Anomalous in-plane magneto-optical anisotropy of self-assembled quantum dots

0.22 MB·English

by  T. Kiessling

Anomalous in-plane magneto-optical anisotropy of self-assembled quantum dots
T. Kiessling1, A. V. Platonov2, G. V. Astakhov1,2,∗ T. Slobodskyy1,
S. Mahapatra1, W. Ossau1, G. Schmidt1, K. Brunner1, and L. W. Molenkamp1
1Physikalisches Institut (EP3) der Universit¨at Wu¨rzburg, 97074 Wu¨rzburg, Germany
2A.F.Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St.Petersburg, Russia
6 (Dated: February 6, 2008)
0
We report on a complex nontrivial behavior of the optical anisotropy of quantum dots that is
0
inducedby amagnetic field in theplane of thesample. Wefindthat theoptical axis eitherrotates
2
intheoppositedirectiontothatofthemagneticfieldorremainsfixedtoagivencrystallinedirection.

n A theoretical analysis based on the exciton pseudospin Hamiltonian unambiguously demonstrates
a that these effects are induced by isotropic and anisotropic contributions to the heavy-hole Zeeman
J term, respectively. The latter is shown to be compensated by a built-in uniaxial anisotropy in a

2 magnetic field BC =0.4 T, resulting in an optical response typicalfor symmetric quantumdots.
1
PACSnumbers: 78.67.Hc,78.55.Et,71.70.-d

]
i
c
s Selfassembledsemiconductorquantumdots(QDs)at- a second spherical harmonic component (i.e., π-periodic
- tract much fundamental and practical research interest. oscillationsunder sample rotation)in the polarizationof
l
r E.g.,QDopticalpropertiesareexploitedinlow-threshold emissionfromnarrowquantumwells (QWs). Thisresult
t
m lasers, while QDs are also proposed as optically con- was explained in terms of a large in-plane anisotropy of
t. trolledqubits [1,2]. Animportant,andfrequentlysome- the heavy-hole g-factor ghxhx = −ghyyh . Subsequently, this
a whatneglectedaspectofQDsistherelationshipbetween interpretationwassubstantiatedusingamicroscopicthe-
m theirsymmetryandtheiropticalproperties. Werecently ory [9, 10].
showed that extreme anisotropy of QDs can lead to effi-
-
d cient optical polarization conversion[3]. Here,wereporttheobservationofafourthharmonicin
n
Self assembledQDs grownby molecular beam epitaxy themagneto-opticalanisotropy(i.e. π/2-periodicoscilla-
o
c generally possess a well-defined (z) axis along the [001] tions in the polarization of the emitted light under sam-
[ growth direction which serves as the spin quantization ple rotation) from CdSe/ZnSe self assembled QDs. We

axis. It thus makes sense to assume the dots have D demonstrate that this effect is quite general. An impor-
1 2d
point-groupsymmetry,andtodescribeoftheinfluenceof tantconsequence is that the polarizationaxis hardly fol-
v
0 an external magnetic field using an isotropic transverse lowsthe magneticfielddirection,thusthe classicalVoigt
7 heavy-hole g-factor g⊥ in the plane of the sample (xy). effectisnotobservedforQDemissionassociatedwiththe
hh
2 Many actual QDs, however, exhibit an elongated shape heavy-holeexciton. Moreover,incontrasttoearlierstud-
1 intheplaneandasimilarstrainprofile,theirsymmetryis ies of in-plane magneto-optical anisotropies, we consider
0 reduced to C or below. In this case the in-plane heavy the contributions of the electron-hole exchange interac-
2v
6
hole g-factoris no longerisotropic[4]. Moreover,evenin tion,whichhavebeenignoredforQWs[8,9]andarezero
0
zerofieldthedegeneracyoftheradiativedoubletislifted for charged QDs [4]. An anisotropic exchange splitting
/
t duetotheanisotropicexchangesplitting[5,6,7]. Anyof may lead to the occurrence of a compensating magnetic
a
m these issues will give rise to optical anisotropy, resulting field BC. When the externally applied field equals BC,
inthelinearpolarizationofthephotoluminescence(PL). theamplitudeofthesecondharmoniccrosseszero,result-
-
d In this manuscript, we discuss the optical anisotropy inginahighlysymmetricalopticalresponseofextremely
n anisotropic QDs. For our QDs we find B 0.4 T. In-
of QDs in the presence of a magnetic field. Classically, C
o triguingly, at this condition the polarization≈axis rotates
the polarizationaxisof the luminescence of a givensam-
c
away from the magnetic field direction.
: pleshouldbecollinearwiththedirectionofthemagnetic
v
field. Thisso-calledVoigteffect,impliesthatwhenrotat-
i
X ing the sample over an angle α while keeping the direc- In order to measure anisotropy we used the detection
tion of the magnetic field fixed, one observes a constant scheme presented in Fig. 1a. The direction of the in-
r
a polarization (in direction and amplitude) for the lumi- plane magnetic field is fixed, while the sample is rotated
nescence. Mathematically one can express this behavior overanangleα. The degreeoflinearpolarizationis now
as following a zeroth order spherical harmonic depen- defined asργ(α)=(Iγ Iγ+90◦)/(Iγ+Iγ+90◦). Here the
−
dence on α. This situation changes drastically for low angle γ corresponds to the orientation of the detection
dimensionalheterostructures because of the complicated framewithrespecttothe magneticfieldandIγ isthe in-
valence band structure. Kusrayev et al. [8] observed tensityofthePLpolarizedalongthedirectionγ. Within
anapproximationofweak magneticfields, a sample that
has C point symmetry in general may have only three
2v
spherical harmonic components [8] linking the polariza-
∗Electronicaddress: [email protected] tionoftheemissiontothesamplerotationangleα. Thus,
2

