Topology driven g-factor tuning in type-II quantum dots

J. M. Llorens,1 V. Lopes-Oliveira,2 V. Lopez-Richard,3 E. R. Cardozo de Oliveira,3 L. Wewior,1

J. M. Ulloa,4 M. D. Teodoro,3 G. E. Marques,3 A. Garcıa-Cristobal,5 G.-Q. Hai,6 and B. Alen1

1Instituto de Micro y Nanotecnologıa, IMN-CNM,CSIC (CEI UAM+CSIC) Isaac Newton, 8, E-28760, Tres Cantos, Madrid, Spain

2Universidade Estadual de Mato Grosso do Sul, Dourados, Mato Grosso do Sul 79804-970, Brazil3Departamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos, Sao Paulo 13565-905, Brazil

4Institute for Systems based on Optoelectronics and Microtechnology (ISOM),Universidad Politecnica de Madrid, Ciudad Universitaria s/n, 28040 Madrid, Spain

5Instituto de Ciencia de Materiales (ICMUV), Universidad de Valencia, Paterna E-46980, Spain6Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, Sao Paulo 13560-970, Brazil

(Dated: April 18, 2019)

We investigate how the voltage control of the exciton lateral dipole moment induces a transitionfrom singly to doubly connected topology in type-II InAs/GaAsSb quantum dots. The latter causesvisible Aharonov-Bohm oscillations and a change of the exciton g-factor which are modulated bythe applied bias. The results are explained in the frame of realistic k · p and effective Hamiltonianmodels and could open a venue for new spin quantum memories beyond the InAs/GaAs realm.

I. INTRODUCTION

III-V semiconductor quantum dots (QDs) are a fun-damental resource for quantum optical information tech-nologies, from integrated quantum light sources to quan-tum optical processors.1 Within this broad field, mostknowledge arises from the InAs/GaAs system, with manyother compounds being still relatively unexplored. III-SbQDs and rings are a good example. They can be grownby several epitaxial methods and can strongly emit in allthe relevant telecom bands, yet, III-Sb infrared quantumlight sources are to be developed. III-Sb compounds alsohave the largest g-factor and spin-orbit coupling (SOC)constant of all semiconductors but, with the exception ofInSb nanowires,2 this advantage has yet to be exploited inspin based quantum information technologies. One ad-vantage of III-V-based quantum nanostructures is thatoptical initialization and read-out can be done in a fewnanoseconds using, for instance, the singly charged exci-ton lambda system with electrons3 or holes4. Using thehole brings the additional benefit of its p-orbital charac-ter, meaning that the overlap between the atomic orbitalsand the nucleus is significantly smaller than for the s-likeelectrons, diminishing the decoherence caused by the hy-perfine interaction.4,5

For quantum information processing in the solid state,the possibility to modify the g-factor through an externalbias is its most outstanding resource.6 Thanks to spin or-bit coupling terms in the Hamiltonian, the application ofan electric field can modify the g-factor magnitude andsign through changes in the orbital part of the QD wave-functions7–9. A similar effect can be obtained changingthe elastic strain around the QD.10 Since the g-factor andwavefunctions are anisotropic, the observed modulationis different in Faraday11–14 and Voigt15 configurations.With InAs/GaAs QDs, external bias in the range of tensof kV cm−1 is necessary and, to prevent electron tunnel-ing out of the QD, the introduction of blocking barriers is

advisable.16,17 Alternately, one could use semiconductornanostructures with larger spin orbit coupling constantsto produce larger modulations at lower bias.

p-GaAs

i-GaAs

n-GaAs

Doubly-connected topologySingly-connected topology

FORWARDREVERSE

+

-+

-

Fz

Fz

FIG. 1. Depiction of the p-i-n diode operational principle act-ing on the hole wave function. Depending on the polarizationof bias the applied electric field (Fz) points upwards (left) ordownwards (right) changing the topology of the wave functionfrom singly-connected to doubly-connected. The size of theelements is not to scale.

Meanwhile, quantum nanostructures with a ring-shaped topology for electrons, holes or both and type-Ior type-II confinement have been the object of intensetheoretical and experimental research.18–28. When twoparticles travel around these quantum rings (QR) cir-cumventing a magnetic flux, the Aharonov-Bohm (AB)effect changes the relative phase factor of their wave func-tions.29 The optical Aharonov-Bohm effect (OABE) canbe detected at optical frequencies through the relativechanges in the electron and hole orbital angular mo-menta. At zero magnetic field, electron-hole exchangeinteraction typically leads to a dark ground state dou-blet split by few hundred µeV from the bright excitondoublet. With increasing applied magnetic field, higherenergy states with non-zero orbital angular momentumcross this quadruplet becoming the new ground state ofthe system. These states are hence optically forbidden

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and must produce a fade-out of the emitted intensity.30

In actual samples such a reduction on the emitted lightis not observed because of the reduced symmetry of thesystem, i.e. the orbital angular momentum is not a goodquantum number any longer. As a consequence, oscil-lations in energy and intensity are observed instead ofa quench.30 The evolution of charged excitons becomesmore complex in either case.31–33

Given the different orbital confinement found in singlyand doubly connected potentials, nanostructures withelectrically tunable topology might be of great interest tocouple spin and orbital degrees of freedom.33,34 In the fol-lowing, we investigate such possibility focusing on type-II InAs/GaAsSb QDs grown on GaAs and embedded ina p-i-n diode structure. For Sb molar fractions beyond16%, the band alignment changes from type-I to type-II, with the electron confined inside the InAs and thehole delocalized in the GaAsSb overlayer.35–37. The re-sulting wave function configuration brings two importantassets. Firstly, the weak localization of the hole results ina larger exciton polarizability or, in other words, a highertunability of the exciton energy. Voltage control of theexciton dipole moment in the vertical direction thus al-lows large tuning of the radiative lifetime.34,38 Secondly,we shall see that, pushing or pulling the hole against thebounding interfaces of the overlayer, also brings a topo-logical change in the hole ground state wave function asschematically shown in Figure 1. By this manipulationwe are in fact engineering the wave function density ofprobability from dot- to ring-like as originally discussedin Ref. 23

The rest of the paper is organized as follows. Thedetails of the sample and optical characterization aresummarized in Section II. We present in Section III themagneto-optical photoluminescence results. We first findthat both, the diamagnetic coefficient and the g-factor,can be modulated by the applied bias. Then, we reportoscillations of the emission intensity and the degree of cir-cular polarization (DCP ) with the applied magnetic fieldwhich occur only under forward bias. We discuss theseeffects in Section IV, relying on two different theoreticalmodels. The first one is described in Section IV A. Itconsists of a multiband k · p axisymmetric model whichprovides a quantitative evolution of the orbital relatedeffects: the diamagnetic coefficient dependence and theintensity oscillation. The limitation of the axial symme-try of this model is complemented with an effective massmodel in Section IV B. It is based on a parabolic con-finement which can analytically evolve from a dot-like toa ring-like potential and includes an eccentricity param-eter. From the solutions of this model, we discuss thespin related results (g-factor and DCP ) in Section IV Cby including the Rashba contribution in the Hamiltonian.

II. EXPERIMENTAL DETAILS

A. Sample and device

A p-i-n diode was grown by molecular beam epitaxy(MBE) on a n-type GaAs (001) substrate. The intrin-sic GaAs region spans 400 nm embedding in its center asingle layer of self-assembled InAs QDs. After the for-mation of the QDs, they were covered by a 6-nm-thickGaAsSb layer with 28% nominal content of Sb. Moredetails about the growth recipe and the morphologicalchanges induced by rapid thermal annealing (RTA) treat-ment of these QDs can be found elsewhere.40 After theRTA treatment, mesas of different sizes and ohmic con-tacts were defined by conventional optical lithographytechniques.

B. Optical characterization

To investigate OABE in our sample, magneto-photoluminescence (MPL) spectra were recorded at 5 Kin the Faraday configuration up to 9 T. To avoid un-necessary manipulations, the light polarization was an-alyzed in the circular basis reversing the magnetic fielddirection. After the circular polarization analyzer, theemission was coupled into a multimode optical fibre andthen imaged into the 100-µm-wide slits of a 300 mm fo-cal length monochromator (1200 lines/mm grating). Thelight intensity was measured with a peltier cooled In-GaAs photomultiplier connected to a lock-in amplifier.For every magnetic field, a PL spectrum was recordedfor V (V) = −2,−1, 0, 1 and 1.35 (corresponding to 88,63, 38, 13 and 4.2 kV cm−1, respectively). V < 0 (>0) corresponds to reverse (forward) bias as indicated inFig. 1 The power of the temperature stabilized 690 nmdiode laser was registered simultaneously being its valueconstant within ±4% during the whole experiment andwithin ±0.2% while scanning the voltage at fixed mag-netic field. Each spectrum was normalized by its corre-sponding excitation power and then analyzed by Gaus-sian deconvolution.

