Rashba spin splitting of the minibands of coupled In As ∕ Ga As pyramid quantum dotsXiu-Wen Zhang, Qiang Xu, Wei-Jun Fan, Jun-Wei Luo, Shu-Shen Li, and Jian-Bai Xia Citation: Applied Physics Letters 92, 143113 (2008); doi: 10.1063/1.2907690 View online: http://dx.doi.org/10.1063/1.2907690 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/92/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electronic states and intraband terahertz optical transitions in InGaAs quantum rods J. Appl. Phys. 111, 073110 (2012); 10.1063/1.3692069 Quantum well thickness dependence of Rashba spin–orbit coupling in the InAs/InGaAs heterostructure Appl. Phys. Lett. 98, 202504 (2011); 10.1063/1.3589812 Miniband structure and photon absorption in regimented quantum dot systems J. Appl. Phys. 109, 074303 (2011); 10.1063/1.3562160 Composition, volume, and aspect ratio dependence of the strain distribution, band lineups and electron effectivemasses in self-assembled pyramidal In 1−x Ga x As/GaAs and Si x Ge 1−x / Si quantum dots J. Appl. Phys. 91, 389 (2002); 10.1063/1.1410318 Conduction band spectra in self-assembled InAs/GaAs dots: A comparison of effective mass and an eight-bandapproach Appl. Phys. Lett. 71, 3239 (1997); 10.1063/1.120302

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Rashba spin splitting of the minibands of coupled InAs/GaAs pyramidquantum dots

Xiu-Wen Zhang,1,2,a� Qiang Xu,1,2 Wei-Jun Fan,1 Jun-Wei Luo,2 Shu-Shen Li,2 andJian-Bai Xia2

1School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798,Singapore2State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors,Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, People’s Republic of China

�Received 27 November 2007; accepted 17 March 2008; published online 11 April 2008�

The Rashba spin splitting of the minibands of coupled InAs /GaAs pyramid quantum dots isinvestigated using the k · p method and valence force field model. The Rashba splitting of the twodimensional miniband in the lateral directions is found due to the structure inversion asymmetry inthe vertical direction while the miniband in the vertical direction has no Rashba spin splitting. As thespace between dots increases, the Rashba coefficients decrease and the conduction-band effectivemass increases. This Rashba spin splitting of the minibands will significantly affect the spintransport properties between quantum dots. © 2008 American Institute of Physics.�DOI: 10.1063/1.2907690�

Low dimensional systems such as semiconductor quan-tum dots exhibit interesting electrical and opticalproperties.1–4 Recently, InAs /GaAs pyramid quantum dotsmade by the Stranski–Krastanov growth method have be-come of great interest.5–7 The strain distribution inInAs /GaAs pyramid quantum dot was investigated using thevalence force field �VFF� method.8,9 The electronic structureof these quantum dots was theoretically studied using em-pirical pseudopotentials9 and the effective-mass method,6 re-spectively.

Nowadays, much of the research in semiconductor phys-ics has been shifting toward spintronics.10,11 One of the mostimportant spin-based devices was proposed by Datta andDas,12 which makes use of the Rashba spin-orbit coupling13

in order to perform controlled rotations of a field-effect tran-sistor. The Rashba spin-orbit coupling is caused by the struc-ture inversion asymmetry �SIA�.13 The Rashba spin splittingsin semiconductor quantum wells14,15 and quantum wires16–18

were extensively studied.Periodic arrays of quantum dots were constructed using

stacks of Stranski–Krastanov InAs /GaAs islands,19,20 whose

periodicities may be accurately controlled.20 The miniband inthe vertical direction of these arrays of dots has been calcu-lated using the simple one dimensional models20 and fullthree dimensional �3D� models,1 respectively. The results arein agreement with the photoluminescence experiments.1,20

In this letter, we will calculate the minibands in the lat-eral and vertical directions of coupled InAs /GaAs pyramidquantum dots. We represent the eight-band Hamiltonian ofzinc-blende semiconductors in the presence of strain in thebasis functions �S�↑, �11� ↑, �10� ↑, �1–1� ↑, �S�↓, �11� ↓, �10�↓, �1–1� ↓ as

Heb = �Hint 0

0 Hint� + Hso + V0, �1�

where Hso is the spin-orbit coupling Hamiltonian ��so is thespin-orbit splitting energy� and V0 is the confining potentialof the quantum dot, which is zero in the dot and is finitelyhigh in the barrier.