2.0
(a) asaaaampZle BY (bs ) a m pflreamone axisB a jq = j[[111_000]] PL intensuty detectTBi=o=1n0(.T6aK) (b) 1.5
d e t e c tfiroanm e or g olarizatia2=a4=0 a„0„„„ 0 2.60 2E.n65erg y (2e.V70) 2.75 eld (T)
g X P 0 -th harmonic %)1 BBCC (c) c fi
(cs ) a m pflreame B q =j - 45° (ds ) a m pflreameB j q = j- mplitudes (-10 a0 a4 BC 0 Magneti.5
Polarizatioan0 = aax4i=s0 a„2„„„ 0 Polarizationa 0a=xais2=0 a„4„„„ 0 Harmonic a--32 a2 0.0
2-nd harmonic 4-th harmonic 0 1 2 3 0 90 180 270 360
Magnetic field (T) Rotation angle (°)

FIG. 1: (Color online) (a) Schematic layout of the angle-
FIG. 2: (Color online) (a) PL emission spectrum of the
resolvedexperimentswhereαistherotationangleofthesam-
CdSe/ZnSeQDsample. (b)3Dplotofthelinearpolarization
0p◦le.orT4h5e◦d)ewteitchtiornesfpreacmtetoργthisermotaagtnedetbicyfiaenldanBg.le(γb)(-e(qdu)aDlitfo- ρ45(α,B) as function of the rotation angle α and magnetic
inductance B. Light (red) and dark (blue) areas correspond
ferentscenariosformagneto-opticalanisotropy(showninthe
to positive and negative values of ρ45(α,B), respectively. (c)
sample frame). (b) The polarization axis follows the mag-
Amplitudes of the zeroth (a0), second (a2) and fourth (a4)
netic field and θ = ϕ, leading to a zeroth harmonic in the
sphericalharmonicsvsB. Thesymbolsareexperimentaldata
anglescan. (c)Thepolarization axisisfixedatθ=const(ϕ),
and the solid lines result from calculations. The arrows in
resulting in a second harmonic in the angle scan. (d) The
panels (b) and (c) indicate thecompensating field BC where
polarization axis rotates away from the magnetic field with
θ=−ϕ, leading to a fourth harmonic in theangle scan. thelinear polarization is π/2-periodic (a2 =0).
one has for γ =0◦ and γ =45◦ observed for the fourth harmonic a4 a0 , a2 , when
| | ≫ | | | |
the polarization ρ (α) in the detection frame changes
γ
ρ (α)=a +a cos2α+a cos4α twice as fast as any polarization linked to the sample
0 0 2 4
frame. This implies that the polarization axis turns
ρ (α)=a sin2α+a sin4α, (1)
45 2 4
in opposite direction to that of the magnetic field, and
with a , a and a the amplitudes of the zeroth, second θ(ϕ)= ϕ (Fig. 1d). In other words, it is collinear with
and fo0urth2 harmo4nic, respectively. α is measured with the mag−netic field when B [100],[010]and perpendicu-
respect to the [110] crystalline axis. lar to the magnetic field whken B [110],[110].
k
For an analysis of the various orientation effects it is We studied the magneto-optical anisotropy of CdSe
more convenientto turn from the detection frame to the QDs in a ZnSe host. The samples were fabricated on
sample frame. Now the sample orientation is fixed and (001)GaAssubstratesusingmolecularbeamepitaxyand
the magnetic field rotates by the same angle α but in self-assembly after depositing one monolayer of CdSe on
opposite direction (Figs. 1b-d). The orientation of the a50nmthickZnSelayer. TheQDswerethencappedby
magnetic field in the spin Hamiltonian, which we dis- 25nmofZnSe. Foropticalexcitationat2.76eVweused
cussbelow,canbe incorporatedmoreconvenientlyusing astilbenedye-laserpumpedbytheUV-linesofanAr-ion
a basis along the [100], [010] and [001] crystalline axes. laser. A typical PL spectrum is shown in Fig. 2a. For
Therefore we introduce the angle ϕ between the mag- detection of the linear polarization we applied a stan-
netic field direction and the [100] axis, as ϕ α 45◦. dard technique using a photo-elastic modulator and a
≡ −
We then consider the orientationofthe polarizationaxis two-channel photon counter [3]. The angle scans were
described by an angle θ in the same [100] basis. performedwiththesamplesmountedonarotatingholder
Wearenowinapositiontodiscusssomelimitingcases controlled by a stepping motor with an accuracy better
of Eq. (1). When the zeroth order spherical harmonic than 1◦. Magnetic fields up to 4 T were applied in the
dominates, a a , a , the polarization axis of the sample plane (Voigt geometry), optical excitation was
0 2 4
| | ≫ | | | |
emission coincides with the magnetic field direction for done using a depolarized laser beam. The degree of lin-
any orientation of the sample, θ(ϕ)=ϕ (Fig. 1b). For a ear polarization of the luminescence of the QDs was de-
dominantly second harmonic response, a a , a , termined at 2.68 eV at a temperature T =1.6 K.
2 0 4
| | ≫ | | | |
thepolarizationaxisisfixedtoadistinctsampledirection The result of angle scans of ρ for various magnetic
45
anddoesnotdependonthemagneticfieldorientation. In fieldstrengthsisshowninFig.2b. Inzeromagneticfield
our experiments, the fixed polarization axis is [110] and the linear polarization is π-periodic (see also Fig. 3a).
θ(ϕ)= 45◦ (Fig. 1c). The most interesting behavior is This demonstrates the low (C ) symmetry of the QDs
2v
−
3

