [ k + N k z + k z N ′ k + ] S † + = − √ [ k z N k − + k − N ′ k z ] C = − ( k z ( N − N ′ ) k − − k − ( N − N ′ ) k z ) C † = − ( k + ( N − N ′ ) k z − k z ( N − N ′ ) k + ) (5)with L, M, N, N ′ being the Stravinou-van Dalen mass parameters.By setting the parameters constant, Eq. (4) Hamiltonian turns into the Luttinger KohnHamiltonian: The Stravinou-van Dalen parameters are then related to the Luttinger param-eters γ , γ , γ as L − M = − γ , 3 L + M = − γ − γ , N − N ′ = 1 + γ − γ − γ and N + N ′ = − γ . H [001] B represents the terms coming from a magnetic field applied along the growth ( z )direction, B : H B = − (cid:18) B x + y ) + B xk y − yk x ) (cid:19) (cid:18) ( γ − γ ) I + γ J z (cid:19) + κµ B B J z . (6)with I a rank-4 identity matrix, J z the angular momentum z-component matrix ( J = 3 / κ = 4 / and µ B the Bohr magneton. H [001] strain is the strain Hamiltonian, formally identical to Eq. (4) with the products k i k j replaced by the strain tensor component ǫ ij . e and φ pz are the hole charge andstrain-induced piezoelectric potential -which is also diagonal-, F an axial electric field and V ( r ) the band-offset potential. For the triangular QDMs we use a structure like thatobtained by metallorganic vapor deposition . The structure is illustrated in Fig. 1, usingsimilar geometry and composition to the pyramidal QDMs of Refs. . Thus, the QDs aremade of GaAs, the barrier of Al . Ga . As and the vertical wire connecting the dots of3 n m n m n m n m d GaAsAl Ga As Al Ga As FIG. 1: QDM formed by triangular QDs. Vertical electric and magnetic fields are applied. Al . Ga . As. We note that the vertical wire plays no critical role in the phenomena wedescribe. The robustness of the results are checked by carrying out additional calculationswith triangular round edges and also breaking the symmetry with two round and a sharpedge (not shown). For comparison, we also study QDMs made of hexagonal QDs. The sidesof the hexagons are taken to have the same dimensions as those in Fig. 1, and a hexagonalwire is used which preserves the C symmetry.For [111] grown QDs, the hole Hamiltonian becomes: H [111] = H [111] BF + H [111] B + H [111] strain + (cid:0) V ( r ) + e ( φ [111] pz ( r ) − F z ) (cid:1) I . (7)where we have obtained H [111] BF and H [111] strain from Eq. (2) by writing k i and J i appearing inthe Hamiltonian as a function of the new coordinates k ′ i and J ′ i according to, k = M k ′ J = MJ ′ (8)where M is the rotation matrix: M = √ − √ √ √ √ √ − q √ . (9)Once we reach the new matrices, the prime is removed from k i for the sake of a betterpresentation. The Burt-Foreman Hamiltonian in the [111] direction then reads: H [111] BF = − ˆ P ′ − ˆ S − ˆ R − ˆ S −† ˆ P ” ˆ C ˆ R ˆ R † ˆ C † ˆ P ” ∗ ˆ S + † R † ˆ S + ˆ P ′∗ (10)4here,ˆ P ′ = [ k x ( γ + γ ) k x + k y ( γ + γ ) k y + k z ( γ − γ ) k z ] − i [ k x ( γ − γ − γ ) k y − k y ( γ − γ − γ ) k x ]ˆ P ” = [ k x ( γ − γ ) k x + k y ( γ − γ ) k y + k z ( γ + 2 γ ) k z ] − i [ k x ( γ − γ − γ ) k y − k y ( γ − γ − γ ) k x ]ˆ S − = − √ (cid:8) ( k − γ k z − k z γ k − ) + √ k x ( γ − γ ) k x − k y ( γ − γ ) k y + i ( k x ( γ − γ ) k y + k y ( γ − γ ) k x )] − k − ( γ + γ ) k z − k z γ k − ] } ˆ S + = − √ (cid:8) ( k + γ k z − k z γ k + ) + √ k x ( γ − γ ) k x − k y ( γ − γ ) k y − i ( k x ( γ − γ ) k y + k y ( γ − γ ) k x )] − k + ( γ + γ ) k z − k z γ k + ] } ˆ R = − √ (cid:8) k − ( γ + 2 γ ) k − − √ k z ( γ − γ ) k + + k + ( γ − γ ) k z ] (cid:9) ˆ C = − [ k z ( γ − γ − γ ) k − − k − ( γ − γ − γ ) k z ] (11)The H B Hamiltonian rotation requires a special care. It was obtained in by initiallydisregarding the effect of the remote bands and later enclosing it, replacing the mass bythe effective mass. Then, since we rotate the crystalline structure keeping the axes fixedand then the Bloch functions, the form of H B does not change. However, the effect of theremote bands does change, so that one should replace γ by γ in the expression of theeffectives masses. This modification must be introduced in H [001] B to reach H [111] B .To calculate the strain ǫ ij in H [111] strain , we take the elastic constants of [001] grown het-erostructures, C [001] , and rotate the axes. The resulting elastic constants C [111] are relatedto C [001] by: C [111] ijkl = X a,b,c,d M ia M jb M kc M ld C [001] abcd (12)Likewise, for the piezoelectric potential we rotate the axes and obtain: p i = X k e [111] ijk ǫ jk (13)with e [111] ijk = P a,b,c M ia M jb M kc e [001] abc .Hamiltonians (1) and (7) are solved numerically after obtaining the strain tensors andpiezoelectric fields using the Comsol . , except for the crystal density, dielectric constant and piezo-electric coefficient, which are obtained from Ref. . Linear interpolations are used for allalloys parameters. Luttinger parameters are inferred from the linearly interpolated masses.5 I. CHARACTER TABLE AND PRODUCT TABLE FOR THE DOUBLE GROUP ¯ C h The character table we use is:¯ C h E C +3 C − σ h S +3 S − ¯ E ¯ C +3 ¯ C − ¯ σ h ¯ S +3 ¯ S − basis A ′ J z E ′ + ω ω ∗ ω ω ∗ ω ω ∗ ω ω ∗ x + iyE ′− ω ∗ ω ω ∗ ω ω ∗ ω ω ∗ ω x − iyA ′′ − − − − − − zE ′′ + ω ω ∗ − − ω − ω ∗ ω ω ∗ − − ω − ω ∗ J x + iJ y E ′′− ω ∗ ω − − ω ∗ − ω ω ∗ ω − − ω ∗ − ω J x − iJ y E − / − ω ω ∗ i − iω iω ∗ − ω − ω ∗ − i iω − iω ∗ J − / E / − ω ∗ ω − i iω ∗ − iω − ω ∗ − ω i − iω ∗ iω J / E / − ω ω ∗ − i iω − iω ∗ − ω − ω ∗ i − iω iω ∗ E − / − ω ∗ ω i − iω ∗ iω − ω ∗ − ω − i iω ∗ − iωE − / − i − i i − − − i i − iE / − − i i − i − − i − i i (14)where ω = e i π and J i is the i -th component of the angular momentum.The table of products of ¯ C h irreducible representations is:6 ′ E ′ + E ′− A ′′ E ′′ + E ′′− E − / E / E / E − / E − / E / A ′ A ′ E ′ + E ′− A ′′ E ′′ + E ′′− E − / E / E / E − / E − / E / E ′ + E ′− A ′ E ′′ + E ′′− A ′′ E − / E / E / E − / E − / E / E ′− E ′ + E ′′− A ′′ E ′′ + E − / E / E / E − / E − / E / A ′′ A ′ E ′ + E ′− E / E − / E − / E / E / E − / E ′′ + E ′− A ′ E / E − / E − / E / E / E − / E ′′− E ′ + E / E − / E − / E / E / E − / E − / E ′′− A ′ E ′− A ′′ E ′′ + E ′ + E / E ′′ + A ′′ E ′ + E ′− E ′′− E / E ′′− A ′ E ′ + E ′′ + E − / E ′′ + E ′′− E ′− E − / A ′′ A ′ E / A ′′ (15) III. SYMMETRY OF THE HAMILTONIAN AND THE WAVE FUNCTIONS The employed H BF Hamiltonian is mass-position-dependent. Actually, in our systemthe mass parameters are constant within every domain, and have a sudden jump at theedge between neighboring domains. For the sake of easiness, to discuss on symmetry, weconsider the Luttinger-Kohn constant-mass parameters H LK limit of H BF .The symmetry of the H LK Hamiltonian (eqs. 4, 10 employing constant mass parameters)including a triangular confining potential and an axial magnetic field is C . However, withinthe axial approximation ( γ = γ ) it reaches C h . The symmetries of their matrix elementoperators can then be calculated from the above character table and the expressions on eqs.5, 11 (assuming