Hint is written as

Hint =Eg + Pe �i/2�p0�kx� + iky�� ip0kz� �i/2�p0�kx� − iky��

− �i/2�p0�kx� − iky�� P1 S T

− ip0kz� S� P3 S

− �i/2�p0�kx� + iky�� T� S� P1

� , �2�

where Eg is the band gap, p0=�EP /2m0, EP is the matrixelement of Kane’s theory,

kx� = kx − �xxkx − �xyky − �xzkz, �3a�

ky� = ky − �xykx − �yyky − �yzkz, �3b�

kz� = kz − �xzkx − �yzky − �zzkz, �3c�

Pe =�2

2m0�c�kx

2 + ky2 + kz

2� + ac��xx + �yy + �zz� , �3d�a�Electronic mail: zhxw99@semi.ac.cn.

APPLIED PHYSICS LETTERS 92, 143113 �2008�

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P1 = −�2

2m0�L� + M�

2�kx

2 + ky2� + M�kz

2 + �av +

b

2���xx + �yy� + �av − b��zz, �3e�

P3 = −�2

2m0�M��kx

2 + ky2� + L�kz

2� + �av − b���xx + �yy�

+ �av + 2b��zz, �3f�

S* = −�2

2m0� 1

2N��kx + iky�kz + 6d��xz + i�yz� , �3g�

T* = −�2

2m0�L� − M�

2�kx

2 − ky2� + iN�kxky +

3

2b��xx − �yy�

+ i23d�xy , �3h�

where �xx ,�yy , . . . are the strain components, ac ,av ,b ,d arethe deformation potentials, �c=m0 /mc− �EP /3��2 /Eg

+1 / �Eg+3���, L�=L−EP /Eg, M�=M, N�=N−EP /Eg, and�=�so /3. For InAs, mc=0.02226m0, L=53.15, M =2.93, N=55.74, and EP=22.2 eV.21 The other parameters of InAsand GaAs are listed in Table I.21 The conduction-band offsetis assumed to be 70% of the band-gap difference.2

We expand the wave function of the electron and holestates in plane wave functions �1 /LxLyLz�ei�knxx+knyy+knzz�,where knx=kx0+nxKx, kny =ky0+nyKy, knz=kz0+nzKz, Kx=2� /Lx, Ky =2� /Ly, Kz=2� /Lz, Lx, Ly, and Lz are the peri-ods of the quantum dot array, and kx0, ky0, kz0 are the wavevectors. The energies as functions of kx0, ky0, and kz0 repre-sent the energy dispersion of the periodic array of dots in thex ,y ,z directions, respectively. For InAs /GaAs pyramidquantum dots, we assume that the pyramid axis is along thez direction, and the x and y directions are the lateral direc-tions, and there is a two monolayer wetting layer in the xyplane.

The shape of the InAs /GaAs pyramid dot is shown inFig. 1. We only show the xz plane because the y axis is

equivalent with the x axis. The base width of the pyramid istwice its height. The height of the pyramid is 8.48 nm. Thespace between quantum dots in the x �or y� direction �2S� ischangeable; S is denoted in Fig. 1. The dot is 3D periodicallyrepeated to form an array.

Figures 2�a� and 2�b� show the minibands in the x direc-tion �as functions of kx0� of coupled InAs /GaAs dots withS=0.565 nm. It is interesting to find that the minibands,which are spin degenerated at zero kx0, split at nonzero kx0.This is the spin splitting in coupled InAs /GaAs pyramidquantum dots. The spin splitting is obvious in the hole mini-band and is small in the electron miniband so that it is hardto see. The Rashba spin splitting is caused by the SIA of thesemiconductor nanostructures. There is a SIA along the ver-tical direction due to the asymmetrical shape of the dot �seeFig. 1�. For clarity, we show the Rashba spin splitting ener-gies ��E� for the electron and hole in Figs. 2�c� and 2�d�,respectively. We see that as �kx0� increases, the spin splittingenergies almost linearly increase when �kx0� is small. In thislinear range, we can approximately use an effective linearRashba term H

Ra* =���V�k0� ·, where k0= �kx0 ,ky0 ,kz0�, to

represent the Rashba spin splitting, similar to the quantumwell case.14 � is the Rashba coefficient, which can be calcu-lated as �= �1 /2����E /�k0�, where �E is the Rashba split-ting energy. As �kx0� increases further, the Rashba spin split-ting energies decrease.

We also calculated the electron and hole minibands inthe vertical direction of coupled InAs /GaAs pyramid quan-tum dots with S=0.565 nm, as shown in Fig. 3. We see thatthe minibands in this direction show no Rashba spin split-ting, which is in agreement with the former works.1,20 Thereason is that the dot is symmetrical in the lateral directions�see Fig. 1�.