III. BIAS DEPENDENTMAGNETO-PHOTOLUMINESCENCE

EXPERIMENTS

Figure 2(a-e) shows the evolution of the σ−-polarizedMPL spectra in vertical electrical fields (Fz) between 88and 4.2 kV cm−1. To highlight the magnetic field effects,the lower panels of Figure 2 show the evolution after sub-traction and normalization by the spectrum at 0 T. Inour previous work, we investigated the electric field re-sponse at 0 T of the type-II InAs/GaAsSb QD system.34

A large reverse bias increases the electron tunneling rateout of the InAs quantum dot potential towards the GaAsbarrier. This leads to a noticeable intensity quenching

3

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88 kVcm-1 63 kVcm-1 38 kVcm-1 13 kVcm-1 4.2 kVcm-1

a

f

b

g

c

h

d

i

e

j

DI s-

(%)

I s-

(arb

. un

its)

Energy (eV)

0 T

9 T

FIG. 2. (a-e) Evolution of the σ−-polarized magneto-photoluminescence in vertical electrical fields between 88 and 4.2 kVcm−1. To highlight the magnetic field effects, (f-j) show the evolution after subtraction and normalization by the spectrum at0 T. Corresponding figures for σ+ detection and details about the Gaussian deconvolution procedure are given in Ref. 39

moving to the left in the upper panel of Figure 2. Theelectric field also reduces the electron-hole overlap andred shifts the peak emission energy at 0 T. In the fol-lowing, we focus on the changes produced by Fz in themagnetic response of the sample. To maintain an unifieddescription, the excitation conditions were kept approxi-mately the same in the experiments of Ref. 34 and here.Thus, the PL emission comprises two bands which aresplit by ≈36 meV at 0 V and 0 T. These bands arisefrom the ground state and bright excited state recombi-nations, both inhomogeneously broadened. Their inte-grated intensity ratio varies with Fz and is always largerthan 7.

The magnetic properties will be discussed attendingsolely to the evolution of the PL near the ground stateof the system. For every Fz, the σ+ and σ− MPL spec-tra were recorded up to 9 T. Gaussian deconvolution wasthen applied to extract the overall evolution of the groundstate emission. From the deconvoluted MPL peak energy,Eσ, and integrated intensity, Iσ, we obtain the averageenergy shift, Eα = (Eσ+ + Eσ−)/2, unpolarized inte-grated intensity, I = (Iσ+ + Iσ−)/2, Zeeman splitting,∆Ez = Eσ+ − Eσ− , and degree of circular polarization,DCP = (Iσ+−Iσ−)/(Iσ+ +Iσ−) for the low energy bandalone.

Figure 3 gathers the diamagnetic energy shift ∆Eα =Eα(B) − E(0) and Zeeman splitting for different elec-tric fields. Solid lines are quadratic and linear fittingsto ∆Eα(B) = αDB

2 and ∆Ez(B) = µBgB, respectively,where µB is the Bohr magneton, and αD and g repre-

sent the diamagnetic shift coefficient and Lande g-factorfor the type-II exciton, respectively.41 Both functions de-scribe accurately the data, except for a linear deviationfor ∆Eα in the strong tunneling regime corresponding torelatively large electric fields in Figs. 3(a-b). We observethat both αD and g are affected by the external electricfield and also that oscillations in magnetic field are ab-sent for the ground state peak energy. In this bias range,the g-factor variation is ≈ 80% and shall be attributedto a voltage modulation of the spin-orbit interaction asit will be discussed in detail in Section IV C.

The αD variation is even more pronounced and ex-hibits a clear diminishing trend reducing the reverse bias.For a magnetic field applied in the growth direction, thediamagnetic shift is usually associated with the excitonwave function extension in the growth plane.42 However,this simple correlation between exciton diameter and dia-magnetic shift magnitude is lost when electronic levelswith different angular momentum cross each other in thefundamental state. These level crossings are the sourceof the interference between different quantum paths lead-ing to AB magneto-oscillations in quantum rings.18 In anensemble experiment, they might be obscured by inhomo-geneous effect but might reveal themselves as a flatteningof ∆Eα(B) and the observed evolution of αD vs. Fz. Inour case, the tunneling also becomes more important asthe positive field increases. In these conditions, the elec-tron wave function penetrates in the barrier and mightalso affect the value of αD. These issues shall be dis-cussed in detail in the next sections.

4

0 2 4 6 80 2 4 6 80 2 4 6 8

0 2 4 6 8

0 2 4 6 8

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92.3 (0.3)

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38.6 (0.3)

-1.27 (0.04)

33.62 (0.06)

-1.07 (0.04)

32.03 (0.01)

-1.27 (0.03)

63 kVcm-1 38 kVcm-1 13 kVcm-1 4.2 kVcm-1DE

z(m

eV

)DE

a(m

eV

)

Magnetic Field (T)

a b c d e

f g h i j

FIG. 3. (a-e) Dependence of the diamagnetic shift on the ap-plied electric field. Solid lines stand for parabolic fits obtainedwith the indicated diamagnetic shift coefficients, αD. (f-j) Ex-perimental Zeeman splittings (symbols) and corresponding g-factors extracted from their linear dependence (solid lines).Shaded areas correspond to the standard errors arising fromgaussian deconvolution of the emission band.

The evolution of the unpolarized integrated intensityafter subtraction and normalization by the 0 T value, ∆I,and the DCP are depicted in Figure 4. At large electricfields, ∆I increases with B exhibiting a strong magneticbrightening up to 75% [Figure 4(a)]. The dependencefollows a fixed linear slope with small oscillations whichare only apparent in the first derivative of the data (notshown). By reducing Fz, the magnetic brightening effectdiminishes and, below 13 kV cm−1, an oscillatory behav-ior develops [Figure 4(b)]. In the same bias range, theDCP is positive, as expected from the negative g-factor,and evolves with magnetic field from a rather monotonicdependence at 83 kV cm−1 to a clearly oscillatory one at4.2 kV cm−1.

IV. DISCUSSION

In the quantum ring literature, MPL intensity oscilla-tions are related to changes in the ground state angularmomentum arising from the OABE.18 Voltage modula-tion of such oscillations have been reported for type-IInAs/GaAs single quantum rings,43,44 and also predictedfor 2D materials.45 The modulation must be associatedto changes in the effective quantum ring confinement po-tential. In the type-II InAs/GaAsSb QD system, holesare spatially localized within the GaAsSb layer near theInAs QDs.34,37,46–49 The strain distribution and electron-hole Coulomb interaction create a net attractive poten-tial for the hole that is balanced by the InAs/GaAsSband GaAs/GaAsSb valence band offsets. An externalelectric field modulates the geometry of this potentialand, in turn, a change of the hole wave function topol-

FIG. 4. Evolution of the unpolarized integrated intensity anddegree of circular polarization in the high (a, c) and low (b,d) bias regime. The top panels represent the qualitative evo-lution of the hole probability density between both regimes.Shaded areas correspond to the standard errors arising fromgaussian deconvolution of the emission band.

ogy from singly to doubly connected is expected. Theinset in Figure 4 represents the qualitative evolution ofthe hole probability density according to the theoreticaldescription developed on Ref. 34.

A. Axially symmetric k · p model

To analyze our data quantitatively, we start with anaxially symmetric model for the type-II InAs/GaAsSbQD system. Under this approximation, the total angularmomentum is a good quantum number for the excitonstates and allows a proper labeling scheme. The wholek ·p Hamiltonian including the strain is forced to be ax-ially symmetric. Hence, even in the multiband case, thestates are labeled according to the z-projection of the to-tal angular momentum (M). In case of small band mix-ing, the state can also be labeled with the z-projection ofthe orbital angular momentum of the envelope functionassociated with the main Bloch component (m). Fur-ther details and explicit expressions can be found in theAppendix A.

The numerical results are obtained for a quantum dotof the same size and composition as those analyzed inour previous studies34,40. The geometry of the InGaAsQD is a lens shape of radius RQD = 11 nm and heightHQD = 3 nm. It sits on top of a 0.5 nm InGaAs wettinglayer (WL) and is surrounded by a GaAsSb shell of 6nm thickness which plays the role of the overlayer. Thecomposition of the nanostructure is Ga0.25In0.75As andthat of the overlayer GaAs0.80Sb0.20.

