Electrically tunable electron g factors in coupled InAs/GaAs pyramid quantum dotsJiqing Wang, Huibing Mao, Jianguo Yu, Qiang Zhao, Hongying Zhang, Pingxiong Yang, Ziqiang Zhu, andJunhao Chu Citation: Applied Physics Letters 96, 062108 (2010); doi: 10.1063/1.3300879 View online: http://dx.doi.org/10.1063/1.3300879 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/96/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantum-confined Stark effects in coupled InAs/GaAs quantum dots Appl. Phys. Lett. 102, 213101 (2013); 10.1063/1.4807770 Anisotropy of electron and hole g-factors in (In,Ga)As quantum dots Appl. Phys. Lett. 99, 221914 (2011); 10.1063/1.3665634 Electron and hole energy levels in InAs/GaAs quantum dots: Size and magnetic field effects J. Appl. Phys. 109, 033703 (2011); 10.1063/1.3524519 Observation of an electrically tunable exciton g factor in InGaAs/GaAs quantum dots Appl. Phys. Lett. 96, 053113 (2010); 10.1063/1.3309684 g-factor and exchange energy in a few-electron lateral InGaAs quantum dot Appl. Phys. Lett. 95, 192112 (2009); 10.1063/1.3264053

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Electrically tunable electron g factors in coupled InAs/GaAs pyramidquantum dots

Jiqing Wang,a� Huibing Mao, Jianguo Yu, Qiang Zhao, Hongying Zhang, Pingxiong Yang,Ziqiang Zhu, and Junhao ChuKey Laboratory of Polarized Materials and Devices, East China Normal University, Shanghai 200062,People’s Republic of China

�Received 8 December 2009; accepted 8 January 2010; published online 9 February 2010�

The electron g factors of coupled InAs/GaAs quantum dots under external magnetic and electricfields are investigated by using the eight-band k • p model. The resonant coupling between the twodots remains under electric fields below 8.2 mV/nm, and is broken above the critical field due to thequantum Stark effect. By applying electric fields, a sign reverse of g factors is observed, and anelectric field tunable zero g factor is found in the quantum dot molecules. Spin-orbit interactionsnicely explain the transition mechanism of g factors under external electric fields. © 2010 AmericanInstitute of Physics. �doi:10.1063/1.3300879�

Nowadays, much of the research in low dimensionalsemiconductor has been shifting toward semiconductorsingle quantum dot �QD�.1,2 Electronic spin in quantum dotsis one of the important properties to be harnessed in spin-tronic devices, as well as in quantum computers for the logi-cal units.3,4 The key quantity needed in understanding thespin effects is the g factor which is the coefficient connectingspin moment with magnetic one. Recently, the dependence ofelectron g factors on the shape and size of QD nanostructureshave been widely investigated both experimentally andtheoretically.5,6 From the view of practical application, alarge or zero g factor is preferable in quantum informationprocessing and receiving. Zhang et al.7,8 reported giant gfactors in diluted magnetic semiconductor QDs. Further-more, zero g factor has also been obtained by an externalelectric field in InAsN nanostructures.

Recently, coupled QDs or periodic arrays of QDs wereconstructed using strain-driven self-assembly growthmode.9,10 The coupled QDs form the bonding and antibond-ing nature of molecular states, and thus provide more roomfor shifting the electron and hole wave functions between thetwo QDs.11,12 Accordingly, the g factor modulation by exter-nal electric fields is an order of magnitude larger than that insingle quantum dot.13 Indeed, distinctive electrically tunedresonances have been discovered in recent experiments forthe exciton g factors in vertically stacked QD molecules.12

However, the manipulation of g factor is mainly limited inthe hole states, since hole coupling between two QDs ismuch stronger due to the smaller barrier height.13 It is ex-pected that the pronounced tuning behavior of g factor alsooccurs in the electron states of QD molecules, which is veryinteresting and useful in quantum computation becauseelectron has smaller effective mass and thus higher responsevelocity.

