VOLUME 82, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 22 FEBRUARY 1999

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Electron and Hole g Factors and Exchange Interaction from Studies of the ExcitonFine Structure in In0.60Ga0.40As Quantum Dots

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov,* V. B. Timofeev,* F. Schäfer, and J. P. ReithmaTechnische Physik, Universität Würzburg, Am Hubland, D-97094 Würzburg, Germany

T. L. Reinecke and S. N. Walck†

Naval Research Laboratory, Washington, D.C. 20375(Received 5 October 1998)

Self-assembled In0.60Ga0.40As quantum dots (QD’s) have been studied by single dot magnetophotoluminescence spectroscopy (B # 8 T ). At B ­ 0 a splitting of the exciton (X) emission is observedwhich we ascribe to an asymmetry of the confinement potential. With increasingB the emission splitsinto a quadruplet corresponding to them ­ 62 and61 X states, which originates from a reduction ofthe cubic symmetry of the QD’s. From the spectroscopic data we obtain values for the electron (e)and hole (h) g factors. We also determine theX singlet-triplet splitting which is found to be enhancedover bulk values by about an order of magnitude due to the increase of thee-h overlap in the QD’s.[S0031-9007(99)08567-1]

PACS numbers: 71.35.Ji, 71.70.Ej, 78.55.Cr

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During the past several years spectroscopic studieelectronic confinement effects in quantum dots (QDhave evolved into a very active research area ranging fbasic physics regarding, e.g., properties of zero dimsional X ’s to technological applications like QD laserSemiconductor QD’s fabricated by self-assembled grohave attracted particular interest due to their high optquality [1–3]. However, a major obstacle in accessfully detailed information on the electronic properties hbeen the inhomogeneous broadening of spectral linesto size and composition variations over ensembles of dRecently several experimental techniques have been doped which permit the study of individual QD’s for whicinhomogeneous broadening effects are suppressed [1

Because of the possibility of performing sensitive sptroscopy of single QD’s, experiments giving clear insiginto their electronic states may be realized. The electrofine structure of the QD levels, which arises from the cpling of the spin and the single particle electronic stagives particularly detailed information. For example,studies of single QD’s formed at GaAsyAl xGa12xAs in-terfaces a splitting of theX states has been reported [6Very recently studies of the Coulomb interactions btween carriers forming multi-X complexes have been peformed in self-assembled QD’s [7,8].

From the Zeeman splitting ofX ’s in QD’s formedby laser induced diffusion [10] theX g-factor has beenobtained. However, to date it has not been possibledetermine the underlyinge and h g-factors separatelyin QD systems. Theseg factors are very sensitive tband mixing and furnish detailed insight into the confinelectronic structure. To our knowledge, at present this no detailed understanding ofg factors in quantum wiresand dots, either experimentally or theoretically.

In the present work we have used small mesa structto study single In0.40Ga0.60As QD’s formed by self-

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assembled growth. We observe at highB emission fromheavy holeX angular momentum statesm ­ 61 andm ­ 62 (bright and darkX ’s). The latter emissionresults from a reduced symmetry of the QD’s which wattribute to a shape asymmetry. From the splitting ofX lines in magnetic field we determine thee and thehg-factors. In addition, we find that theX singlet-tripletsplitting is 200 meV, which is drastically increased ovethe bulk value.

The QD’s were grown on GaAsf001g substrates at atemperature of470 ±C by depositing the equivalent o4.8 monolayers of In0.60Ga0.40As. Stransky-Krastanowgrowth [11] starts after deposition of about three monolers. The QD’s were capped without growth interruptioAn average dot density of1 3 1010 cm22 is determinedfrom scanning electron microscopy of a sample withocap layer. The QD’s have an average lateral size of ab20 nm. In low excitation photoluminescence (PL) temission from QD arrays has a spectral width of ab25 meV. Lithography was used to fabricate small mestructures with lateral sizes down to 100 nm, for whiwe obtain an average number of one dot per mesa.optical studies only those mesas were selected whoseexcitation spectra consist of a single emission line. Frthis observation we conclude that only one QD is cotained in such a mesa which is to be contrasted with ostructures, whose spectra consist of a few lines.

The samples were held at a temperature of 1.5 Kthe helium insert of an optical split-coil magnetocryos(B # 8 T). The magnetic field was aligned normal to theterostructure (z direction). The samples were opticalexcited by a cw Ar1 laser using very low excitationdensities (,0.1 Wycm22). The emission was disperseby a double monochromator and detected by a chacoupled-device camera. Its polarization could be analyby a quarter-wave-retarder and a linear polarizer.

