Anticorrelation between the splitting and polarization of the exciton fine structure in single
self-assembled InAs/GaAs quantum dots
Chia-Hsien Lin,1Wen-Ting You,1Hsiang-Yu Chou,1Shun-Jen Cheng,1Sheng-Di Lin,2and Wen-Hao Chang1,*
1Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan 2Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan
(Received 30 September 2010; revised manuscript received 16 December 2010; published 28 February 2011) We report on a systematic correlation between the fine-structure splitting and polarization anisotropy of excitons in InAs/GaAs quantum dots, but with an unexpected reversal in order of the polarization eigenaxes. Such an anticorrelation is explained by a large valence-band-mixing induced splitting due to the shape and strain anisotropies. The strength and phase of valence band mixing are also found to play an important role in the tuning of fine structure splitting using an in-plane magnetic field.
DOI:10.1103/PhysRevB.83.075317 PACS number(s): 78.67.Hc, 78.55.Cr
I. INTRODUCTION
The biexciton-exciton cascade process in semiconductor quantum dots (QDs) has been considered as a promising source of entangled photon pairs1–7 for quantum cryptography8 and
quantum teleportation.9 Practical applications, however, are
limited by the fine-structure splitting (FSS) in the intermediate bright exciton state due to the anisotropic electron-hole exchange interactions.10To erase the “which path” information in the radiative cascade, tremendous effort has been made to reduce the FSS by applying external perturbations, including magnetic field,2,11,12electric field,13–16and uniaxial stress17–19.
Although entanglement has been successfully restored by a number of tuning schemes,2,16 the underlying tuning
mech-anisms remain far from clear, due to the lack of a detail understanding of the key factors controlling FSS in QDs.
The physical origins of FSS in QDs have been elucidated theoretically either within the framework of envelop function approximation20,21 or based on the atomistic pseudopotential
approach.22,23 In general, FSS in QDs is a consequence of
symmetry reduction. Two main sources of in-plane asymmetry have been recognized: (i) the intrinsic atomistic asymmetry of the underlying lattice and (ii) the extrinsic dot shape asymmetry due to a preferential elongation developed during QD growth. The atomistic asymmetry can lead to a nonzero built-in FSS even in a shape-symmetric dot.22The asymmetric shape, on the other hand, induces a nonzero long-range exchange splitting when the exciton envelop function shows in-plane asymmetry.20,21Besides, a reduced symmetry will allow mixing of isospin states of heavy and light holes in the valence band.22,24–26This mixing will further contribute a splitting into
FSS due to the exchange interaction coupled via the light-hole admixture.22,25 All of the aforementioned effects on FSS are
closely related to the size, shape, and composition profile of individual dots, which are, however, poorly known due to the quite limited information available from morphological techniques. Since it is also unlikely to access both the structural asymmetry and the optical spectra for each dot embedded in host materials, experimental verifications of the correlation between FSS and structural asymmetry of QDs are still absent thus far.
In this work we present an alternative approach to access both FSS and in-plane asymmetry of individual dots via
measurements of optical anisotropy.24–28 We show that FSS and polarization anisotropy of exciton in InAs/GaAs self-assembled QDs are correlated, but with an unexpected reversal in order of the polarization eigenaxes. Such an anticorrelation indicates that the splitting induced by the exchange interactions coupled via valence-band mixing (VBM) play an important role in the excitonic FSS in elongated QDs. The effect of an in-plane magnetic field on VBM and hence the tuning of FSS is also discussed.