n (%) 4 (a) (b) B» BC awnhderheePaˆvyis-htohlee bdaipnodlse amreomh±en12teulm|Pˆ|op±er23ahthori.=W∓eed±efi[7n]e,
zatio 2 e± = (ex ±iey)/√2 with ex k [100] and ey k [010] are
polari 0 ufonritthveecftoourrs.pWosesitbhluesofipntidcafolrtrtahnesoitpiotincsalmatrixelements
ar -2
Line-4 r45 B=0T r45 B=0.4T hΨηel|Pˆ|Ψµhhi∝−e++ηµei(θhh−θel)e−. (3)

%) 4 (c) (d) [110] These matrix elements thus predict a linear polarization
n ( [1_10] withanaxisthatisrotatedoveranangleθ = 1(θ θ )
atio 2 from [100] when η = µ (η,µ = 1), and ro2tathehd−oveelr
polariz 0 [100] [110] θ =Fo21r(aθnhhel−ecθterlo)n+in90a◦n−wexhteenrnηal=mµa±.gneticfieldB,wecan
near -2 r r write the Zeeman Hamiltonian as
Li-4 45 B=1T 0 B=4T
1
⊥
0 Rot9at0ion a1n8g0le, a27 (0°) 0 Ro9ta0tion1 a8n0gle,2 a7 0(°) 360 Hel = 2gelµB(σxBcosϕ+σyBsinϕ) , (4)
where g⊥ is the electron g-factor, yielding directly θ =
FIG. 3: (Color online) Angle scans of the linear polariza- el el
ϕ. For D symmetry, the interaction of holes with an
tionoftheluminescence. Thesymbolsareexperimentaldata, 2d
in-plane magnetic field B can be described by [12]
solidlinesrepresentcalculations basedontheHamiltonianof
Eq. (7). (a) Built-in linear polarization (B = 0 T). (b) A
=q g µ J3Bcosϕ+J3Bsinϕ , (5)
highly symmetrical optical response (i.e. π/2-periodic) ap- Hhh 1 0 B(cid:0) x y (cid:1)
pears at a compensating field Bc ≈ 0.4 T. (c) Angle scan in
amagneticfieldB=1TexeedingBC. (d)Anglescaninthe whereq1 isaconstant. Itsufficestonotethatthepartof
saturationregime,B=4T.Thedetectionframeργ in(a)-(c) matricesJx3 andJy3,relatedtotheangularmomentumof
is rotated by an angle γ = 45◦; in (d) γ = 0◦. The arrows heavy holes, behave as 3σ and 3σ , respectively [7].
indicatethepointswheretheorientationofthemagneticfield Oncomparingtheresulti4ngxHamil−to4niaynwithEq.(2)this
aligns with high-symmetry directions in thesample frame.
implies θ = ϕ for the eigenstate,yielding [cf. Eq.(3)
hh
−
and the text thereafter] a rotation over an angle θ = ϕ
orθ = ϕ+90◦ forthe polarizationoftheluminescen−ce.
(the dots are elongatedin the plane) resulting in a finite −
InaccordancewithFig.1dthiscorrespondstothefourth
value of a in Eq.(1). One can regardthis measurement
2 spherical harmonic.
asreflecting the ’built-in’ linearpolarizationofthe array
The second harmonic may appear for structures with
of dots. For high magnetic fields a changes sign and
2 C symmetryorbelow. Inthiscasethereisacorrection
an additional fourth harmonic signal appears (see also 2v
to the magnetic interaction, given by [12]
Fig. 3c). At an intermediate magnetic field B 0.4 T
C
≈
the second harmonic crosses zero, while the fourth har- ′ =q g µ J3Bsinϕ+J3Bcosϕ , (6)
monic remains finite (see also Fig. 3b). The resulting Hhh 2 0 B(cid:0) x y (cid:1)
π/2-periodic optical response corresponds to the higher
where q is C invariant. From this corection one finds
2 2v
(D2d) symmetry expected for symmetric (i.e., not elon- θ =ϕ 90◦, leading to a rotation of the luminescence
hh
gated) QDs. This observation clearly demonstrates that polarizat−ion over θ = 45◦, which implies (Fig. 1c) a