constant mass parameters):Γ H LK = A ′ E ′′− E ′ + E ′′ + A ′ E ′ + E ′− A ′ E ′′− E ′− E ′′ + A ′ . (16)Accordingly, the symmetry of the envelope bonding/anti-bonding ground state functions7ust be: A ′ ( b ) E ′′ + ( a ) E ′− ( b ) A ′′ ( a ) and A ′′ ( a ) E ′ + ( b ) E ′′− ( a ) A ′ ( b ) (17)where the labels ( a ) , ( b ) indicates the bonding/anti-bonding character of the components.These envelope components, combine with the Bloch functions yielding the wave function.The Bloch functions are built as the symmetry-adapted product of J = 1 angular momentumfunctions and the J = 1 / σ h allows to employ bonding ( χ ( σ h ) = 1) and anti-bonding ( χ ( σ h ) = − J = 1 angularmomentum functions. For example, employing anti-bonding angular momentum functionswe have: | / , / i = − √ | ( X + i Y ) ↑i | / , − / i = √ | ( X − i Y ) ↓i| / , / i = q | Z ↑i − √ | ( X + i Y ) ↓i | / , − / i = q | Z ↓i + √ | ( X − i Y ) ↑i (18)The bonding J = 1 angular momentum functions are like the antibonding where X, Y, and Z are replaced by J x , J y , and J z . As a result, the table of products allows us to determinethe symmetries of the Bloch functions: | / , / a i → E / | / , / b i → E − / | / , / a i → E − / | / , / b i → E / | / , − / a i → E / | / , − / b i → E − / | / , − / a i → E − / | / , / b i → E / (19)The Bloch functions symmetries required to combine with the envelope components eq. 17(left) yielding E / and E − / , and those required to combine with eq. 17 (right) yielding E / and E − / , are: E / E − / E / E − / and E − / E / E − / E / ; E − / E / E − / E / and E / E − / E / E − / (20)8 V. SPIN MIXING IN QDMS GROWN ALONG [001] In the paper we have considered triangular QDs grown along the [111] direction becausethey are formed naturally in that direction. The physics leading to hole spin mixing ishowever connected with the envelope symmetry, and does not depend on the crystal orien-tation. To illustrate this point, in Fig. 2 we plot the expectation value of the Bloch angularmomentum h J z i as a function of the interdot distance d for the upper Zeeman doublet ofGaAs/Al . Ga . As QDMs (states | i and | i ). All the parameters are taken as in Fig. 3(d)of the paper, except that now the QDM is grown along [001] -i.e. Hamiltonian H [001] insteadof H [111] -. As can be seen, the picture is qualitatively the same as that obtained for [111] - < Jz > FIG. 2: Calculated spin purity of states | i and | i for different interdot distances at the resonantelectric field for a triangular QDM grown along [001]. grown QDMs (Fig.3(d) in the paper). The main difference is that the critical distance d c ,where the bonding-antibonding reversal takes place and hole spin mixing is maximized, isnow shifted towards longer interdot distances. This is because the effective masses of HHalong [001] are lighter than those along [111]. Therefore, the tunneling is stronger.Indeed, if we take constant mass parameters in H [001] BF , eq. 4, we obtain the correspondingLuttinger-Kohn Hamiltonian: 9 [001] LK = − ˆ P + ˆ Q − ˆ S ˆ R − ˆ S † ˆ P − ˆ Q R ˆ R † P − ˆ Q ˆ S R † ˆ S † ˆ P + ˆ Q (21)with ˆ P ± ˆ Q = [( γ ± γ )( k x + k y ) + ( γ ∓ γ ) k z )] / R = [ −√ γ ( k x − k y ) + i √ γ k x k y ] / S = γ √ k x − i k y ) k z (22)Also, if we take constant mass parameters in H [111] BF , eq. 10, we obtain: H [111] LK = − ˆ P + ˆ Q − ˆ S ˆ R − ˆ S † ˆ P − ˆ Q R ˆ R † P − ˆ Q ˆ S R † ˆ S † ˆ P + ˆ Q (23)with ˆ P ± ˆ Q = ( γ ± γ )( k x + k y ) + ( γ ∓ γ ) k z ˆ R = − √ ( γ + 2 γ ) k − + √ √ ( γ − γ ) k + k z ˆ S = − √ √ ( γ − γ ) k + √ (2 γ + γ ) k − k z (24)By comparing ˆ P + ˆ Q in Eq. (22) with Eq. (24), one can note the different HH effectivemasses in the z direction: 1 / ( γ − γ ) vs 1 / ( γ − γ ).It is worth noting that the ˆ R operator in H [001] LK does not have C rotational symmetry(the C character table can be obtained from that of C h by just considering rotations andkeeping rows 1-3,7-8 and 12). However, for many III-V materials including GaAs, γ ≈ γ and one can approximate them both by ¯ γ = ( γ + γ ) / R and ˆ R † matrix elements,thus yielding H [001] LK with axial symmetry, that is reduced to C (or C h ) symmetry by thetriangular confining potential and the magnetic field. On the other hand, Hamiltonian H [111] LK does have C symmetry, but –as discussed in the paper– the axial approximation in both R and S matrix elements is needed to display exact C h symmetry.In short, in both crystallographic directions the Hamiltonian has approximate C h sym-metry, which becomes exact if the axial approximation is assumed. Similar considerationson the axial approximation hold for the strain terms.10 . SPIN MIXING IN INAS/GAAS QDMS Next, we consider InAs/GaAs QDMs grown along [001], similar to those obtained by self-assembled growth but with triangular (pyramidal) QD shape. Unlike for GaAs/AlGaAs,no interdot wire is present in this case. On the other hand, strain and piezoelectricity nowplay a significant role.Figure 3 shows the spin purity of the four fist hole states as a function of an externalelectric field. One can see that also in this case there is a strong spin mixing ( |h J z i| ≪ / J z = +3 / J z = − / - < J z > |2|3 |4 (b) -14-13-12-11 E ne r g y ( m e V ) 35 36 37F (kV/cm) |1 ( a) FIG. 3: (a) and (b): hole energy levels and Bloch angular momentum expectation value of atriangular InAs/GaAs QDM grown along [001], as a function of a vertical electric field. The insetsin (a) show the hole localization for each doublet. 11t is remarkable that the spin mixing here takes place for bonding and antibonding statessplit by more than 2 meV. Besides, the spin mixing takes place over a wide window of electricfields (wider than for GaAs). This is because of the stronger SOI of InAs as compared toGaAs. These results suggest that the eventual design of triangular InAs/GaAs QDMswould also form a promising system for hole spin manipulation. VI. PERTURBATIONAL ESTIMATE OF THE SPIN MIXING STRENGTH In this section we expand the discussion of Fig. 3 based on perturbation theory. The goalis to show that the spin in triangular QDMs is much stronger than in circular QDMs.The Hamiltonian describing the hole states in a QDM, be it H [001] or H [111] , can be splitas: H = H + H ′ . (25)Here H are the diagonal terms, whose eigenfunctions are single-band HH or LH states: | Ψ HH ⇑ ( X p ) i = f / ( X p ) , | Ψ LH ⇑ ( X p ) i = f / ( X p ) | Ψ LH ⇓ ( X p ) i = f − / ( X p ) , | Ψ HH ⇓ ( X p ) i = f − / ( X p ) (26)where f J z ( X p ) is the envelope function of X rotational symmetry and p parity. In turn, H ′ represents the off-diagonal terms of H , coming from H BF and H strain . This term isresponsible for the band coupling. Without loss of generality, because the symmetry of H BF and H strain is the same, in what follows we consider GaAs/AlGaAs QDMs, where H strain isnegligible. The analysis is further simplified replacing H BF by its constant mass