II. EXPERIMENT
The QD sample was grown by molecular beam epitaxy.29 A layer of InAs self-assembled QDs (2.0 monolayers) was grown on GaAs at 480◦C without substrate rotations, yielding a gradient in dot density on the wafer ranging from 108
to 1010 cm−2. The QDs were finally capped by a 100-nm
undoped GaAs layer. A layer Al metal mask with arrays of apertures (φ∼1–2 μm) was fabricated on the sample surface for isolating single QD emissions. Single dot spectroscopy was performed in a micro-photoluminescence (μPL) setup combined with a 6-T super-conducting magnet. The PL signals were analyzed by a 0.75-m grating monochroma-tor combined with a liquid-nitrogen-cooled charge-coupled device (CCD) camera. Polarization-resolved measurements were carried out by using a linear polarizer combined with a half-wave retarder. The system has been carefully calibrated, such that polarization artifacts arising from the anisotropic responses of all optical components used can be excluded. By using Lorentzian lineshape fittings to the exciton peaks and sinusoidal fittings to the polarization-dependent intensity, the FSS, major polarization axis, and linear polarization degree can be determined with accuracy better than±5μeV, ±2◦, and ±2%, respectively.
III. RESULTS AND DISCUSSION
Figure1shows polarized PL spectra of three different QDs. Each exciton (X) line consists of a doublet|x and |y, with linear polarization eigenaxes close to the [110] (x) and [1¯10] (y) crystallographic axes (±5◦). The FSS, FS≡ Ex− Ey,
range from+100 to −30 μeV and vary from dot to dot. Each
1344.0 1344.5 1347.5 1348.0 1396.0 1396.5 1397.0 1397.5 Energy ( meV ) 1401.5 1402.0 1402.5 1403.0 PL intensity ( a rb. u nits) X XX X XX X XX (a) (b) (c) QD-A QD-B QD-C (d) QD-A [110] [110] 0 90 180 270 360 X In te n s it y ( arb. un its )
Polarization angle φ ( degree )
X+ 0 30 60 90 120 150 180 210 240 270 300 330 [110] [110] (e) Iy Ix Ix+Iy
FIG. 1. (Color online) (a),(b),(c) Polarization resolved PL spectra along [110] and [1¯10] directions for three different QDs. (d) Polar plots of the intensities of the X doublet Ix, Iy and the total X-line
intensity Ix+ Iyfor QD-A as a function of the polarization angle φ.
(e) A comparison between the polarization anisotropy of X and X+ in QD-A.
high-energy component of X shows a stronger intensity. The total-X intensity exhibits a linear polarization degree P≡
(Ix− Iy)/(Ix+ Iy) up to 26%, with a major polarization axis
close to the x axis [Fig.1(c)]. QD-B shows a smaller P=
12%. The low-energy component of X now has a stronger intensity, while the major polarization axis remains close to the x axis. For QD-C, the PL intensity is nearly isotropic. The extracted Pis negative, but with a polarization degree below
our detection accuracy (±2%). We consider P≈ 0 for this
particular dot.
Figures2(a)and (b) show the measured FSand P as a
function of the X-emission energy EX for a series of QDs.
A systematic decrease of FSwith EXand a reversal in sign
when EX 1.4 eV are observed. The sign of FSis closely
related to the order of X and XX lines, as has been reported previously.30,31In Fig.2(b), a systematic decrease of Pwith EXis also observed, but only for EX<1.4 eV. For QDs with EX>1.4 eV, which are close to the wetting layer energy of
1.425 eV, Pare scattered. For all the investigated QDs (except
QD-C), the major polarization axis aligns within±5◦ along the x axis.
The physical origin of polarization anisotropy in QDs is a combined effect of the asymmetric confinement, strain anisotropy, piezoelectricity, as well as atomistic asymmetry.24–28 For a polarization degree up to 26%, the
dominant source would be the dot-shape elongation, giving rise to an anisotropic envelop function elongated along the long axis. Theoretical calculations based on an empirical tight-binding approach27,28predicted that a polarization degree
up to 26% would correspond to a shape elongation ratio (Lx/Ly) at least up to∼1.7. The decreasing Pin Fig. 2(b)
thus indicates a reducing shape asymmetry with the reducing dot size. In Fig.3(a), a plot of FS vs P is displayed. We
found that FSS is correlated with the polarization anisotropy for larger QDs emitting at EX<1.4 eV (solid symbols).