Figure 5 shows the results in the exciton picture. Eachelectron (hole) state is defined by the quantum num-

5

0 2.5 5 7.5

B (T)

0 2.5 5 7.5

B (T)

0 2.5 5 7.5

B (T)

-2

0

2

4

6

8

0 2.5 5 7.5

E (

me

V)

B (T)

0 2.5 5 7.5

B (T)

40 kV/cm

10 kV/cm -30 kV/cm60 kV/cm115 kV/cm

FIG. 5. Exciton energy levels relative to the ground state energy at B = 0 T. Blue lines are states with M = −1, red linesM = 1, light-red lines M > 1, light-blue M < −1 and black M = 0. The spot size is proportional to the radiative rate and isbrought to the same scale throughout the pannels. The contour plot on top of each plot depicts the probability density of thehole ground state at B = 0 T. The axes dimensions are 50 nm and 12.5 nm.

ber Me(h) introduced in Eq. (A2). Hence, the excitonstates are defined by the addition of the total angularmomentum of the constitutive particles M = Me + Mh

(z-projections). The only optically active exciton statesare those characterized by M = 0,±1. The emitted pho-tons are polarized along the z direction for M = 0 andcircularly polarized σ± for M = ±1. We have selectedfive values of the electric field which fully cover the topo-logical transition from dot to ring. For clarity, we haveadded a contour plot of the hole probability density ontop of each energy level dispersion. One can easily dis-tinguish the localization of the hole particle on top of thenanostructure (115 kV cm−1) and its drift towards thebase as the electric field diminishes. The correspondingexciton states are indicated only for the central panel asXM,n, being M = 0,±1,±2, . . . the exciton total angu-lar momentum z-projection and n = 1, 2, . . . the principalquantum number of that particular M manifold. We firstfocus on the dispersion for Fz = 115 kV cm−1 (large re-verse bias). At B = 0 T, the ground state is composed ofthe fourth-fold degenerated quadruplet with n = 1 andM = ±1 and ±2. The magnetic field splits this level intofour branches. The M = ±1 can be distinguished by thenon-zero radiative rate (dots over the line) and more in-tense colored lines used in the representation. These fourexciton states result of the combination of two electron

and two hole states whose dominant Bloch amplitudesare |1/2,±1/2〉 and |3/2,±3/2〉. The envelope functionof each dominant Bloch amplitudes is defined by m ≈ 0.This explains the weak dependence of the internal struc-ture of the quadruplet to the electric field and hence tothe topology of the hole wave function. Irrespective ofwhether the hole is located above the apex of the QD(F = 115 kV cm−1) or close to its base (F = −30 kVcm−1) the energy levels are weakly perturbed. On thecontrary, the quantization energy of the excited statesis significantly perturbed by the electric field. The firstexcited state at B = 0 T is out the plot for F = 115kV cm−1, while at F = −30 kV cm−1 the energy split-ting with the ground state is ≈ 1.5 meV. This clearlyillustrates how the electric field efficiently modifies theangular and radial confinement exerted on the hole wavefunction. In theses states, the dominant envelope wavefunctions of the hole is characterized by |m| > 0 andtherefore are more sensitive to the vertical position withinthe overlayer.

Figure 5 illustrates that the applied electric field caneffectively isolate the ground state from the excited statesfor F > 60 kV cm−1 or bunch excited and ground statestogether for F < 60 kV cm−1. In the bunching regime,the model predicts the crossing between states of differ-ent M , and thus a change in the ground state orbital

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Dia

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ic c

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Electric Field (kV/cm)

αXαeαh

FIG. 6. Diamagnetic coefficient for the electron (dashed red),hole (solid blue), and the exciton (thick-solid black) states.

confinement. More explicitly, the X3,1 high energy stateand the X1,1 and X2,1 low energy states cross at B = 5.4

and 7.8 T for F = 10 kV cm−1, respectively. These cross-ings occur at fields within the range of the experimentalvalues presented before. This would result in the possi-bility of observing experimentally, at least, the crossingbetween X3,1 and the states of lower energy and thusan associated optical AB oscillation. The precise cor-respondence with the experimental results is difficult toestablish given that its observation would depend on thethermalization of the excited carriers and the eventualrelaxation of the selection rules in the actual QDs, as ex-plained below. As expected for AB related effects, thenumber of crossings and their actual positions depend onthe exciton in-plane dipole moment which, in our case,decreases with the electric field. In view of these results,the oscillatory dependence observed for the unpolarizedintegrated intensity and DCP in Fig. 4 is a consequenceof a change in the total angular momentum of the groundstate and hence a signature of the OABE.

The inhomogeneous broadening of the MPL spectrain Sec. III prevents the observation of the individual en-ergy levels shown in Fig. 5, thus peak energy magneto-oscillations cannot be detected in our experiment.28 Thisnotwithstanding, our analysis of the emission energyshifts in Figures 3(a-e) revealed a three fold increase ofthe diamagnetic shift coefficient with the electric field.From the numerical results shown in Figure 5, we cancalculate this magnitude for the bright exciton states.The resulting coefficients for the electron αe, hole αh andexciton αX are shown independently in Figure 6.

The hole diamagnetic shift decreases from ∼15 to ∼5µeV/T2 by increasing the electric field from -30 to 115kV cm−1. This is expected since, during the hole drift

from the base towards the apex of the overlayer, its den-sity of probability shrinks decreasing αh (see contour plotinsets in Fig. 5).42 In the same bias range, αe goes fromfrom ∼23 to ∼32 µeV/T2, increasing with the reversebias. This is a consequence of the spill-over of the elec-tron wave function in the GaAs barrier underneath. ForFz > 60 kV cm−1, electron tunneling out of the InAsQD becomes noticeable.34 The numerical model catchesthis situation only approximately and, for the largestbias, the calculated αX is only 37 µeV/T2, while exper-imentally αD is 92.3 µeV/T2. Indeed, for Fz > 63 kVcm−1, the experimental dispersion becomes increasinglylinear. Linear dispersion is characteristic of the Landauregime. In our context, it corresponds to a weaker con-finement as a result of the delocalization of the electronwave function as the tunnel regime becomes dominant[see Figs. 3(a,b)]. In the opposite bias regime, the corre-spondence with the experimental values is much better.αD (32.62 µeV/T2) is found at F = 4.2 kV cm−1 whileαX renders 36.05 µeV/T2. As shown in Fig. 5, in thisbias range, the ground state of the electron-hole systemis composed mostly of states with angular momentum|M | > 1. Under the axially symmetric approximation,these states are purely dark and do not contribute toαX . If the symmetry of the actual confinement poten-tial is broken, they can achieve oscillator strength, lowerfurther the diamagnetic shift and produce AB intensityoscillation, as discussed in the next Section.

B. In plane asymmetry effects

It has been profusely reported that quantum dot elon-gation can take place after capping. The strain fieldsthat build up during this process provoke the anisotropicsegregation of In atoms leading to an eccentricity in-crease of previously cylindrical systems.50,51 Analogousprocess is also triggered during the synthesis of type-IIInAs/GaAsSb quantum dots which also might lead toanisotropic piezoelectric fields.48,52 Thus, it is a goal ofour following discussion to assess the effects of in-planeconfinement anisotropy in the electronic properties. Tothis end, we introduce an effective mass description ofboth the conduction and valence bands, and incorporatethe effects of confinement asymmetry for electrons andholes in a model that can emulate quantum dots andrings within the same framework, as well as the resultingRashba spin-orbit coupling fields arising from confine-ment and external fields.

The eigen-value problem for the conduction band willbe solved by expanding the corresponding wave functionsin the basis of the eigen-solutions of the following effectivemass Hamiltonian,

H =~2

2µ∗k2 + V0(z) + V0(ρ) +

g∗µB2

B · σ , (1)

with µ∗ the effective mass, k = −i∇+e/~A and the mag-netic field pointing along the growth z-direction, where

7

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0 2 4 6 8 1 00 . 0

0 . 5

1 . 0

C

BA

δ = 0

Energ

y (me

V)

( a )

M a g n e t i c F i e l d ( T )

( b )

m C = 0

m C = 2

m B = - 1 m A = 2

m A = 0m B = 1

|Cα m|2

FIG. 7. (a) Calculated energy levels in a quantum ring for thetop valence subband: solid curves correspond to an eccentricring with δ = 2 meV/100 nm2 while the dashed curves arecalculated for a cylindrical geometry with δ = 0. (b) Calcu-lated weight coefficients of the A, B and C energy levels in (a)as a function of the magnetic field strength. (c) Calculatedoscillator strength for the interband transitions involving theconduction band ground state and the admixed valence bandlevels A and C.

σ is the Pauli matrix vector, and A = B/2ρϕ. For theunperturbed basis, V (z) will be assumed as a rigid wallpotential profile while the in-plane confinement in polarcoordinates takes the form:53

V0(ρ) =a1

ρ2+ a2ρ

2 − 2√a1a2, (2)

which allows obtaining an exact solution that covers boththe quantum ring and the quantum dot confinements.The parameters a1 and a2 define the structure shape andfor a1 6= 0, a ring with radius RQR = (a1/a2)1/4 is ob-tained. In turn, by setting a1 = 0, a parabolic quantumdot with effective radius R2

QD = ~/(2π√

2a2µ∗) can beemulated. This potential allows us to describe effectivelythe change of topology on the hole as a result of the ap-plied electric field.

For the valence band basis, used to expand the Lut-tinger Hamiltonian eigen-solutions, we use an analo-gous separable problem yet assuming anisotropic effec-tive masses in Eq. (1), and replacing g∗/2 and σz, by−2κ and the angular momentum matrix for j = 3/2, Jz,respectively.