In this letter, we explore the properties of molecule elec-tron g factors under a wide range of applied bias in coupledInAs/GaAs pyramid quantum dots. Taking into account thespin-orbit interaction, the piezoelectric polarization and theZeeman splitting effect, we use the eight-band k • p model tocalculate electron g factors of the QD molecules under ex-

ternal magnetic and electric fields. It is found that electricfields can transform electron g factors from negative to posi-tive values, and zero g factor can be achieved in a certainexternal bias. There are two effects of electric fields on the gfactors: resonant coupling between two dots under low fieldsand Stack shift after critical electric field.

Figure 1�a� shows the vertically stacked InAs/GaAs cou-pling QD structure studied in this letter. Each truncated pyra-mid with �011� side facet is characterized by the quantum dotheight h, the dot width w, and the dot separation distance d.Here, we take h=5, w=15, and d=2 nm. For simplicity, thewetting layer below each dot is neglected. We calculate theenergies of electron ground and first-excited states of theentire mesoscopic system using an eight-band envelop func-tion method that has been implemented into the simulationsoftware NEXTNANO.14

The potential along the growth direction and the wavefunctions for the two lowest energy levels without any ap-plied fields are plotted in the Fig. 1�b�. The wave function of

a�Electronic mail: jqwang@ee.ecnu.edu.cn.

(a) (b)-4 0 4 8 12 16

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FIG. 1. �Color online� �a� Structure of coupled InAs/GaAs quantum dotsstudied in this letter. �b� Corresponding potential profiles along the growthdirection and the wave functions for the bonding �blue line� and antibonding�red line� levels without any applied fields. �c�. Magnetic field dependenceof the energy of the spin-splitted ground state. c+�c−�denotes the spin-up�down� energy level. �d� The energy difference of the spin-splitted groundstate as a function of magnetic fields.

APPLIED PHYSICS LETTERS 96, 062108 �2010�

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the ground state mainly distributes in the lower dot due to thedecreasing potential induced by the piezoelectric effect,whereas that of the first excited state shifts to the higher dot,revealing a resonance behavior associated with the formationof bonding and antibonding electron states. Figure 1�c� dis-plays the calculated molecular energies of the bonding stateunder external magnetic fields �B�. It can be seen the degen-erate ground states split into doublets due to the Zeemaneffect. In addition, the spin-up levels are lower than the spin-down ones, which stems from the spin-orbit interactions ineight-band k • p calculation.15 The energy difference of the c+and c� states is shown in Fig. 1�d�. The �E increase linearlywith B. Based on the linear relationship, we define the elec-tron g factor, ge= �E�c+�−E�c−�� /�bB.

In Fig. 2, we plot the molecule electron g factors forboth the ground and first-excited states in the presence ofexternal electric fields �F�. It may be noted that the g factorsfor the bonding and antibonding states remains almost con-stant for F�6.0 mV /nm, in agreement with the result ofAndlauer and Vogl.13 The reason can be attributed to the lesspronounced variation of wave functions in both dots in caseof low fields. When 6.0�F�8.2 mV /nm, the g factors ofthe two lowest states have opposite variation trends with thebiases, demonstrating that the resonant coupling between thebonding and antibonding states strongly enhances.12,13 Theprobability in intermediate GaAs barrier for the bondingstate is higher than that for the antibonding state. As a result,for the present coupled quantum dots, the formation of bond-ing �antibonding� states as a function of external electricfields remarkably increases �decreases� the overlap of themolecular states with the GaAs barrier in between thecoupled QDs which leads to a increase �decrease� in the mo-lecular electron g factors. To further analyze the resonantbehavior in this case, the wave functions for the lowest twostates are given in the inset of Fig. 2. As shown in the inset,at an external field of over than 6.0 mV/nm, the bondingelectrons are mainly located in the upper dot, whereas theantibonding electrons concentrate in the lower dot. The mainparts of wave function for the ground state shift from thelower to the upper QD in the range of F�6.0 mV /nm sincelager electric fields pull down the potential barrier of theupper dot and make it lower than that of the lower dot. More-over, in this region, although the g factors of the ground state�bonding state� still increase, the values are always negative.