However, no correlation was found for higher energy QDs (open symbols) near the wetting layer energy, indicating that
1320 1340 1360 1380 1400 1420 0 100 200 300 400
X emission energy ( meV )
δ0 ( μ eV ) -50 0 50 100 Δ FS ( μ eV ) (a) (b) A B C XX> X XX < X FS < FS> 0 10 20 30 P (%) A B C (c) A C B
FIG. 2. (Color online) (a),(b) The measured FSand Pof X line
as a function of EXfor a series of QDs. The data from QD-A, -B, and
-C are also indicated. (c) The measured bright-to-dark X splitting δ0
as function of EX.
the shape elongation for these weakly confined dots has no systematic size dependence.
If the dominant source of FSS is the long-range exchange interaction induced by an anisotropic envelop function, the low-energy component of X will be polarized along the dot elongation axis.32,33 Strikingly, our data suggest that, for
those QDs with FS>0, the polarization direction of the
low-energy X line is perpendicular to the major polarization axis defined by the elongation axis. Therefore, there must be a mechanism that tends to reverse the order of polarization eigenaxes, leading to the anticorrelation between FSS and polarization anisotropy. -5 0 5 10 15 20 25 30 -100 -50 0 50 100 150 200 Energy Splitting ( μ eV ) P (%) (a) A B C (b) ΔVBM -50 0 50 100 Δ FS ( μ eV ) C A 1 δ x y
FIG. 3. (Color online) (a) The measured FSas a function of P for the investigated QDs. The line is a linear fit to the data. (b)
The estimated δ1 and the VBM induced splitting VBM= 2Pδ0 as
a function of P. For clarity, data from weakly confined dots [open
The polarization anisotropy can be described by the mixing between heavy-hole and light-hole states with projections of Jz= ±3/2 and ±1/2.24–26 Because the light-hole states
are shifted in energy by the confinement and strain, the lowest hole state is still dominated by a heavy hole, with a light-hole component added as a perturbation, which reads as |ψ±
h | ± 3/2 − (ρe±2iθP/hl)| ∓ 1/2, where hl is the
heavy-light-hole energy splitting, ρ and θPare the strength and
the phase of the mixing.35Because the light-hole component
contributes a polarization opposite to that of the heavy hole, the resulting polarization will be elliptical with a linear polarization ratio|P| = 2γ /(1 + γ2) and a major polarization
axis defined by θP, where γ ≡ √ρ
3hl. From Fig. 2(b), we evaluate γ 0–0.13. The phase factor θP is associated with
the off-diagonal elements in the Luttinger-Kohn and the Bir-Pikus Hamiltonians due to anisotropic confinement26,27
and strain,25,27 respectively. For strained InAs/GaAs QDs, since both the strain and confinement are associated with the shape asymmetry,27 the resulting θ
P will thus align with the
dot elongation axis.
Now we consider the influences of VBM on FSS. Restrict-ing the consideration on the bright X subspace and considerRestrict-ing the light-hole component as a perturbation, the effective spin Hamiltonian for the exchange interactions can be expressed as10,25,32,34 ˆ HexX 1 2 δ0 δ1+ 4δ0γ δ1+ 4δ0γ∗ δ0 , (1)
in the basis of |+1, | − 1, where δ0 (δ1) is the exchange
splitting between the bright and dark states (the two bright states) of X with pure heavy-hole characters, and 4δ0γ with
γ = γ e−2iθP is induced by VBM due to the coupling of exchange splitting δ0 via the light-hole admixture.25,34 Here,
the effect of exchange interactions on VBM has been ignored. This can be verified from the polarization anisotropy of trion lines (X+), which are identical to that of X lines in the same dot [Fig.1(e)]. In the absence of light-hole mixing (γ = 0), the two bright X states coupled into |x = |+1+|−1√
2 and
|y = |+1−|−1√
2 eigenstates, resulting in two equally intense
emission lines separated by FS= Ex− Ey= δ1. Since the
dot shape and the envelop function are elongated along x, the long-range exchange interaction predicted that the low-energy eigenstate is|x,32i.e., δ
1<0. This is clearly contrary to our
result of FS>0 shown in Fig.2(a). In the presence of VBM
(γ = 0), the intensities, FSS, and directions of polarization eigenaxes will be modified, as has been detailed in Ref.25. For θP = 0◦according to the major polarization axis, FSS is
given by
FS δ1+ 4δ0γ . (2)
Here, an additional splitting induced by VBM, VBM= 4δ0γ,
is added to (subtracted from) δ1 when P>0 (P<0).