analogue, H LK , Eqs. (21) or (23) within axial approximation. This leads to the following expressionfor H ′ : 12 ′ = − − S R − S † RR † S R † S † (27)Within this approximation the matrix element operators R and S are just proportional to k − and k − k z , respectively.Considering H ′ as a perturbative term, the mixing of states up to second order is given by: | Ψ (2) k i = X i = k X j = k h Ψ (0) i | H ′ | Ψ (0) j i E k − E i h Ψ (0) j | H ′ | Ψ (0) k i E k − E j ! | Ψ (0) i i (28)Note that the coupling between | Ψ HH ⇑ ( X p i i ) i and | Ψ HH ⇓ ( X p k k ) i via H ′ requires | Ψ LH ⇑ ( X p j j ) i as intermediate state, yielding the contribution: − h X p i i | S | X p j j ih X p j j | R | X p k k i ∆ E hh ∆ E lh (29)where ∆ E hh is the energy difference between the HH states | Ψ HH ⇓ ( X p k k ) i and | Ψ HH ⇑ ( X p i i ) i ,and ∆ E lh that between | Ψ HH ⇓ ( X p k k ) i and the LH state | Ψ LH ⇑ ( X p j j ) i .Alternatively, this coupling can also be achieved with | Ψ LH ⇓ ( X p j j ) i as intermediate state,yielding the contribution: h X p i i | R | X p j j ih X p j j | S | X p k k i ∆ E hh ∆ E lh . (30)The matrix elements in the numerator of Eqs. (29), (30) determine the selection rules inthe band coupling process. E.g. in the C h group, the matrix element operator S has E ′′− symmetry ( R has E ′ + ), see Eq. (16). Then, a totally symmetric A ′ h Ψ HH ⇑ | ground statemust couple, via S , with a | Ψ LH ⇑ i state of symmetry E ′′ + ( A ′ ⊗ E ′′− ⊗ E ′′ + = A ′ , otherwise theintegral is zero). Next, h Ψ LH ⇑ | with symmetry E ′′− (the complex conjugate of that of | Ψ LH ⇑ i )must couple, via R , with | Ψ HH ⇓ i of A ′′ symmetry ( E ′′− ⊗ E ′ + ⊗ A ′′ = A ′ , otherwise the integralis zero). Likewise, h Ψ HH ⇑ | of A ′ symmetry (note that A ′ and A ′′ are reals and then coincidewith their complex conjugates) can couple via R with | Ψ LH ⇓ i of E ′− symmetry. Then, h Ψ LH ⇓ | of E ′ + symmetry will couple, via S , with | Ψ HH ⇓ i of A ′′ symmetry ( E ′ + ⊗ E ′′− ⊗ A ′′ = A ′ ).The above reasonings lead us to define the C h allowed couplings, represented by thickvertical lines on the left side of Fig. 4. Blue and yellow arrows correspond to either con-tribution, eqs. (29) and (30). In the figure we have simplified the notation of the states in13q. (26) to ( X p , J z ), where X is the rotational symmetry of the envelope function ( A , E + and E − in C or M z in C ∞ ), p = ′ or ′′ represents the even/odd parity, and J z indicates thenon-zero component of the four-fold spinor.One can now compare with the case of circular QDMs, where the group is C ∞ h and theenvelope functions are labeled by M z and parity. In this group, the matrix element operator R ( S ) is even (odd) and has M z = − M z = − h Ψ HH ⇑ | ground state of0 ′ symmetry can couple via S ( − ′′ ) with a | Ψ LH ⇑ i of symmetry 1 ′′ . Then, h Ψ LH ⇑ | ofsymmetry − ′′ couple, via R ( − ′ ), with | Ψ HH ⇓ i of 3 ′′ symmetry (because − − ′′ ⊗ ′ ⊗ ′′ = ′ ). h Ψ HH ⇑ | ′ may also couple, via R ( − ′ ) with | Ψ LH ⇓ i ′ . In turn, h Ψ LH ⇓ | of − ′ symmetrycouples, via S ( − ′′ ), with | Ψ HH ⇓ i of 3 ′′ symmetry.Taking into account Fig. 4 and Eqs. (29) and (30) we can see that ∆ E hh in the denominatorinvolved in C h is quite smaller than that involved in C ∞ h and, therefore, the interactionshould be much stronger.As a matter of fact, the strong spin mixing at the resonant electric fields is a singularbehavior of triangular QDMs. In C nh symmetries