Diamagnetic Response of Exciton Complexes in Semiconductor Quantum Dots
Ming-Fu Tsai,1Hsuan Lin,2Chia-Hsien Lin,2Sheng-Di Lin,1Sheng-Yun Wang,2Ming-Cheng Lo,1Shun-Jen Cheng,2 Ming-Chih Lee,2and Wen-Hao Chang2,*
1Department of Electronic Engineering, National Chiao Tung University, Hsinchu, 300 Taiwan 2
Department of Electrophysics, National Chiao Tung University, Hsinchu, 300 Taiwan (Received 11 June 2008; published 24 December 2008)
We report measurements of diamagnetic shifts for different exciton complexes confined in small InAs quantum dots. The measured diamagnetic responses are sensitive to the number of carriers in the exciton complexes, with systematic differences between neutral excitons, biexcitons, and trions. Theoretical calculations suggest that such systematic differences arise from very different extents of electron and hole wave functions confined in small quantum dots. The measured magnetic response of Coulomb energies is found to vary with the cube of the wave function extent, and can be a sensitive probe to the electron-hole wave function asymmetry.
DOI:10.1103/PhysRevLett.101.267402 PACS numbers: 78.67.Hc, 78.55.Cr
Excitons confined in semiconductor quantum dots (QDs) have been proven to be well suited for photonics-based quantum information processing, such as single-photon emitters [1] and quantum logic gates [2]. To ma-nipulate the charge and spin of excitons confined in a QD, it is necessary to understand the Coulomb interactions among the constituent charge carriers, as well as their responses to a magnetic field B. For neutral excitons (X), the exciton energy increases quadratically with B, i.e., the diamagnetic shift (E ¼ B2) [3–6]. The measured
dia-magnetic coefficient reflects not only the QD’s spatial confinement, but also the interparticle Coulomb interac-tions, because the magnetic field squeezes the exciton wave function, which in turn enhances the binding energy and hence reduces the overall diamagnetism. This picture is well established, and has long been used to analyze the diamagnetic shifts of X confined in various nanostructures [4–9]. However, the magnetic responses of confined ex-citon complexes, such as biexex-citons (XX) and trions (either Xþ or X) consisting of more charged particles than a
neutral exciton, are still less well known to date. The behavior of XX and X are potentially much more inter-esting, because their singlet spin configuration and the lack fine-structure splitting are of great importance for many quantum optics applications, such as the optical prepara-tion of pure spin states [10,11], as well as the generation of entangled photon pairs [12]. Although a few data for the diamagnetic shifts of XX and X have been reported [13– 15], a quantitative understanding for the magnetic response of an exciton complexes has yet to be established.
Here, we report systematic measurements of diamag-netic shifts for different exciton complexes strongly con-fined in small InAs self-assembled QDs. We demonstrate for the first time that the diamagnetic responses are sensi-tive to the number of carriers in the exciton complexes, with systematic differences between neutral excitons, biex-citons, and trions. Theoretical calculations indicate that
such systematic differences could be observed only when the confined electron and hole wave functions exhibit a large difference in their lateral extents. Furthermore, the magnetic response of Coulomb energy was found to scale as the cube of its single-particle wave function extent, and can be a sensitive probe to the electron-hole wave function asymmetry.
The sample (LM4596) was grown by molecular beam epitaxy. A layer of InAs self-assembled QDs (2.0 MLs) was grown on GaAs at480C without substrate rotations, yielding a gradient in dot density on the wafer ranging from 108 to1010 cm2. The QDs were finally capped by a
100-nm undoped GaAs layer. Atomic force microscopy of uncapped samples reveals that the InAs QDs are lens-shaped, with an average height and diameter of 2 ð0:5Þ and 15 nm, respectively. For embedded dots, the dot sizes are expected to be even smaller. Single dot spectroscopy were performed in a micro-photo-luminescence (-PL) setup combined with a 6-T super-conducting magnet. The PL signals were analyzed by a 0.75-m grating monochromator combined with a liquid-nitrogen-cooled charge-coupled device (CCD) camera.
The Coulomb interactions between electron-electron (e-e), electron-hole (e-h), and hole-hole (h-h) in individual QDs were first investigated by -PL measurements. In Fig. 1(a), the PL spectra measured at T¼ 8 K for six different QDs are displayed. In spite of the wide spread in emission energy (1330–1390 meV), the spectral feature is very similar. Four emission lines associated with the recombination of X, XX, Xþ and X can be observed. These emission lines have been identified based on power-dependent and polarization-resolved PL measurements. X and XX lines were first identified by their linear and quad-ratic power dependencies of intensities [see Fig.1(b)]. The neutral and charged excitons were then distinguished by polarization-resolved measurements. Emissions from X and XX are known to exhibit polarization doublets with PRL 101, 267402 (2008) P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2008week ending
the same fine-structure splitting (FSS) due to the e-h exchange interaction [16]. The lack of measurable FSS can thus be used to distinguish Xþ and X states from the X state.