the in-plane optical anisotropy of QDs can be compen- ±
response following a second spherical harmonic.
satedbyin-planeZeemanterms. Wealsofindthattheze-
Foraquantitativeanalysisamoredetailedapproachis
rothharmonicsignalisweak. Thisisclearlyseene.g. for
required which we provide below. The essential experi-
theρ anglescaninFig.3dwhichwastakenforB =4T,
0 mentaldataaresummarizedinFig.2c,whereweplotthe
where the field-induced alignment is saturated.
amplitudesofthesphericalharmonic,i.e. thecoefficients
We now model our observations using a pseudospin
a ,a anda ,extractedfromthefitsoftheexperimental
0 2 4
formalism [9, 11]. We denote by | + Spi and | − Spi data on ργ(α) using Eq. (1) vs magnetic field (symbols).
the wavefunctions of an electron (p = el) or heavy-hole
In QDs, the electron-hole exchange interaction is sig-
(p = hh) with pseudospin projection S along the z
direction. WedefineapseudospinHamil±tonianinmatrix nificant and a corresponding term Hex must be taken
into account [13], resulting in an exchange splitting δ
form for each band, as follows 0
between the L = 1 and L = 2 exciton states,
z z
| ± i | ± i
where L is the projection of the angular momentum of
δ z
p
Hp = 2 (σxcosθp+σysinθp), (2) the exciton. Moreover,Hex leads to a (smaller) splitting
δ < δ of the nonradiative exciton states. When the
2 0
where σ and σ are the Pauli matrices. This Hamilto- symmetry is lowered to C , an additional anisotropic
x y 2v
nianhaseigenfunctions Ψ±1 +Sp eiθp Sp . The exchange term ′ [7, 14] appears and results in an ad-
p ∝| i± |− i Hex
opticalmatrixelements foratransitionbetweenelectron ditional splitting δ of the radiative doublet.
1
4
In the basis of the exciton states Φ = L = 1 sicallydeterminesthesaturationlevelofthepolarization
1,2 z
| ± i
and Φ = L = 2 , the final spin Hamiltonian = at high magnetic fields.
3,4 z
+ +| ′ +± i + ′ is given by the followHing
Hel Hhh Hhh Hex Hex
matrix [7, 14]
Using this approach,we have calculatedthe linear po-
′ ′
larizations ρ and ρ in the sample frame using
δ iδ δ δ (100) (110)
0 1 el hh
H= 21 δiδδ∗e∗1l −δδδh0h −δδh∗δh0 δδe∗δ2l  . (7) ρρE′(′q1.10()9cc)oo.ssF22ααro.m+Wρte′(h1wi0s0e,)rwseienab2folαeuntaodnrdtehpρer4o5pd(oαul)caer=iazallρte′(io1x1np0s)ersρii0mn(2αeαn)t−=al
hh el 2 − 0 (100)
datausingauniquesetofparametersforgivenexcitation
Here, δel = µBge⊥lB+ and δhh = µB(cid:0)ghihB++ighahB−(cid:1) powerandexcitationenergy. Thecalculationsweredone
arein-planeZeemanterms,gi = 3g q andga = 3g q taking a bath temperature T = 1.6 K and a coefficient
hh 2 0 1 hh 2 0 2
are the isotropic and anisotropic contributions to the K =0.04. Fromthebestfitswefoundexchangeenergies
heavy-hole g-factor. B± = Be±iϕ are effective magnetic δ0 =2.9meV,δ2 =0.1meV,δ1 =0.2meVandg-factors
fields. Analytical solutions for the energy eigenvalues g⊥ =1.1, gi = 0.5, ga =0.6 .
e hh − hh
E and the normalized eigenfunctions Ψ = ΣV Φ of
j j nj n
Hamiltonian (7) can be found in the high magnetic field
An exemplary result of the calculations are the solid