© 1999 The American Physical Society

VOLUME 82, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 22 FEBRUARY 1999

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PL spectra from different single QD’s can be dividinto two classes with different behavior in magnetic fieThe solid lines in Fig. 1 show typical PL spectra froa single QD belonging to one class. The spectrathese QD’s show a splitting into four components wincreasingB. Because of the weak excitation, all linare associated with the recombination of singlee-h pairsin the dot. The two outer lines have strong intensitwhereas the two inner ones have comparatively wintensities. The spectra shown by dashed lines forB ­ 0and 8 T belong to a dot from the second class and exthe well-known behavior ofX transitions in magneticfield, namely a splitting of the emission line into a doubdue to the Zeeman effect.

Figure 2 shows polarized PL spectra of a QD whgives the four transitions in the unpolarized specof Fig. 1. At zero magnetic field (lower panel) thluminescence is linearly polarized parallel either tof110g or the f110̄g direction [6]. The emission lineare split from one another by about150 meV. WithincreasingB the polarization of the emission changfrom a linear into a circular one, as seen from the spe

1,311 1,312 1,313 1,314

B=8 T

B=7 T

B=6 T

B=5 T

B=4 T

B=3 T

B=2 T

B=1 T

B=0 T

energy [eV]

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FIG. 1. Solid traces: PL spectra of an asymmetric QDvarying B. Dashed lines atB ­ 0 and 8 T give spectra oa symmetric QD. To facilitate comparison of magnetic fieffects the emission of the symmetric QD which occurs1.3195 eV atB ­ 0 is shifted by 7.2 meV to lower energies.

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at B ­ 8 T (upper panel). In contrast, the emissiospectra of dots with a doublet splitting at highB showno significant linear polarization atB ­ 0.

Figure 3 gives the observed transition energies fosingle QD with a quadruplet splitting versusB. Themagnetic field dependences of the transition energies refrom the diamagnetic shift of theX plus the Zeeman energdue to the interaction of theX spin withB, which dependson the spin orientation. In strongly confined QD’s theXdiamagnetic shift is determined by the single particle wafunctions and is independent of theX spin orientations.The differentB dependences of the emission lines thereforesult from their Zeeman splittings. Both the splittinbetween the spectral lines with strong emission intensiand that between the weak lines increase linearly withBat high fields (inset of Fig. 3). The splitting between thtwo strong spectral lines is 1.4 meV atB ­ 8 T, whereasfor the weak lines the splitting is only 0.7 meV. At lowfields, however, a deviation from the linear dependencobserved for the strong lines. There the behavior canwell described by a square dependence onB.

The heavy holeX is fourfold degenerate atB ­ 0neglecting thee-h exchange interaction [12]. TheXstates are characterized by the total angular momenm along thez direction,m ­ Se,z 1 Jh,z ­ 61, 62 withthe e spin Se,z ­ 61y2 and the h spin Jh,z ­ 63y2.The spin Hamiltonian for anX in magnetic field is givenby [13–16]

1,3120 1,3125 1,3130

energy [eV]

πyπx

B=0

norm

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1,311 1,312 1,313 1,314

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σ+

B=8T

FIG. 2. Spin-polarized PL spectra of a QD with reducsymmetry.

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VOLUME 82, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 22 FEBRUARY 1999

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FIG. 3. Magnetic field dependence of the spin-polarizedXtransition energies of In0.60Ga0.40As QD’s. The inset showsthe spin-splittings of the bright and darkX ’s versusB.

H ­ azJh,z 3 Se,z 1X

i­x,y,z

biJ3h,i 3 Se,i

1 mB

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gh,z

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where the first two terms give thee-h exchange energand the third one the interaction of theX spin withB. AtB ­ 0, the radiative statesm ­ 61 and the nonradiativeones,m ­ 62, are split byEex ­ 1.5az 1 3.375bz dueto exchange interaction (singlet-triplet splitting) [17].

An asymmetry of the confinement potential in the Qplane (bx fi by) results in an exchange energy splittiof the jmj ­ 1 X states (besides modifying thejmj ­ 2splitting). The split states are linear combinationsthe m ­ 61 X ’s, jL1y2l ­ 1y

p2 sj11l 6 j21ld. The

mixing results in a linear polarization of the QD emissioas observed for the QD’s in Fig. 2 atB ­ 0. From thesplitting of the two polarized spectral lines we obtainasymmetry splittingEas,61 ­ 0.75jbx 2 byj of 150 meV.