The symmetry constrains can be subsequently relaxedby reshaping the in-plane confinement in the followingway, V (ρ, ϕ) = V0(ρ) + δ · ρ2 cos2 ϕ. This is an exten-sion of the profile proposed in Refs. 54 and 55, where theterm controlled by the parameter δ determines the eccen-tricity of the outer rim of the confinement. The valuesof the eccentricity are given by e =

√1− a2/(a2 + δ),

for δ > 0, that corresponds to an elliptical shrinking,or e =

√1− (a2 + δ)/a2 for δ < 0, that leads to an el-

liptical stretching. Additionally, an electric field alongthe growth direction can be considered by inserting theterm eFz into V (z) that will couple the wave functioncomponents of different parity. The details on the solu-tion of the Schrodinger equation are in Appendix B. Theasymmetric solution is given by the expansion

Ψ =∑

α,n,m,l

Cαn,m,lψαn,m,l(ρ, ϕ, z), (3)

where α labels the basis functions at the Brillouin zonecenter in the Kane model (|1/2,±1/2〉, |3/2,±3/2〉 and|3/2,±1/2〉), m is the z-projection of the orbital angularmomentum, n is the radial quantum number and l is thevertical quantum number. Thus, the expansion coeffi-cients, Cαn,m,l, determine the spin character of the finalstate, namely, the degree of hybridization of the variousm components.

The eccentricity breaks the cylindrical symmetry. Fig-ure 7(a) shows the impact of the symmetry break on thevalence band energy levels for the first few states con-fined in a QR with RQR = 12 nm and height hQR = 7nm. For this calculation, we have omitted the spin split-ting. Dashed lines render the cylindrical case (δ = 0meV) mimicking the energy dispersion found in the pre-vious section. Solid lines stand for an eccentric ring withδ = 2 meV. The only relevant energy levels for the dis-cussion are represented with bold lines and labeled asA, B and C. The A and B levels cross at B=4 T, andthe A and C levels anti-cross at B=7.5 T. The mixingof different m-components reduces the effective diamag-netic shift of the valence band ground hybrid state. Thismixing is quantified in Fig. 7(b), where the expansion

coefficients∣∣Cα0,m,1

∣∣2 resulting from the diagonalizationof the valence band Hamiltonian have been depicted. Inthe symmetric case, the ground state experiences a sharpchange at the crossing from a m = 0 character to a m = 1(not shown). In contrast, in the eccentric case there areno discontinuities, reflecting a soft evolution of the wavefunction character with the magnetic field. The statescorresponding to the energy levels A and C swap theircharacter from m = 0 to m = 2 at the anti-crossing.The energy level B acquires a dominant m = 1 characterbeing insensitive to the crossing and anti-crossing.

8

0

40

80

120

0 2 4 6 8 10

-12

-8

-4

0

ΔI (

%)

Magnetic Field (T)

A

C

FIG. 8. Calculated oscillator strength for the interband tran-sitions involving the conduction band ground state and theadmixed valence band levels A and C (thin red and blacklines, respectively). Computed relative intensity ∆I with arate equations model (thick grey line). The electrons arecalculated for a QD of RQD=15 nm and hQD=7 nm. Theeccentricity is e=0.53.

The changes of the electronic states energy and char-acter modulate also the emission intensity, which can becalculated from the oscillator strength (OS). As the elec-tron ground state has m=0 character, the state B is op-tically inactive within the model, being its OS negligible.States A and C, however, show an exchange of oscillatorstrength as they approach the anti-crossing as shown inFigure 8. The smooth increase of the OS at high mag-netic fields is attributed to the magnetic brightening. Inthis situation, we can calculate the emission intensity ofthe system solving a simple rate equations model basedon Ref. 56 as explained in Ref. 39. The radiative ratesare parameters of the model and scale with the OS calcu-lated for the states A and C. For state B, the pure darkcondition is relaxed, since dark states must eventuallyrecombine due to hybridization processes not consideredhere (e.g. electron-hole exchange). As shown in Fig-ure 8, our model predicts an intensity oscillation around7 T with similar phase but slightly larger amplitude thanthe experimental evolution displayed in Figure 4(b). Itmust be noted that the position of the crossings andthe energy difference between states presents fluctuationsamong QDs. This shall lead to a shallower experimentaloscillation resulting from the ensemble average of manycurves like the ones shown in Figure 8.26

C. Spin-orbit coupling effects

We have just shown that the symmetry reduction in aquantum ring lowers the effective diamagnetic shift of theground state as the angular momentum states become in-termixed. In turn, including spin-orbit interaction terms,the electrical modulation of the lateral and vertical con-finement also changes the spin states.10–14,16,17,57 Ac-cording to the results presented in Fig. 3(f-j), the excitonenergy spin splitting increases with the device reversebias. Meanwhile, within the GaAsSb overlayer, the holewave function geometry and topology evolve. Both obser-

FIG. 9. Calculated first order correction of the conductionband Lande factor as function of the eccentricity for a quan-tum dot according to Eq. (4)

vations can be connected once the spin orbit interactionis incorporated to the model represented by Eqs. (1)-(3).The interaction is introduced through the Rashba contri-bution to the total Hamiltonian Hc

R = αcσ · (∇V × k),for the conduction band, or Hv

R = αvJ ·(∇V ×k), for thevalence band. The expression for the spin orbit Hamil-tonian assuming the confinement profile that includes allthe asymmetry terms and the resulting energy correc-tions are given in Appendix C.

In turn, we can separate the spin-orbit effects inducedby confinement and asymmetry into first or higher or-der contributions. The latter ones are produced by thecoupling of previously unperturbed levels and appearstrongly when these levels approach inducing (or enhanc-ing) anticrossings, mostly at higher fields. The first or-der terms appear already at vanishing fields and mayprovoke, for instance, the tuning of the effective Landefactor. We shall focus the discussion on these ones.

The first order correction to the conduction band Zee-man splitting induced by the spin-orbit coupling can becalculated exactly for the ground state of a quantum dotaccording to:58

g(1)c =

2m0αcµ∗R2

QD

[1 + sign(δ)

e2

2− [1 + sign(δ)] e2

]. (4)

This result is depicted in Fig. 9 and illustrates how theLande factor correction grows as the quantum dot vol-ume is reduced by shortening RQD or by shrinking thelateral confinement into an ellipse (δ > 0 corresponds toan elliptical shrinking).

For a quantum ring, such a modulation is not trivialand depends on the way the confinement shape is modi-fied as shown in Fig. 10. Note that by fixing a2 and in-creasing a1, the quantum ring radius increases as a resultof widening the inner rim [Fig. 10(a)]. Such a modulationof the confinement profile can either increase or decreasethe valence band Lande factor correction according to the

9

FIG. 10. (a) In-plane confinement profiles for fixed a2 andvarying a1. (b) Calculated first order correction of the Landefactor for fixed a2 and varying eccentricity as function of a1.(c)In-plane confinement profiles for fixed a1 and varying a2. (d)Calculated first order correction of the Lande factor for fixeda1 and varying eccentricity as function of a2.

value of the eccentricity and the sign of δ as displayed inFig. 10(b). For large enough rings, the eccentricity mayeven lead to an absolute reduction of the Lande factor.If in contrast, the parameter a1 is fixed while increas-ing a2, the confinement potential shrinks as a result ofthe reduction of the external radius. This condition isplotted in Fig. 10(c) while the corresponding first ordercorrection to the g-factor appears in panel (d). Such a re-duction of the quantum ring radius provokes a monotonicincrease of this correction factor regardless the value ofthe eccentricity.

Through this model, the non-monotonic increase ofthe g-factor displayed in Fig. 3 can be ascribed to thebias dependent shrinkage and symmetry reduction of thehole wave function. To assess the absolute values of theg-factor correction, and the relevance of antimonides inthat respect, we may contrast systems with different spin-orbit coupling: GaAs, GaSb and InSb, respectively. Forthe valence band of these materials, αGaAs

v = −14.62 A2,αGaSbv ≈ 3.3αGaAs

v , αInSbv ≈ 38αGaAs

v . Their calculationis detailed in Appendix C. According to these values, theunits used in Fig. 10 for the valence band Lande factorcorrections are -0.012, for GaAs, -0.038, for GaSb, and-0.442, for InSb. In turn, for the conduction band Landefactor correction, plotted in Fig. 9 the units are radiusdependent and for a pure InAs QD with RQD = 8 nmthey equal 1.6.

For electron-hole pairs confined in type-II nanostruc-tures, the unperturbed effective Zeeman splitting can be

obtained from, ∆E(0)c −∆E

(0)v = (g0

c + 6κ)µBB. In thecase of an electron in InAs and a hole in GaAs0.8Sb0.2,

the effective Lande factor is g(0)e−h = g

(0)c (InAs) +

6κ(GaAs0.8Sb0.2) = −3.62. By introducing the effectsof confinement, the Lande factor can be corrected as

ge−h = g(0)e−h + g

(1)c − g(1)

v . Note that according to thevalues of αc > 0 and αv < 0, the correction leads to aLande factor increase in relative terms towards more pos-itive values. For low fields, the electrons and holes areconfined in QD shapes with a1 = 0 and a2=5 meV/100nm2 for holes while RQD=8 nm for electrons. This leads

to Lande factor corrections of g(1)c =1.6 and g

(1)v =0.129

for electrons and holes, respectively, leading to a totalLande factor ge−h=-1.891. At higher electric fields thehole wave function moves downwards, where we can ex-pect a radius increase of an eccentric QR (R2

QR=√a1/a2)

with e = 0.53. This translates into the model by a grow-ing a1 from 0 to 10 meV·100 nm2, with a2=5 meV/100nm2, corresponding to the QR with RQR=12 nm charac-terized in Fig. 7. According to Fig. 10(b), this results in

an increase on the correction of g(1)v =0.172. In turn, the

electron is confined inside of the QD and it could moveupwards, with a smaller effective RQD=6.8 nm with re-spect to the situation of larger positive fields. In Eq. (4),

g(1)c is inversely proportional to RQD and therefore in-

creases its value to g(1)c =2.21. This translates into a total

Lande factor of ge−h=-1.23. The change in the estima-tion of the g-factor between low and high fields agreeswith the observed experimental change in g from -1.8 to-1.3.