The binding energy of XX, Xþand X, with respect to X, are shown in Fig.1(c). In all cases, the biexciton binding energy EbXXð EX EXXÞ is negative (i.e., antibinding),
varying from 1 to 6 meV for different dots, whereas the X binding energy is almost fixed at about EbX 6:2 0:4 meV. Interestingly, the Xþ binding energy Eb
Xþ also shows a variation similar to that of XX, but with a larger binding energy ranging from7 to 12 meV. This result highlights the importance of the imbalanced Coulomb repulsions and attractions caused by the different spatial extents of electron (le) and hole (lh) wave functions. The sign of these binding energies indicates that lh< le; i.e., the holes are more localized in the QDs. Quantita-tively, the direct Coulomb interaction between e-e (Vee), e-h (Veh), and h-h (Vhh) can be treated as perturbations to the single-particle state in the strong confinement regime. If the correlation effect is excluded as a first approxima-tion, the binding energy for X (Xþ) is given by EbX ¼ Veh Vee (EbXþ¼ Veh Vhh), while the XX binding
en-ergy is given by EbXX¼ 2Veh Vhh Vee. Because lh<
le, a small change in lhcan lead to a large variation in Vhh.
Therefore, it can be inferred that the variation in the XX and Xþ energies arise mainly from the fluctuation in lh
among different dots. As shown in Fig.1(c), we can see that EbXX EbXþ EbX, as expected for the first-order
approximation in the strong confinement regime.
The wave function extents and their responses to the applied magnetic field B were investigated by magneto-PL measurements. A typical result is shown in Fig. 2. The
magnetic field was applied along the growth direction (Faraday geometry). In this geometry, each line splits into a doublet by the Zeeman effect. The deduced g factor of X is g¼ 3:0 0:1 for all investigated QDs, and is identical with that of XX and X within our detection accuracy. The diamagnetic shifts reveal a quadratic depen-dence on B, as expected in the weak-field limit. As illus-trated in Fig.2(b), the average energy of each doublet was fitted to B2, from which the diamagnetic coefficient can be obtained for each exciton complex. Figure3displays the diamagnetic coefficients of X and XX for different QDs as a function of the X emission energy. All the investigated QDs show a small exciton diamagnetic coefficient X,
ranging from 7 to 11 eV=T2, and getting larger with the increasing X energy. These X values are significantly
smaller than the bulk value (109 eV=T2) and the two-dimensional (2D) quantum limit value (20 eV=T2) [6], indicating a strong confinement. In addition, we found that the biexciton diamagnetic coefficient XX is30ð10Þ%
smaller than X. A nearly identical reduction in Xþ was also observed, as shown in Fig.3.
To understand the underlying mechanism of the reduced diamagnetism, theoretical model calculations were per-formed. Because the QD shape is rather flat, the 3D exciton problem can be reduced to an effective 2D problem by using the adiabatic approximation [8,17]. For simplicity, we take a 2D parabolic potential as the effective lateral confinement for both electrons and holes with a quantiza-tion energy of @! and a wave function extent of l¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @=m
!
q
, where denotes e or h and mare the effec-tive masses. The use of parabolic model simplifies the problem considerably, because of the availability of ana-lytical formula for all Coulomb interactions being parame-terized by l. The interacting Hamiltonian can thus be
constructed using the generalized formulations for the
σ+ σ−
σ+ σ−
µ
µ
FIG. 2 (color online). (a) Magneto-PL spectra for QD2 mea-sured under B¼ 0–6 T in the Faraday geometry. (b) The B-dependent energy shifts of the X and XX lines. Solid and dashed lines are fitting curves using quadratic B dependence.
µ
FIG. 1 (color online). (a) PL spectra taken from six different QDs. The energy scale is relative to the X peak energy at 1343.7, 1359.2, 1368.4, 1370.5, 1378.4 and 1380.0 meV from QD1 to QD6, respectively. (b) Power-dependent PL intensity of X, XX, Xþand Xfor the QD3. (c) Binding energies as a function of the
X emission energy. The open symbols are Eb
XX EbXþ that
coincides very well with EbX.
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single-particle energy and the Coulomb matrix elements given explicitly in Refs. [18,19]. Predominant Coulomb matrix elements for X, X, Xþ, and XX states are those involving the lowest s orbital,
V ¼ e 2 40r ffiffiffiffi 2 r 1 l;
where lqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl2þ l2Þ=2with , ¼ e, h. Because we just focus on the diamagnetic behavior, the Zeeman terms are omitted here for brevity. The energy spectra of neutral and charged exciton complexes were then calculated using the standard configuration-interaction (CI) method within the basis constructed from the low-lying electron and hole shells [18,20].