From Fig. 2, we can also see that a further increase ofelectric fields �F�8.2 mV /nm� will make the molecular

states localized within the individual upper dot. In this case,the molecular coupling plays no role because electric fieldinduced Stark effects further decrease the potential of theupper QD. Therefore, the bonding and antibonding states dis-appear, and both the ground and first excited states are con-fined in the upper QD �see the inset of Fig. 2�. Here, we takethe critical bias Fc=8.2 mV /nm to characterize the transi-tion of the molecular system from the coupled resonant stateto the Stark-induced single QD state under external biases.To explore the mechanism of the transition, we calculate theenergy gap �Eg� of the nanostructure as a function of electricfields �Fig. 3�. The band gap starts to decrease when F�8.2 mV /nm, which implies that electron Stark shift ismore obvious after the critical field. Accordingly, the reso-nant coupling between the two dots is broken, and the Starkeffect dominates within the bias region. Furthermore, an ob-vious property for the region is that the spin splitting of theexcited state �p-shell� is much larger than that of the groundstate �s-shell� due to nonzero mesoscopic angularmomentum.16

An interesting phenomenon of zero g factor for theground state is clearly observed at an external field of about11 mV/nm in Fig. 2. With the increasing fields, the energygap decreases continually, thus, the conduction band feelsmore coupling from the valence band �VB�, which increasesorbit contribution to the electron g factors. In our computa-tion condition, the heavy hole state of the valence band ex-hibits positive g factors. Additionally, electron g factors canbe tuned to positive values under electric fields exceeding 11mV/nm. For comparison, we also calculate the g factors forsingle QD with the same shape and size, it is found that thesign of g factor dose not change under a large bias region.This means that QD molecules provide more room or degreeof free to manipulate electron spin due to much more physi-cal nature in the system, Therefore, with a fixed magneticfield, we can use the electric field to tune the electron spin tobe polarized, unpolarized �g=0�, or antipolarized. As com-pared with the magnetic field controllable g factor, the elec-tric field tune is independent of the temperature and magneticfield, and thus is more stable.

To further understand the evolution of electron g factorsunder electric fields, we investigate the dependence of thehole state �p state� components in the CB Bloch state. Theresults are plotted in Fig. 4. It should be noted that the CBBloch function is not pure s state and is influenced by the

0 2 4 6 8 10 12 14-0.2

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FIG. 2. �Color online� Calculated electron g factors for ground �black line�and first excited states �red and blue lines� as a function of external electricfields lying in the vertical �001� direction.

0 2 4 6 8 10 12 140.0

0.1

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Electric field (mV/nm)

Eg

FIG. 3. Energy gap of coupled QD molecules as a function of electric fields.

062108-2 Wang et al. Appl. Phys. Lett. 96, 062108 �2010�

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hole Bloch states due to the multicomponent properties ofthe envelope functions, then the interactions between CB andVB will significantly influence the weight of s state in CB.As can be seen from Fig. 4, the p state components in thetwo lowest electron states have almost similar electric fielddependent behaviors with that of g factors �Fig. 2�. With theincreasing biases, the p state components in the lowest stateincrease gradually when 6.0�F�8.2 mV /nm, and moreevidently at electric fields above 8.2 mV/nm where the inter-action between CB and VB enhances due to the decrease ofEg �Fig. 3�. This kind of mixture of different states increasesthe spin-orbit interactions and gives rise to the sign reverseof the electron g factors. The effect is also found inInSb1−xNx nanowire and InGaAs QD by decreasing bandgap.15,17 Hence, the evolution of electric tunable g factor canbe attributed to the amount of orbit contribution to electronstates which can be controlled by external electric fields.

In summary, we studied the electron g factors of coupledQDs under a wide bias range using the eight-band k • pmodel. The resonant coupling between the two dots is main-

tained under electric fields below 8.2 mV/nm and is brokenabove the critical field due to the quantum Stark effect. It isalso found that the g factors can be changed from negative topositive values. Especially, under a fixed magnetic field, wecan use the electric field to tune the electron spin in QDmolecules to be polarized, unpolarized, or antipolarized. Theamount of orbit contribution to CB electron states dominatesthe variation of electron g factors under external electricfields.

This work was supported by National Science Founda-tion of China �Grant Nos. 60876067 and 60990312� and Ma-jor State Basic Research Development Program of China�Grant No. 2007CB924902�.

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0 2 4 6 8 10 12 14

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FIG. 4. �Color online� The dependence of the p state components for theground �black line� and first excited states �red and blue lines� on externalelectric fields.

062108-3 Wang et al. Appl. Phys. Lett. 96, 062108 �2010�

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