In order to indicate the energy dispersion, we calculatedthe conduction-band effective mass �m

e*� by fitting a parabola

to the lowest electron miniband energy E�kz0� around kz0=0.The calculated m

e* in Fig. 3�a� is 0.516m0, which is quite

larger than that of InAs bulk material, 0.02226m0, becausethe coupling between neighbor dots in the z direction issmall. We also calculated the conduction-band effective massof the miniband in the x direction, as shown in Fig. 2�a�. Wesee that the m

e* of the miniband in the x direction is quite

TABLE I. Parameters of InAs and GaAs used in the calculation.

Material a0 �nm� Eg �eV� �so �eV� ac �eV� av �eV� b �eV� d �eV�

InAs 0.605 83 0.418 0.38 −5.08 1.0 −1.8 −3.6GaAs 5.565325 1.519 0.341 −7.17 1.16 −1.7 −4.55

FIG. 1. The shape of the InAs /GaAs pyramid quantum dot.

FIG. 2. The minibands in the x direction and the spin splitting energies ofcoupled InAs /GaAs pyramid quantum dots with S=0.565 nm. �a� Electronminiband. �b� Hole miniband. �c� Electron spin splitting energy. �d� Holespin splitting energy.

143113-2 Zhang et al. Appl. Phys. Lett. 92, 143113 �2008�

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smaller than that in Fig. 3�a�, which is because the spacebetween dots in the x direction is much smaller than that inthe z direction �see Fig. 1� and the coupling between neigh-bor dots decreases as the space of dots increases.

Figure 4�a� shows the conduction-band effective mass ofthe minibands in the x direction of coupled InAs /GaAs pyra-mid quantum dots as functions of S. As the two minibandshave Rashba spin splitting �see Figs. 2�a� and 2�c��, we usetheir average values to fit the effective mass. We see that them

e* in the x direction increases as S increases because the

coupling between dots decreases. Figures 4�b� and 4�c� showthe Rashba coefficients of the lowest electron miniband andhighest hole miniband, respectively, as functions of S. Wefind that the Rashba coefficients decreases as S increases,which is because the SIA in the vertical direction is smallerfor larger S �see Fig. 1�. As S increases, the ratio of theasymmetrical part �pyramid� decreases. The hole Rashba co-efficient can be 6.4 meV nm, which is relatively large.18

The Rashba spin splitting of the minibands of coupledquantum dots will significantly affect the spin transport prop-erties in the xy plane. The spin Hall effect, which happens in

semiconductor quantum wells with Rashba spin splitting,will also happen in a layer of coupled InAs /GaAs pyramidquantum dots in which the wetting layer �see Fig. 1� mayhelp the carriers to transport. There may be Bloch oscilla-tions when the carriers move in the minibands.1 The Rashbaspin splitting of the minibands may introduce an interestingspin-dependent Bloch oscillation. In order to use the Rashbaspin splitting in spin transport, one needs the carrier trans-portation in the lateral direction, which requires regular pe-riodic dots in the lateral direction. The quantum dots synthe-sized by electron beam lithography, x-ray lithography, and soon, can be periodically positioned in the lateral plane, whilecurrently, most experimental self-assembled quantum dotsare randomly positioned in the lateral plane.

In summary, the Rashba spin splitting and conduction-band effective mass of the minibands of coupled InAs /GaAspyramid quantum dots were studied. The Rashba splitting ofthe two dimensional miniband in the lateral directions isfound due to the SIA in the vertical direction. As the spacebetween dots increases, the Rashba coefficients decreasesand the conduction-band effective mass increases due to thedecrease in the coupling between dots. The Rashba spinsplitting of the minibands of coupled quantum dots will sig-nificantly affect the spin transport properties between quan-tum dots. The spin Hall effect and spin-dependent Bloch os-cillation may happen in coupled InAs /GaAs pyramidquantum dots.

Wei-Jun Fan would like to acknowledge the supportfrom A*STAR �Grant No. 0621200015� and AcRF RGM1/07. Jian-Bai Xia and Shu-Shen Li would like to acknowl-edge the support from the National Natural Science Founda-tion of China �Nos. 90301007 and 60521001�. Xiu-WenZhang would like to acknowledge Professor Kai Chang, Pro-fessor Jingbo Li, and Dr. Wen Yang for their helpful discus-sions. Xu Qiang would like to thank Professor Lin-WangWang for his VFF strain calculation programs.

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FIG. 3. The minibands in the z direction of coupled InAs /GaAs pyramidquantum dots with S=0.565 nm. �a� Electron miniband. �b� Hole miniband.

FIG. 4. The Rashba coefficients � and conduction-band effective mass me*

of the minibands in the x direction of coupled InAs /GaAs pyramid quantumdots as functions of S. �a� m

e*. �b� � of the lowest electron miniband. �c� �

of the highest hole miniband.

143113-3 Zhang et al. Appl. Phys. Lett. 92, 143113 �2008�

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