V. CONCLUSIONS

In summary, we have presented experimental evi-dence on the tuning of the electronic structure of type-IIInAs/GaAsSb quantum dots under vertical electric andmagnetic fields. Induced by the external bias, the driftof the hole wave function in the soft confinement of theGaAsSb layer encompasses a wide range of geometricaland topological changes in the effective potential. Thesechanges cannot be easily induced in type-I systems andprovide a rich variety of insights on the way geometryaffects the electronic structure and thus the optical re-sponse. The observed effects have been studied with the-oretical approaches that include electronic confinement,strain fields and spin effects on the same footing. Inparticular, we report how the application of an externalbias tunes the hole confinement geometry independentlyof the electron. Under certain bias conditions, the holeconfinement topology changes and magnetic field oscil-lations are observed. The oscillations follow the orbitalquantization of the hole state and are thus susceptibleto the confinement size and eccentricity. Although fur-ther work is needed to put these ideas at work in single

10

InAs/GaAsSb nanostructures, this modulation of hole or-bital quantization, accompanied by the tuning of the ex-citon g-factor, might pave the way for the voltage controlof spin degrees of freedom as required by several quantumtechnologies.

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial sup-port from EURAMET through EMPIR program17FUN06-SIQUST, from Spanish MINEICO throughgrants TEC2015-64189-C3-2-R, MAT2016-77491-C2-1-R, EUIN2017-88844 and RYC-2017-21995, from Comu-nidad de Madrid through grant P2018/EMT-4308, andfrom CSIC through grants I-COOP-2017-COOPB20320and PTI-001. Support from Brazilian agencies is alsoacknowledged through FAPESP grant 2014/02112-3 andCNPq grant 306414/2015-5.

Appendix A: Details of the multi-bandaxisymmetric model

The electronic structure of the InAs/GaAsSb QDs iscomputed by the k ·p method folowing Trebin et al.59,60

notation. The final Hamiltonian is the result of addingthree contribution, the k · p term (Hk·p), the strain in-teractions (Bir-Pikus Hε) and the magnetic interactions(HB):

H = Hk·p +Hε +HB . (A1)

We have neglected the linear ki terms in the valence-valence interaction, the quadratic kikj and εij termsin the conduction-valence interaction and the kiεjk inthe strain-induced interactions. To simplify the analy-sis, we have imposed axial symmetry in all the termsinvolved in the Hamiltonian. This approximation alsoaffects the strain distribution forcing us to consider thematerials as elastically isotropic. Compact expressionsare obtained with the Eshelby’s inclusions method61 andFourier transform of the strain tensor62. With this for-malism we succeeded in explaining Raman frequencyshifts induced by the strain63,64 and strain distributionsextracted from middle energy ion scattering65 experi-ments. The solution of the problem is obtained consider-ing that only the band edges are discontinuous across ma-terial interfaces. To describe electrons and holes properlyin type-II QDs, we have decoupled the conduction andvalence bands in H. Hence, the electrons are describedby a single band model, while a 6×6 Hamiltonian is usedfor the holes.

Axial symmetry allows us to define a total angular mo-mentum of z component M , which is the result of addingthe corresponding components of the envelope functionorbital angular momentum (m) and Bloch’s amplitude

total angular momentum (jz): M = m + jz. Each elec-tronic state is described by the wave function

Ψ(M)(r) =

8∑

k=1

F(M)m,k (r)uk

=

8∑

k=1

1√2πeimφF (M)

m,k (ρ, z)uk,

(A2)

where uk is the Bloch amplitude of the k band at the

origin of the Brillouin zone and F(M)m,k (r) is the envelope

function associated to the k Bloch component. The enve-lope function in expressed cylindrical coordinates definedby a phase factor characterized by m and a two dimen-

sional component F (M)m,k (ρ, z).

The solution of the Schrodinger equation is obtainedby expanding Eq. (A2) in a complete basis. The basis isdefined by the eigenfunctions of a hard-wall cylinder:

F (M)m,k =

∑

α,µ

N (m)α Jm

(k(m)α ρ/R

)

×√

2/Z sin [µπ (z/Z − 1/2)] , (A3)

where Jm(x) is the Bessel function of order m, k(m)α is its

zero number α = 1, 2, . . ., µ = 1, 2, . . ., R and Z are theradius and height of the expansion cylinder and

N (m)α =

√2

R∣∣∣Jm+1

(k

(m)α

)∣∣∣(A4)

is the normalization of the radial part. This definitionsensure the orthonormality of the expansion basis. A sim-ilar procedure was followed by Tadic et al. in Refs. 66and 67. Further details can be found in Ref. 68.

In Figure 11, we show an outline of the QD embeddedin the overlayer enclosed by the expansion cylinder. Thenumber of geometrical parameters is therefore reduced tosix.

Appendix B: Details of the EMA non-axisymmetricmodel

The solution for the 3D Schrodinger equation,Φ(ρ, θ, z), corresponding to the potential profile in Eq. (2)is given by

ψ(e/h)n,m,l(ρ, ϕ, z) = φ(e/h)

n,m (ρ, ϕ)χ(e/h)l (z)ue/h, (B1)

where χ(e/h)l (z) is the wave function for a rigid square

well and ue/h = |j,mj〉 are the basis functions at theBrillouin zone center in the Kane model: |1/2,±1/2〉 and|3/2,±3/2〉 for the electron and heavy hole, respectively.

11

GaAsSb

InGaAs

GaAs

GaAs

FIG. 11. Depiction of the geometrical model to describe aQD of radius RQD and height hQD, a overlayer of thickness tand a wetting-layer of thickness dWL embedded in hard-wallcylinder of radius R and height Z.

The planar wave function has the form

φ(e/h)n,m (ρ, ϕ) =

1

λ(e/h)

(Γ[n+M(e/h) + 1]

2M(e/h)n!(Γ[M(e/h) + 1])2

)1/2

(B2)

×(

ρ

λ(e/h)

)M(e/h) e−imϕ√2π

e− 1

4

(ρ

λ(e/h)

)2

×1 F1

(−n,M(e/h) + 1,

1

2

(ρ/λ(e/h)

)2),

where 1F1 is the confluent hypergeometric function, n =0, 1, 2, . . . is the radial quantum number, l = 1, 2, . . . isthe vertical quantum number, and m = 0,±1,±2, . . . la-bels the angular momentum. The corresponding eigen-energies for the 3D problem are

E(e/h)n,m,l,sz

=

(n+

1

2+M(e/h)

2

)~ω(e/h)

− m

2~ω∗c(e/h) −

µ∗(e/h)

4ω0(e/h)

2RQR2

+

(l2π2~2

2µ∗(e/h)L2

)+ g∗(e/h)µBB · sz, (B3)

with M(e/h) =

√m2 +

2a1µ∗(e/h)

~2 , ω∗c(e/h) = eB/µ∗(e/h),

ω0(e/h) =√

8a2/µ∗(e/h), ω(e/h) =√ω2c(e/h) + ω2

0(e/h) and

λ(e/h) =√

~µ∗(e/h)

ω(e/h).

Appendix C: Details of the SOC non-axisymmetricmodel

The expression for the spin orbit Hamiltonian assum-ing the confinement profile that includes all the asymme-try terms is given by:

HcR = 2αcσz

[(−a1

ρ2+ a2ρ

2 + δρ2 cos2(ϕ)

)(eB

2c~− i

ρ2

∂

∂ϕ

)− iδ sin(ϕ) cos(ϕ)

(1 + ρ

∂

∂ρ

)]

−αc∂V

∂z

{σ+

[e−iϕ

(∂

∂ρ− i

ρ

∂

∂ϕ+eB

2c~ρ+

1

ρ

)]−σ−

[eiϕ(∂

∂ρ+i

ρ

∂

∂ϕ− eB

2c~ρ+

1

ρ

)]},

(C1)

with σ± = 1/2(σx ± σy), being σi the components ofthe Pauli matrices. The spin orbit Hamiltonian for thevalence band can be readily obtained by replacing thesematrices with those for angular momentum j = 3/2 andαc by αv.

We can extract from Eq. (C1) the diagonal terms thatcontribute to the first order renormalization of the spin-splitting of the ground state withm = 0 for the limitB →0, ∆Espin = ∆E(0) + ∆E(1), in the basis of unperturbedstates introduced by Eq. (B1). The zero order value isessentially the Zeeman splitting given by

∆E(0)c = g∗µBB, (C2)

in the case of the conduction band, while for the valenceband stands as

∆E(0)v = −6κµBB. (C3)

In the case of a conduction band electron confined withina QD potential, V = a2ρ

2 + δ · ρ2 cos2 ϕ, the first order

contribution of the spin-orbit interaction, in the limit oflow fields, can be reduced to

∆E(1)c =

4m0αc~2

× a2

{1 + sign(δ)

e2

2− [1 + sign(δ)] e2

}⟨ρ2⟩µBB.