limit (δ , δ δ , δ , δ ), as follows
hh el 0 1 2 lines in Fig. 3, which follow the experimental data very
| | | |≫| | | | | |
closely. For a large number of similar fits we have eval-
E = δ δ ; E = δ δ ,
1,4 ∓| el|∓| hh| 2,3 ±| el|∓| hh| uated the harmonic amplitudes (a0, a2 and a4) using
Fouriertransformation. Theresultsasafunctionofmag-
neticfieldareplottedassolidlinesFig.2c. Fromthisplot
1 1 1 1 we find a compensating field BC of 0.42 T, which coin-
1 e2iθ e2iθ e2iθ e2iθ  cides within the detection error with the experimental
V = 2 e−iθel e−−iθel −e−iθel e−iθel , (8) value. We observe that as a general trend the isotropic
−eiθhh eiθhh −eiθhh eiθhh  hole g-factor ghih is responsible for the fourth harmonic
− − andtheanisotropicg-factorga isessentialforthesecond
hh
weih(θehrhe−θeeilθ)elha=veδthele/|sδaeml|,eemiθehahnin=g aδhs∗hi/n|δEhqh.|(a3n).dEeq2si.θ(8=) hnaetrimcoQnWic.sT[8h]eanladttoeurrwfiansdrinepgosrftoeldlowpretvhieougselnyerfaolrtmreangd-
of previous theoretical approaches [9]. We conclude that
clearly show the same trends expected from the qualita-
the very goodagreementwith experiment provesthe va-
tive analysis presented above [Eqs. (2-6)]. For a given
lidity of our approach.
finite magnetic field, solutions for E and Ψ can easily
j j
be calculated numerically.
Using the solutions to Hamiltonian (7) we may obtain Summarizing, we have observed anomalous behavior
the intensity andpolarizationofthe luminescence. Since of the in-plane magneto-optic anisotropy in CdSe/ZnSe
only the Φ = L = 1 excitons are optically active, QDs, in that the second and fourth spherical harmonics
1,2 z
theopticalmatri|xelem±entiinanarbitrarydirectionefor of the response dominate over the classical zeroth order
eigenfunction Ψ can be written as: M (e) = V e + response. Weshowtheexistenceofacompensatingmag-
j j 1j +
−
V2je− [14]. We then can calculate the intensity of the neticfield,leadingtoasymmetryenhancementofQDop-
luminescence linearly polarized along an axis rotated by tical response. All of these findings could be excellently
an angle ξ with respect to the [100] crystalline axis as modeled using a pseudospin Hamiltonian approach, and
I = M (e ξ)2. provide direct evidence for the existence of a preferred
j,(ξ) j
| || |
The polarization of the PL from an ensemble of QDs crystalline axis for the optical response, and for the im-
mustbe averagedoverthe thermalpopulationofexciton portanceoftheexchangeinteractionfortheopticalprop-
states, and can in the sample frame be expressed as erties ofself assembledQDs. The physics responsible for
our findings is not limited to QDs and can be applied to
ρ′ =KPj Pj(cid:0)Ij,(ξ)−Ij,(ξ+90◦)(cid:1), (9) other heterostructures of the same symmetry where the
(ξ) Pj Pj(cid:0)Ij,(ξ)+Ij,(ξ+90◦)(cid:1) heavy-hole exciton is the ground state.
where Pj e−Ej/kBT is the Boltzmann factor . K is a ThisworkwassupportedbytheDeutscheForschungs-
∝
scaling factor which corrects for spin relaxation, and ba- gemeinschaft (SFB 410) and RFBR.
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