Therefore, the lower-energy component of X is not necessary to be polarized along the QD long axis,25,34 contrary to the case where only pure heavy-hole states are considered.
To estimate VBMand δ1, we have measured the
bright-to-dark splitting δ0 by applying an in-plane magnetic field,10,11
which are shown in Fig. 2(c). Similar to FS, a
system-atic decrease of δ0 with EX is also observed. From the
deduced γ P/2, we estimate VBM 2Pδ0 and δ1 FS− VBM, which are shown in Fig. 3(b). Strikingly, we
obtain a huge VBM, which appears to govern the overall
size dependence of FS. All the estimated δ1 are negative,
consistent with the elongation direction predicted by the major polarization axis. This means that FSS in those QDs with significant shape elongation (P>5%) is dominated
by the VBM induced splitting. It should be mentioned that both the measured P and the coupling of δ0 via VBM also
depend on the overlap integrals of the electron, heavy-hole and light-hole envelop functions.35 If the light hole is less
confined, the actual VBM-induced splitting will be less than 2PBD. Nevertheless, even when the estimated VBM is
reduced by a factor of two due to the less-confined light-hole envelops, the decreasing FS is still controlled by the
decreasing VBM.
The VBM can be modified by an in-plane magnetic field B through the Zeeman interactions on the electron and hole, which are given by10–12,24,25 HˆX
B = μBgeSeˆ · B−
2μB(κ ˆJh· B+ q ˆJh3· B), where ˆSe and ˆJh are the electron
and hole angular momentum operators, κ and q are Luttinger parameters. Since κ q,10,24 the direct coupling between
the heavy-hole states|±3/2 via the ˆJ3
h term is expected to
be weak. However, if the linear term κ ˆJh· B dominates the Zeeman interaction, the magnetic coupling between the|+1 and|−1 bright X states can occur only through the light-hole admixture.12,24 In other words, the Zeeman interaction will affect FSS in a way similar to the coupling of exchange interactions through VBM. For simplicity, considering the light-hole component as a perturbation, the Zeeman interaction on the hole states (|ψ+
h, |ψh−) under a magnetic field B = B(cos ϕ, sin ϕ,0) can be expressed as12,24 i,jSh,ighijBj, where Sh,x= − ˆσx/2 and Sh,y = ˆσy/2 are the pseudospin operators
( ˆσi: Pauli matrix); gijh is the transverse hole g factor tensor,
which is diagonal when the coordinate system is aligned with the major polarization axis, with gh= 12κγ = −gxxh = gyyh.
1365.0 1365.5 1368.0 1368.5 Energy ( meV ) 0 1 2 3 4 5 6 7 0 50 100 150 B ( T ) ΔFS ( μ eV ) 0 1 2 3 4 5 6 7 -50 0 50 100 B ( T ) 0 1 2 3 4 5 6 7 -50 0 50 B ( T ) (a) 0.65 e g= − 0.33 h g = − 0.60 e g= − ge= −0.71 0.30 h g = − gh= +0.03 QD C QD B X XX 1401.5 1402.0 1402.5 1403.0 1403.5 Energy ( meV ) X XX 1396.0 1396.5 1397.0 1397.5 1398.0 Energy ( meV ) X XX [110] B=0 T (b) (c) B=2.5T B=5 T B=0 T B=2.5T B=5 T (d) (e) (f) PL
intensity ( arb. units )
[110]
[110]
[110] [110][110]
FIG. 4. (Color online) The measured magneto-μPL spectra for three different QDs with (a) an initial FS>0, (b) an initial FS<
0, and (c) an initial FS<0, but P≈ 0 (QD-C). (d),(e),(f) The
corresponding FSas function of in-plane B shown in (a), (b), and
(c), respectively. The in-plane g factors ge and gh deduced from μB(ge± gh)B are also indicated.