(C4)

Meanwhile, for the valence band and a quantum ringprofile, V = a1/ρ

2 + a2ρ2 − 2

√a1a2 + δ · ρ2 cos2 ϕ, the

12

expression reads

∆E(1)v =

6m0αv~2

×[a2

{1 + sign(δ)

e2

2− [1 + sign(δ)] e2

}⟨ρ2⟩

−a1

⟨1

ρ2

⟩]µBB.

(C5)

This allows introducing the first order correction to

the Lande factor defined as g(1)c,v ≡ ∆E

(1)c,v/(µBB) and

shown in Fig. 9 and 10 in the article, respectively. Withthem, one may now assess the relative effect of the con-finement geometry and topology change on the spin split-ting. The calculation of the Rashba coefficient appearingin Eq.(C5) is computed from the band parameters in Ta-ble I and the expression:

αv = − eP2

3E20

+eQ2

9

[10

E′20− 7

(E′0 + ∆′0)2

], (C6)

which is defined as r8v8v41 in 60.

Appendix D: Material parameters

The material band parameters are extracted from69.Magnetic related parameters are shown in Table I. The

values of the compounds GaInAs and GaAsSb are ob-tained through linear interpolation when no bowing pa-rameters is reported.

GaAs InAs GaSb

g -0.44 -14.9 -9.25k 1.20 7.60 4.60q 0.01 0.39 0.00

E0 (eV) 1.519 0.418 0.813E′0 (eV) 4.488 4.390 3.3∆0 (eV) 0.341 0.380 0.750∆′0 (eV) 0.171 0.240 0.330P (eVA) 10.493 9.197 9.504P ′ (eVA) 4.780i 0.873i 3.326iQ (eVA) 8.165 8.331 8.121

TABLE I. Parameters related with the spin Zeeman effect andband parameters used in the computation of the Rashba coef-ficient. The values of GaAs and InAs are taken from Ref. 60(page 221). The Zeeman parameters of GaSb from Ref. 70(pages 486 and 491) and the band parameters from Ref. 71.

1 P. Michler, ed., Quantum Dots for Quantum InformationTechnologies, Nano-Optics and Nanophotonics (SpringerInternational Publishing, 2017).

2 S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg,K. Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov,and L. P. Kouwenhoven, Spectroscopy of Spin-Orbit Quan-tum Bits in Indium Antimonide Nanowires, Phys. Rev.Lett. 108, 166801 (2012).

3 M. Atature, J. Dreiser, A. Badolato, A. Hogele, K. Karrai,and A. Imamoglu, Quantum-Dot Spin-State Preparationwith Near-Unity Fidelity, Science 312, 551 (2006).

4 D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wust,K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. War-burton, A Coherent Single-Hole Spin in a Semiconductor,Science 325, 70 (2009).

5 M. Vidal, M. V. Durnev, L. Bouet, T. Amand, M. M.Glazov, E. L. Ivchenko, P. Zhou, G. Wang, T. Mano,T. Kuroda, X. Marie, K. Sakoda, and B. Urbaszek, Hy-perfine coupling of hole and nuclear spins in symmetric(111)-grown GaAs quantum dots, Phys. Rev. B 94, 121302(2016).

6 K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, andL. M. K. Vandersypen, Coherent Control of a Single Elec-tron Spin with Electric Fields, Science 318, 1430 (2007).

7 J. Pingenot, C. E. Pryor, and M. E. Flatte, Method forfull Bloch sphere control of a localized spin via a singleelectrical gate, Appl. Phys. Lett. 92, 222502 (2008).

8 T. Andlauer and P. Vogl, Electrically controllable g tensorsin quantum dot molecules, Phys. Rev. B 79, 045307 (2009).

9 J. Pingenot, C. E. Pryor, and M. E. Flatte, Electric-field manipulation of the Lande g tensor of a hole inan In0.5Ga0.5As/GaAs self-assembled quantum dot, Phys.Rev. B 84, 195403 (2011).

10 H. M. G. A. Tholen, J. S. Wildmann, A. Rastelli,R. Trotta, C. E. Pryor, E. Zallo, O. G. Schmidt, P. M.Koenraad, and A. Y. Silov, Strain-induced g-factor tuningin single InGaAs/GaAs quantum dots, Phys. Rev. B 94,245301 (2016).

11 F. Klotz, V. Jovanov, J. Kierig, E. C. Clark, D. Rudolph,D. Heiss, M. Bichler, G. Abstreiter, M. S. Brandt, andJ. J. Finley, Observation of an electrically tunable excitong factor in InGaAs/GaAs quantum dots, Appl. Phys. Lett.96, 053113 (2010).

12 V. Jovanov, T. Eissfeller, S. Kapfinger, E. C. Clark,F. Klotz, M. Bichler, J. G. Keizer, P. M. Koenraad, G. Ab-streiter, and J. J. Finley, Observation and explanation ofstrong electrically tunable exciton g factors in composi-tion engineered In(Ga)As quantum dots, Phys. Rev. B 83,161303 (2011).

13 N. Ares, V. N. Golovach, G. Katsaros, M. Stoffel, F. Four-nel, L. I. Glazman, O. G. Schmidt, and S. De Franceschi,Nature of Tunable Hole g Factors in Quantum Dots, Phys.Rev. Lett. 110, 046602 (2013).

13

14 P. Corfdir, Y. Fontana, B. Van Hattem, E. Russo-Averchi,M. Heiss, A. Fontcuberta i Morral, and R. T. Phillips, Tun-ing the g-factor of neutral and charged excitons confined toself-assembled (Al, Ga)As shell quantum dots, Appl. Phys.Lett. 105, 223111 (2014).

15 T. M. Godden, J. H. Quilter, A. J. Ramsay, Y. Wu,P. Brereton, I. J. Luxmoore, J. Puebla, A. M. Fox, andM. S. Skolnick, Fast preparation of a single-hole spin inan InAs/GaAs quantum dot in a Voigt-geometry magneticfield, Phys. Rev. B 85, 155310 (2012).

16 A. J. Bennett, M. A. Pooley, Y. Cao, N. Skold, I. Far-rer, D. A. Ritchie, and A. J. Shields, Voltage tunabilityof single-spin states in a quantum dot, Nat. Commun. 4,1522 (2013).

17 J. H. Prechtel, F. Maier, J. Houel, A. V. Kuhlmann,A. Ludwig, A. D. Wieck, D. Loss, and R. J. Warbur-ton, Electrically tunable hole g factor of an optically ac-tive quantum dot for fast spin rotations, Phys. Rev. B 91,165304 (2015).

18 V. M. Fomin, ed., Physics of Quantum Rings (SpringerBerlin Heidelberg, 2014).

19 A. Lorke, R. Johannes Luyken, A. O. Govorov, J. P. Kot-thaus, J. M. Garcia, and P. M. Petroff, Spectroscopy ofNanoscopic Semiconductor Rings, Phys. Rev. Lett. 84,2223 (2000).

20 A. Fuhrer, K. Ensslin, M. Bichler, S. Luscher, T. Heinzel,T. Ihn, and W. Wegscheider, Energy spectra of quantumrings, Nature 413, 822 (2001).

21 E. Ribeiro, A. O. Govorov, W. Carvalho, and G. Medeiros-Ribeiro, Aharonov-Bohm signature for neutral polarizedexcitons in type-II quantum dot ensembles, Phys. Rev.Lett. 92, 126402 (2004).

22 L. G. G. V. Dias da Silva, S. E. Ulloa, and T. V. Shah-bazyan, Polarization and Aharonov-Bohm oscillations inquantum-ring magnetoexcitons, Phys. Rev. B 72, 125327(2005).

23 N. A. J. M. Kleemans, I. M. A. Bominaar-Silkens, V. M.Fomin, V. N. Gladilin, D. Granados, A. G. Taboada, J. M.Garcıa, P. Offermans, U. Zeitler, P. C. M. Christianen,J. C. Maan, J. T. Devreese, and P. M. Koenraad, Os-cillatory Persistent Currents in Self-Assembled QuantumRings, Phys. Rev. Lett. 99, 146808 (2007).

24 I. L. Kuskovsky, W. MacDonald, A. O. Govorov,L. Mourokh, X. Wei, M. C. Tamargo, M. Tadic, and F. M.Peeters, Optical Aharonov-Bohm effect in stacked type-IIquantum dots, Phys. Rev. B 76, 035342 (2007).

25 I. R. Sellers, V. R. Whiteside, I. L. Kuskovsky, A. O. Gov-orov, and B. D. McCombe, Aharonov-Bohm excitons at el-evated temperatures in type-II ZnTe/ZnSe quantum dots,Phys. Rev. Lett. 100, 136405 (2008).