In a magnetic field theX eigenstates withjmj ­ 1 aregiven by

jL1y2l ­sj11l 1 sr 6

p1 1 r2 dj21ldq

2s1 1 r2 6 rp

1 1 r2 d(2)

where r gives the ratio of the spin-splittingsge 1

ghdmBB to the asymmetry energyEas,61. At highfields (at larger) Eas,61 can be neglected and the twstates transform into circularly polarized states. TB dependence of the splitting between these stis given by DE61 ­

psge 1 ghd2m2

BB2 1 E2as,61.

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Similarly, the j m j­ 2 X ’s are split by DE62 ­psge 2 ghd2m2

BB2 1 E2as,62. At low B the splitting

increases quadratically with magnetic field, and thtransforms into a linear dependence. The changeooccurs at rather small magnetic fields of about 2because the Zeeman energy then becomes considerlarger than the asymmetry splitting.

For the present QD’s we observe a splitting of thX emission into a quadruplet at highB. In magnetoPLexperiments on crystals with cubic symmetry only thheavy holeX ’s with angular momentam ­ 61 can beobserved whenB k f001g, while the X ’s with m ­ 62are dark. Therefore we attribute the observation ofquadruplet splitting to a symmetry reduction of the QD’that is the orientation ofB effectively deviates from [001][18]. In that case the darkX ’s mix with the opticallyactive ones and gain oscillator strength. This mixinghowever, not very strong, as seen from the comparativweak emission intensities of the inner spectral lines.

The different dependences of the Zeeman splittingsthem ­ 61 andm ­ 62 X ’s permit a separate determination of thee andh g factors. Neglecting the exchangenergies at highB the splitting of these doublets is givenby DE61 ­ jge 1 ghjmBB andDE62 ­ jge 2 ghjmBB.From the splitting of them ­ 62 X ’s we obtain agfactor g62 ­ gh 2 ge ­ 21.39 6 0.05, whereas for them ­ 61 X ’s g61 ­ gh 1 ge ­ 23.02 6 0.05. Com-bining both relations we obtainge ­ 20.81 and gh ­22.21 [19].

At high B the splittings between the twos2 polarizedemission lines and between thes1 polarized ones arenot identical. Them ­ 62 spectral features are shiftedto lower energies towards thes1 polarized brightX.This asymmetry of the quadruplet arises from the singtriplet splitting Eex, for which we obtain200 meV fromthe spectra. This exchange energy is 1 order of magtude larger than the singlet-triplet splitting reported fobulk GaAs [20,21]. This comparison shows the stronenhancement of the exchange interaction by the thrdimensional confinement in the QD’s.

The short ranged part of thee-h exchange energy in anX is given by the probability of thee andh being at thesame position [22]

Eex

E3Dex 3 pa3

X,3D­

Zd3r jCs$r , $r dj2. (3)

Here E3Dex is the exchange energy in bulk, andaX,3D is

the bulkX Bohr radius. We have represented the preseQD’s as cylinders of In0.60Ga0.40As having a height towidth ratio of 1:3 embedded in GaAs [23]. The results dhowever, not depend sensitively on the assumed shapthe dots. For calculating the right-hand side of Eq. (3) whave made multiparameter variational calculations of theXwave functions [24,25] in the QD’s. As shown in Fig. 4 bthe solid line, the calculation gives a strong increase of texchange energy with decreasing dot size as a consequof the increasede-h overlap for the smaller dots.

VOLUME 82, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 22 FEBRUARY 1999

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theoretical results In0.60Ga0.40As QD's

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FIG. 4. e-h exchange interaction plotted versus QD size.

The experimental result for the present QD’s is includin Fig. 4 by a square, and the result for bulk GaAs20 meV from Ref. [21] is shown by a circle. We havalso included an estimate for bulk In0.60Ga0.40As obtainedby scaling the bulk GaAs value using the correspondmaterial parameters. The theoretical curve is taken tothrough this bulk value. The calculation describes tenhancement of the exchange energy in good accordthe experimental data.