with n > J z = − / 2) has different rotational symmetry symmetry than the first one ( J z = 3 / C ∞ h . Then, for similar reasons, the coupling betweenthe states belonging to the first two doublets are also forbidden. In the case of C h QDMthe symmetries of S and R are B g and A g respectively. Bonding (anti-bonding) molecularorbitals are of Ag ( B u ) symmetry. Then, the doublet antibonding ground state (bondingfirst excited state) are B u ⇑ , B u ⇓ ( A g ⇑ , A g ⇓ ). Therefore, any coupling among these fourstates is forbidden by symmetry, as can be easily checked with the help of Eqs. (29) and(30). ∗ Electronic address: [email protected]; URL: http://quimicaquantica.uji.es/ B. A. Foreman, Effective-mass Hamiltonian and boundary conditions for the valence bands ofsemiconductor microstructures, Phys. Rev. B , 4964 (1993). J. Planelles and J. I. Climente. Magnetic field implementation in multiband k . p Hamiltoniansof holes in semiconductor heterostructures, J. Phys.: Condens. Matter, , 485801 (2013). IG. 4: Diagram of single band hole energy levels under a longitudinal magnetic field for a QDMwith symmetry C h (left column) and C ∞ h (right column). The labels ( X p , J z ) indicate the sym-metry of the levels. X represents the rotational symmetry of the envelope function ( A , E + and E − in C and M z in C ∞ h ), p = ′ or ′′ represents the even/odd parity and J z indicates the non-zero component of the spinor (e.g. J z = 3 / J z = 1 / J. van Bree, A. Yu. Silov, P. M. Koenraad, M. E. Flatt´e, and C. E. Pryor, g factors anddiamagnetic coefficients of electrons, holes, and excitons in InAs/InP quantum dots, Phys. Rev.B , 165323 (2012). Q. Zhu, K. F. Karlsson, M. Byszewski, A. Rudra, E. Pelucchi, Z. He, and E. Kapon, Hy-bridization of Electron and Hole States in Semiconductor Quantum-Dot Molecules, Small ,329 (2009). F. Michelini, M. A. Dupertuis, and E. Kapon, Novel artificial molecules: Optoelectronic prop- rties of two quantum dots coupled by a quantum wire, in Proceedings of the 14th InternationalWorkshop on Computational Electronics , Pisa, 2010, edited by G. Basso and M. Macucci, p.255. Q. Zhu, J. D. Ganiere, Z. B. He, K. F. Karlsson, M. Byszewski, E. Pelucchi, A. Rudra, andE. Kapon, Pyramidal GaAs/AlzGa1-zAs quantum wire/dot systems with controlled heterostruc-ture potential, Phys. Rev. B , 165315 (2010). J.B. Xia, Effective-mass theory for superlattices grown on (11N)-oriented substrates, Phys. Rev.B , 9856 (1991). I Vurgaftman, JR Meyer, and LR Ram-Mohan, Band parameters for III-V compound semicon-ductors and their alloys, J. Appl. Phys. , 5815 (2001). O. Madelung, Semiconductors: Data Handbook , (Springer-Verlag, Berlin, 2004). L.C Lew Yan Voon and M. Willatzen, The kp method: electronic properties of semiconductors ,(Springer Science & Business Media, 2009). E.A Stinaff, M Scheibner, A.S Bracker, I.V Ponomarev, V.L Korenev, M.E Ware, M.F Doty,T.L Reinecke, and D Gammon, Optical signatures of coupled quantum dots, Science , 636(2006). The spin admixture we report is proportional to the extradiagonal elements effective mass, γ and γ . The value of these parameters is in turn modulated by the strength of the spin-orbitinteraction. Then, the extent of the spin admixture is ultimately due to spin-orbit interaction.. The value of these parameters is in turn modulated by the strength of the spin-orbitinteraction. Then, the extent of the spin admixture is ultimately due to spin-orbit interaction.