26 M. D. Teodoro, V. L. Campo Jr., V. Lopez-Richard,E. Marega Jr., G. E. Marques, Y. Galvao Gobato,F. Iikawa, M. J. S. P. Brasil, Z. Y. AbuWaar, V. G.Dorogan, Y. I. Mazur, M. Benamara, and G. J. Salamo,Aharonov-Bohm Interference in Neutral Excitons: Effectsof Built-In Electric Fields, Phys. Rev. Lett. 104, 086401(2010).

27 S. Miyamoto, O. Moutanabbir, T. Ishikawa, M. Eto, E. E.Haller, K. Sawano, Y. Shiraki, and K. M. Itoh, ExcitonicAharonov-Bohm effect in isotopically pure 70Ge/Si self-assembled type-II quantum dots, Phys. Rev. B 82, 073306(2010).

28 H. D. Kim, R. Okuyama, K. Kyhm, M. Eto, R. A. Tay-lor, A. L. Nicolet, M. Potemski, G. Nogues, L. S. Dang,

K.-C. Je, J. Kim, J.-H. Kyhm, K. H. Yoen, E. H. Lee,J. Y. Kim, I. K. Han, W. Choi, and J. Song, Observa-tion of a Biexciton Wigner Molecule by Fractional OpticalAharonov-Bohm Oscillations in a Single Quantum Ring,Nano Lett. 16, 27 (2016).

29 Y. Aharonov and D. Bohm, Significance of electromag-netic potentials in the quantum theory, Phys. Rev. 115,485 (1959).

30 A. O. Govorov, S. E. Ulloa, K. Karrai, and R. J. War-burton, Polarized excitons in nanorings and the opticalAharonov-Bohm effect, Phys. Rev. B Rapid Comm. 66,081309R (2002).

31 M. Bayer, M. Korkusinski, P. Hawrylak, T. Gutbrod,M. Michel, and A. Forchel, Optical Detection of theAharonov-Bohm Effect on a Charged Particle in aNanoscale Quantum Ring, Phys. Rev. Lett. 90, 186801(2003).

32 R. Okuyama, M. Eto, and H. Hyuga, Optical Aharonov-Bohm effect on Wigner molecules in type-II semiconductorquantum dots, Phys. Rev. B 83, 195311 (2011).

33 J. M. Llorens and B. Alen, Wave-Function Topology Ef-fects on Charged Excitons in Type-II InAs/GaAsSb Quan-tum Dots and Rings, Phys. Status Solidi RRL , 1800314(2018).

34 J. M. Llorens, L. Wewior, E. R. C. d. Oliveira, J. M. Ul-loa, A. D. Utrilla, A. Guzman, A. Hierro, and B. Alen,Type-II InAs/GaAsSb quantum dots: Highly tunable exci-ton geometry and topology, Appl. Phys. Lett. 107, 183101(2015).

35 K. Akahane, N. Yamamoto, and N. Ohtani, Long-wavelength light emission from InAs quantum dots coveredby GaAsSb grown on GaAs substrates, Physica E 21, 295(2004).

36 J. M. Ripalda, D. Granados, Y. Gonzalez, A. M. Sanchez,S. I. Molina, and J. M. Garcıa, Room temperature emis-sion at 1.6 µm from InGaAs quantum dots capped withGaAsSb, Appl. Phys. Lett. 87, 202108 (2005).

37 H. Y. Liu, M. J. Steer, T. J. Badcock, D. J. Mowbray,M. S. Skolnick, P. Navaretti, K. M. Groom, M. Hopkinson,and R. A. Hogg, Long-wavelength light emission and las-ing from InAs/GaAs quantum dots covered by a GaAsSbstrain-reducing layer, Appl. Phys. Lett. 86, 143108 (2005).

38 C. Heyn, A. Kuster, M. Zocher, and W. Hansen, Field-Controlled Quantum Dot to Ring Transformation in Wave-Function Tunable Cone-Shell Quantum Structures, Phys.Status Solidi RRL , 1800245 (2018).

39 See the supplemental material at https://journals.aps.org/prapplied for the MPL and rate equations details.

40 J. M. Ulloa, J. M. Llorens, B. Alen, D. F. Reyes, D. L.Sales, D. Gonzalez, and A. Hierro, High efficient lumines-cence in type-II GaAsSb-capped InAs quantum dots uponannealing, Appl. Phys. Lett. 101, 253112 (2012).

41 M. Hayne and B. Bansal, High-field magneto-photoluminescence of semiconductor nanostructures:High-field magneto-PL of semiconductor nanostructures,Luminescence 27, 179 (2012).

42 K. J. Nash, M. S. Skolnick, P. A. Claxton, and J. S.Roberts, Diamagnetism as a probe of exciton localizationin quantum wells, Phys. Rev. B 39, 10943 (1989).

43 F. Ding, N. Akopian, B. Li, U. Perinetti, A. Govorov,F. M. Peeters, C. C. Bof Bufon, C. Deneke, Y. H. Chen,A. Rastelli, O. G. Schmidt, and V. Zwiller, Gate controlledAharonov-Bohm-type oscillations from single neutral exci-tons in quantum rings, Phys. Rev. B 82, 075309 (2010).

14

44 B. Li and F. M. Peeters, Tunable optical Aharonov-Bohmeffect in a semiconductor quantum ring, Phys. Rev. B 83,115448 (2011).

45 G. O. de Sousa, D. R. da Costa, A. Chaves, G. A. Farias,and F. M. Peeters, Unusual quantum confined Stark effectand Aharonov-Bohm oscillations in semiconductor quan-tum rings with anisotropic effective masses, Phys. Rev. B95, 205414 (2017).

46 J. M. Ulloa, R. Gargallo-Caballero, M. Bozkurt, M. delMoral, A. Guzman, P. M. Koenraad, and A. Hierro,GaAsSb-capped InAs quantum dots: From enlarged quan-tum dot height to alloy fluctuations, Phys. Rev. B 81,165305 (2010).

47 J. M. Ulloa, J. M. Llorens, M. del Moral, M. Bozkurt, P. M.Koenraad, and A. Hierro, Analysis of the modified opti-cal properties and band structure of GaAs1−xSbx-cappedInAs/GaAs quantum dots, J. Appl. Phys. 112, 074311(2012).

48 P. Klenovsky, V. Krapek, D. Munzar, and J. Humlıcek,Electronic structure of InAs quantum dots with GaAsSbstrain reducing layer: Localization of holes and its effecton the optical properties, Appl. Phys. Lett. 97, 203107(2010).

49 A. Hospodkova, M. Zıkova, J. Pangrac, J. Oswald,J. Kubistova, K. Kuldova, P. Hazdra, and E. Hulicius,Type I-type II band alignment of a GaAsSb/InAs/GaAsquantum dot heterostructure influenced by dot size andstrain-reducing layer composition, J. Phys. D: Appl. Phys.46, 095103 (2013).

50 D. Alonso-Alvarez, B. Alen, J. M. Ripalda, A. Rivera,A. G. Taboada, J. M. Llorens, Y. Gonzalez, L. Gonzalez,and F. Briones, Strain driven migration of In during thegrowth of InAs/GaAs quantum posts, APL Mater. 1,022112 (2013).

51 M. D. Teodoro, A. Malachias, V. Lopes-Oliveira, D. F. Ce-sar, V. Lopez-Richard, G. E. Marques, E. Marega, M. Be-namara, Y. I. Mazur, and G. J. Salamo, In-plane mappingof buried InGaAs quantum rings and hybridization effectson the electronic structure, J. Appl. Phys. 112, 014319(2012).

52 V. Krapek, P. Klenovsky, and T. Sikola, Excitonic finestructure splitting in type-II quantum dots, Phys. Rev. B92, 195430 (2015).

53 W.-C. Tan and J. C. Inkson, Electron states in a two-dimensional ring - an exactly soluble model, Semicond. Sci.Technol. 11, 1635 (1996).

54 V. Lopes-Oliveira, Y. I. Mazur, L. D. de Souza, L. A. B.Marcal, J. Wu, M. D. Teodoro, A. Malachias, V. G. Doro-gan, M. Benamara, G. G. Tarasov, E. Marega, G. E. Mar-ques, Z. M. Wang, M. Orlita, G. J. Salamo, and V. Lopez-Richard, Structural and magnetic confinement of holesin the spin-polarized emission of coupled quantum ring-quantum dot chains, Phys. Rev. B 90, 125315 (2014).

55 V. Lopes-Oliveira, L. K. Castelano, G. E. Marques, S. E.Ulloa, and V. Lopez-Richard, Berry phase and Rashbafields in quantum rings in tilted magnetic field, Phys. Rev.B 92, 035441 (2015).

56 M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, H. Ebe,and H. Ishikawa, Quantum-dot semiconductor optical am-

plifiers for high-bit-rate signal processing up to 160 Gbs−1 and a new scheme of 3R regenerators, MeasurementScience and Technology 13, 1683 (2002).

57 L.-W. Yang, Y.-C. Tsai, Y. Li, A. Higo, A. Murayama,S. Samukawa, and O. Voskoboynikov, Tuning of the elec-tron g factor in defect-free GaAs nanodisks, Phys. Rev. B92, 245423 (2015).

58 L. Cabral, F. P. Sabino, V. Lopes-Oliveira, J. L. F.Da Silva, M. P. Lima, G. E. Marques, and V. Lopez-Richard, Interplay between structure asymmetry, defect-induced localization, and spin-orbit interaction in Mn-doped quantum dots, Phys. Rev. B 95, 205409 (2017).