In the weak-field limit, the off-diagonal Zeeman terms can be further treated as perturbations to the bright states, which can be expressed as ˆ HBX μ 2 BB2 2δ0 1 2 g2e+ g2h gegh gegh 1 2 g2 e+ g2h . (3)
Combining ˆHexXand ˆHBX, we obtain
FS δ1+ (4δ0+ βB2)γ , (4) where β= 12κgeμ2B/δ0. It is thus clear that, depending on
the sign of gegh, a quadratic magnetic-field-induced splitting
B(B)= βγ B2will either add to (gegh>0) or subtract from
(gegh<0) the initial FSS.
Figures4(a)to4(c)show the magneto-μPL spectra for three QDs measured at different in-plane magnetic fields B. The corresponding FSS as a function of B are shown in Figs.4(d) to (f). We found that the in-plane B always increase the FSS for the investigated QDs with P>0, as displayed in Fig.4(a)
and (b), indicating that ge and ghare of the same sign. This
also explains why only those QDs with an initial negative FSS can be tuned to zero by an in-plane B, since the magnitude and sign of ghare mainly determined by the strength and phase of
VBM. As anticipated from Eq. (4), an in-plane B cannot affect the FSS without light-hole components (γ = 0) if the ˆJh3term
is negligible. This special case is shown in Fig.4(c), where the measured FSfor QD-C is almost unaffected by the applied B.
The deduced ghfor QD-C is nearly zero because of P≈ 0 for
this particular case. It is worth pointing out that this result is independent of the field direction ϕ, confirming the dominant contribution of the linear term κ ˆJh· Bthrough VBM.
Due to the effect of VBM on FSS, the strategy for reducing FSS should further consider the way of reducing the heavy-light-hole mixing or eliminating the exchange coupling through VBM. Applying a uniaxial stress17–19to modify both
the strength and phase of VBM is very promising. However, the applied external perturbation must be aligned with the major axis defined by VBM. If a phase difference is present, the real and the imaginary parts of the off-diagonal elements in Eq. (1) cannot be eliminated simultaneously, leading to an anticrossing in the polarization eigenstates and a lower bound for FSS tuning.16,18,19 On the other hand, since the effect of
an in-plane magnetic field on FSS has an identical phase with VBM [see Eq. (4)], a truly zero FSS can be achieved regardless the field direction.
IV. CONCLUSION
We observed a systematic correlation between the FSS and polarization anisotropy of an exciton in InAs/GaAs QDs, but with an unexpected reversal in order of the polarization eigenaxes. Such an anticorrelation indicates that the splitting induced by the exchange interactions coupled through VBM play an important role in the excitonic FSS in elongated QDs. This finding will impact the prospective strategy for FSS tuning by external perturbations.
ACKNOWLEDGMENTS
This work is supported in part by the program of MOE-ATU and the National Science Council of Taiwan under Grant No. NSC99-2112-M-009-008-MY2.
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35For simplicity, we assume that the envelop functions for the
heavy-hole (χhh) and light-hole (χlh) states are identical. If not,
a factor ofχeχlh/χeχhh should be included in γ and a factor
ofχ2
eχlhχhh/χe2χ 2
hh in 4δ0γ, where χe is the electron envelop
![FIG. 1. (Color online) (a),(b),(c) Polarization resolved PL spectra along [110] and [1¯10] directions for three different QDs](https://thumb-ap.123doks.com/thumbv2/9libinfo/7504179.116773/2.911.526.780.105.420/fig-color-online-polarization-resolved-spectra-directions-different.webp)