Currently the shape of self-assembled QD’s is contversial [26]. The spectral features observed here afrom asymmetries of the QD confinement potential. Tformation of differently shaped dots most probably orignates from inhomogeneous strain distributions duringgrowth of the QD’s which may result in changes of the dshape and also of the local material composition. Thusasymmetry effects might depend on the growth paramters of the QD’s but they should occur independentlythe QD material. Because of the quantum confinemethese effects become more important for smaller sizethe dot. Single dot spectroscopy can provide detailedsight into asymmetries of the QD confinement potentHowever, further investigations are required to obtainstill more detailed picture of the geometry of the QDand its relation to the optical spectra.

This work was supported by the State of Bavaria athe U.S. Office of Naval Research.

*On leave from the Institute of Solid State Physic142432 Chernogolovka, Russia.

†Present address: Institute of Optics, UniversityRochester, Rochester, NY 14627.

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[1] J.-Y. Marzin et al., Phys. Rev. Lett.73, 716 (1994).[2] R. Leon, P. M. Petroff, D. Leonard, and S. Fafard, Scien

267, 1966 (1995).[3] M. Grundmannet al., Phys. Rev. Lett.74, 4043 (1995).[4] A. Zrenner et al., Phys. Rev. Lett.72, 3382 (1994);

K. Brunneret al., ibid.73, 1138 (1994).[5] R. Steffenet al., Phys. Rev. B54, 1510 (1996).[6] D. Gammonet al., Science273, 87 (1996); Phys. Rev.

Lett. 76, 3005 (1996).[7] E. Dekelet al., Phys. Rev. Lett.80, 4991 (1998).[8] L. Landin et al., Science280, 262 (1998).[9] A. Kuther et al., Phys. Rev. B58, 7508 (1998).

[10] U. Bockelmann, W. Heller, and G. Abstreiter, Phys. ReB 55, 4469 (1997).

[11] See, for example, D. J. Eaglesham and M. Cerullo, PhRev. Lett. 64, 1943 (1990); D. Leonard, K. Pond, anP. M. Petroff, Phys. Rev. B50, 11 687 (1994); R. Leonand S. Fafard, Phys. Rev. B58, R1726 (1998).

[12] Self-assembled QD’s are formed due to strain betwethe QD material and the substrate on which the QD’sgrown. This strain splitting is considerably larger than texchange energy or the Zeeman splitting. Thereforeneglect the light holeX in our discussion.

[13] H. W. van Kasterenet al., Phys. Rev. B41, 5283 (1990).[14] M. J. Snellinget al., Phys. Rev. B45, 3922 (1992), and

references therein.[15] E. Blackwoodet al., Phys. Rev. B50, 14 246 (1994).[16] E. L. Ivchenko and G. E. Pikus,Superlattices and Other

Heterostructures,Springer Series in Solid-State Scienc110 (Springer, Berlin, 1997).

[17] The cubic terms in Eq. (1) also cause an energy splittof the triplet X statesm ­ 22 and 12. This splitting isgiven byEas,62 ­ 0.75jby 1 bx j.

[18] K. Cho et al., Phys. Rev. B11, 1512 (1975); Nirmalet al., Phys. Rev. Lett.75, 3728 (1995).

[19] Quantum confinement causes a strong modification ogfactors. See, for example, Ref. [14] for quantum wells.

[20] W. Ekardt, K. Lösch, and D. Bimberg, Phys. Rev. B20,3303 (1979).

[21] In GaAsyAl xGa12xAs quantum wells the dependencethe exchange energy on the well width was studiin Ref. [15]. There a maximum exchange energy150 meV has been reported for narrow wells. FInxGa12xyGaAs quantum wells a considerably weakenhancement is expected due to the reduced confinempotential height.

[22] T. Takagahara, Phys. Rev. B47, 4569 (1993); R.Romestain and G. Fishman, Phys. Rev. B49, 1774 (1994).

[23] This is approximately the height to width ratio which westimate from scanning electron microscopy micrograpfrom an uncapped sample.

[24] S. N. Walck and T. L. Reinecke, Phys. Rev. B57, 9088(1998); M. Bayeret al., Phys. Rev. B57, 6584 (1998).

[25] The parameters in the calculations are for In0.60Ga0.40As,me ­ 0.0404, mh ­ 0.35, and for GaAs,me ­ 0.0665,mh ­ 0.35. The valence and conduction band offsebetween In0.60Ga0.40As and GaAs are taken to be 468 an230 meV and the dielectric constant is 15.0.

[26] See, for example, J. Kim, L.-W. Wang, and A. ZungePhys. Rev. B57, R9408 (1998).

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