59 H. R. Trebin, U. Rossler, and R. Ranvaud, Quantum reso-nances in the valence bands of zinc-blende semiconductors.I. Theoretical aspects, Phys. Rev. B 20, 686 (1979).

60 R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer BerlinHeidelberg, 2003).

61 J. D. Eshelby, The determination of the elastic field of anellipsoidal inclusion, and related problems, Proc. R. Soc.Lond. A 241, 376 (1957).

62 A. D. Andreev, J. R. Downes, D. A. Faux, and E. P.O’Reilly, Strain distributions in quantum dots of arbitraryshape, J. Appl. Phys. 86, 297 (1999).

63 A. Cros, N. Garro, J. M. Llorens, A. Garcıa-Cristobal,A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin,Raman study and theoretical calculations of strain in GaNquantum dot multilayers, Phys. Rev. B 73, 115313 (2006).

64 N. Garro, A. Cros, J. M. Llorens, A. Garcıa-Cristobal,A. Cantarero, N. Gogneau, E. Sarigiannidou, E. Mon-roy, and B. Daudin, Resonant Raman scattering in self-assembled GaN/AlN quantum dots, Phys. Rev. B 74,075305 (2006).

65 D. Jalabert, J. Coraux, H. Renevier, B. Daudin, M.-H.Cho, K. B. Chung, D. W. Moon, J. M. Llorens, N. Garro,A. Cros, and A. Garcıa-Cristobal, Deformation profile inGaN quantum dots: Medium-energy ion scattering ex-periments and theoretical calculations, Phys. Rev. B 72,115301 (2005).

66 M. Tadic, F. M. Peeters, and K. L. Janssens, Effect ofisotropic versus anisotropic elasticity on the electronicstructure of cylindrical InP/In0.49Ga0.51P self-assembledquantum dots, Phys. Rev. B 65, 165333 (2002).

67 M. Tadic and F. M. Peeters, Intersublevel magnetoabsorp-tion in the valence band of p-type InAs/GaAs and Ge/Siself-assembled quantum dots, Phys. Rev. B 71, 125342(2005).

68 J. M. Llorens Montolio, Estructura electronica ypropiedades opticas de puntos cuanticos auto-organizados,Ph.D. thesis, Institut de Ciencia dels Materials, Universi-tat de Valencia (2007).

69 I. Vurgaftman, J. Meyer, and L. Ram-Mohan, Band pa-rameters for III-V compound semiconductors and their al-loys, J. Appl. Phys. 89, 5815 (2001).

70 F. Meier and B. P. Zakharchenya, Optical Orientation(Modern Problems in Condensed Matter Sciences) (Else-vier Science Ltd, 1984).

71 M. Cardona, N. E. Christensen, and G. Fasol, Relativisticband structure and spin-orbit splitting of zinc-blende-typesemiconductors, Phys. Rev. B 38, 1806 (1988).

Supplementary Material of topology driven g-factor tuning in type-II quantum dots

J. M. Llorens,1 V. Lopes-Oliveira,2 V. Lopez-Richard,3 E. R. Cardozo de Oliveira,3 L. Wewior,1

J. M. Ulloa,4 M. D. Teodoro,3 G. E. Marques,3 A. Garcıa-Cristobal,5 G.-Q. Hai,6 and B. Alen1

1Instituto de Micro y Nanotecnologıa, IMN-CNM,CSIC (CEI UAM+CSIC) Isaac Newton, 8, E-28760, Tres Cantos, Madrid, Spain

2Universidade Estadual de Mato Grosso do Sul, Dourados, Mato Grosso do Sul 79804-970, Brazil3Departamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos, Sao Paulo 13565-905, Brazil

4Institute for Systems based on Optoelectronics and Microtechnology (ISOM),Universidad Politecnica de Madrid, Ciudad Universitaria s/n, 28040 Madrid, Spain

5Instituto de Ciencia de Materiales (ICMUV), Universidad de Valencia, Paterna E-46980, Spain6Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, Sao Paulo 13560-970, Brazil

(Dated: April 18, 2019)

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I. MPL SPECTRAL ANALYSIS DETAILS

Figure S1 shows the σ+-polarized MPL spectra which completes the σ− data shown in Figure 2 of the manuscript.Every spectrum was spectrally analyzed by two-component Gausssian deconvolution. To improve the fidelity of theanalysis, the fit to the experimental data was done after subtraction and normalization by the spectrum at 0 T, asshown in figure S2. Figure S3 shows the result of deconvolution with ground state (GS) and excited state (ES)explicitly indicated. The low energy band (GS) was then further analyzed in the discussion of the magnetic propertiesof the InAs/GaAsSb system.

0.0

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88 kVcm-1 63 kVcm-1 38 kVcm-1 13 kVcm-1 4.2 kVcm-1

a

f

b

g

c

h

d

i

e

j

Energy (eV)

DI s+

(%)

I s+

(arb

. un

its)

PLPL

FIG. S1. (a-e) Evolution of the σ+-polarized magneto-photoluminescence in vertical electrical fields between 88 and 4.2 kVcm−1. To highlight the magnetic field effects, (f-j) show the evolution after subtraction and normalization by the spectrum at0 T.

0

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Experiment

Gaussian fit

DI s+(%

)PL

DI s+(%

)PL

a b c d e

f g h i j

Energy (eV)1.10 1.15 1.20 1.10 1.15 1.20 1.10 1.15 1.20 1.10 1.15 1.20 1.10 1.15 1.20

FIG. S2. (a-e) Evolution of the σ+-polarized magneto-photoluminescence in vertical electrical fields between 88 and 4.2 kVcm−1 after subtraction and normalization by the spectrum at 0 T. (f-j) Same representation for the spectra obtained afterGaussian deconvolution of the data.

3

1.10 1.15 1.20

I PL (

arb

itra

ry u

nit

s)

Energy (eV)1.10 1.15 1.20

Energy (eV)1.10 1.15 1.20

Energy (eV)

GS

1.10 1.15 1.20

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Energy (eV)

Experiment Gaussian fit Gaussian fit

88 kVcm-1 a b c

I PL

(arb

. u

nit

s)

FIG. S3. (a) Evolution of the σ+-polarized magneto-photoluminescence at 88 kV cm−1. (b,c) Results of the two-componentGaussian deconvolution procedure.

II. INTENSITY EMISSION CALCULATION WITH A RATE EQUATIONS MODEL

For the computation of the emission intensity, we simplify the electronic structure around the QD ground state asbeing described by the three levels “A”, “B” and “C” of Fig. 7 in the main text. The radiative rate of each stateis proportional to the oscillator strength of Fig. 7(c). The rate equations model is based on Ref. 1. The first orderdifferential equations system is:

dNR

dt= G− (ΓR→C + ΓR,r)NR (1)

dNC

dt= ΓR→CNR + ΓB→CNB − (ΓC,r + ΓC→B)NC (2)

dNB

dt= ΓC→BNC + ΓA→BNA − (ΓB,r + ΓB→A + ΓB→C)NB (3)

dNA

dt= ΓB→ANB − (ΓA,r + ΓA→B)NA (4)

where Ni is the population of a state i. The transition rate to a state of lower energy is proportional to the occupationof that state Γi→j = (1−Pj)Γ

0i→j . The occupation of a state j is the ratio between its population and the degeneracy

Pj = Nj/gj . The degeneracy is two for the three states. The thermal excitation of carriers from lower towardshigher energy levels Γj→i = (1 − Pi)Γ

0j→i. Under the aproximation of the detailed balance between relaxation and

emission rate: Γ0j→i = Γ0

i→j exp[−(Ej −Ei)/kBT ] where kB is the Boltzmann constant and T the temperature. Sincethe ensemble experiment cannot resolve these individual states, the emitted intensity is computed as the sum of theradiative rate times the population of the three states. We take as parameters vaues Γ0

i→j = ΓR→C = 100 GHz, T= 5 K, ΓR,r = 1 GHz. The maximum radiative rate of the states is Γr,max = 0.25 GHz. The values of ΓA,r andΓC,r are scaled according to the computed wave-function overlap in Fig. 7(c). In the case of state B, pure darkconditions lead to unphysical charge accumulation effects within the recombination model. We relax this conditionassuming that state B can recombine radiatively with a lifetime ΓB,r = (ΓA,r + ΓC,r)/10, i.e. an order of magnitudesmaller that the rates of A and C. This radiative coupling shall arise from hybridization with bright states beyondthe current electronic structure model. Under these approximation, the calculated emission intensity (represented inFig. 8) oscillates around the AB field as experimentally observed. The resulting rates are shown in Fig. S4.

4

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 8 10

Γ i,r(G

Hz)

B (T)

A

B

C

FIG. S4. Radiative rate of states A, B and C considered in the calculation.

1 M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, H. Ebe, and H. Ishikawa, Quantum-dot semiconductor optical amplifiers forhigh-bit-rate signal processing up to 160 Gb s−1 and a new scheme of 3R regenerators, Measurement Science and Technology13, 1683 (2002).