Using the parameters le ¼ 4:5–5:6 nm and lh=le ¼ 0:51–0:55 with electron (hole) effective mass of m
e ¼
0:05m0 (mh¼ 0:5m0), we obtained a general agreement
with the experimental binding energies and diamagnetic shifts for all investigated dots [20]. The magneto-PL spec-tra were evaluated by replacing ! with the B-dependent
hybridized frequencies ðBÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 þ ð!c=2Þ2 q , where !c ¼ eB=m is the cyclotron frequency. The simulated
results of QD2 are plotted in Fig. 4, and compared with experimental diamagnetic shifts. The simulations also pre-dict that the diamagnetic shifts of XX and Xþare reduced by the same amount from that of X, consistent with our experimental findings [Fig. 4(a)]. From the calculations, we found that the fundamental cause of the reduced dia-magnetism is that lh< le. In order to highlight this point, simulations using lh¼ le are also shown in Fig. 4(c), where all exciton complexes show essentially the same diamagnetic shift.
Our calculations make clear how the different leand lh change the overall diamagnetic shifts. The applied mag-netic field superimposes a magmag-netic confinement to the QDs and thereby modifies le and lh accordingly. With the
increasing B, the magnetic confinement will be first expe-rienced by the more extended electron wave function. Therefore, the magnetic response of Vee and Veh
(Coulomb terms involving electrons) will be more
signifi-cant than that of Vhh. In the weak-field limit, where !c
!, the B dependencies of the single-particle energy " ðBÞ
and the Coulomb energies VðBÞ can be expanded
ana-lytically as,
"ðBÞ "ð0Þ þ SPB2þ . . .
VðBÞ Vð0Þ þ Coul B2þ . . .
where SP ¼ e2l2=8m is the single-particle diamagnetic coefficient, and Coul ¼ kðl6þ l6Þ=2l3 accounts for the magnetic response of Coulomb energy, with the constant k defined as ðe2=4
0rÞðe2pffiffiffiffiffiffiffiffiffi=2=16@2Þ. If the
contribu-tions from the Coulomb term Coul were ignored, i.e., only the single-particle energy is considered, XXand Xþare expected to be the same as X ¼ ðSPe þ SPh Þ SPe , which is dominated by the electron wave function extent and varies as l2e. However, the result of X> XX’ Xþ
shown in Fig. 3 indicates that the contributions from Coulomb terms Coul are non-negligible. In fact, one can see that Coul has an even stronger dependence on the wave function extent (/ l3). If we take the Coulomb energies as perturbations in the strong confinement regime, the dia-magnetic shift of X should be corrected as X ¼ ðSPe þ SP
h Þ Couleh , involving both single-particle and Coulomb
contributions. On the other hand, the diamagnetic shift of XX will deviate from that of X by an amount of ¼ X XX¼ ð2Couleh Coulee Coulhh Þ. This accounts for
how the different le and lh can lead to very different contributions of the e-e and h-h Coulomb interactions to the overall diamagnetism. Because of lh< le and hence l3
h l3e, we have Coulee ’ Couleh Coulhh , that leads to
X XX¼ ’ Coulee . The same arguments can be
ap-plied to the Xþ case, where Xþ is also reduced by a similar amount, in agreement with our experimental observations.
Quantitatively, the actual single-particle diamagnetic shift of X can be deduced from ðXþ Þ ’ SPe / l2e. On the other hand, because ’ Coulee / l3e, a plot of vs ðXþ Þ3=2 will result in a straight line with a
> =
FIG. 4 (color online). (a) Experimental diamagnetic shifts of X, XX and Xþ in QD2. (b) Simulated diamagnetic shifts for
different exciton complexes with le¼ 4:5 nm and lh=le¼ 0:52.
(c) The same as (b) but using le¼ lh ¼ 4:5 nm. The curves for
Xþin all cases and that for X in (c) have been offset by25 eV
for clarity.
γ
µ
FIG. 3 (color online). Diamagnetic coefficient of X, XX, and Xþfor different QDs as a function of the X emission energy.
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slope kð8me=e2Þ3=2. Such a plot is shown in Fig. 5, where the experimental data of XX and Xþ for all in-vestigated QDs agree quantitatively with the model pre-diction. This unambiguously demonstrates that the mag-netic response of Coulomb energy Coulee scales as l3e.
Interestingly, the slope shown in Fig.5corresponds to an electron effective mass of me¼ 0:05m0. This value is
close to but somewhat larger than the theoretical value of strained InAs islands (0:042m0) [21]. It implies that the
electron wave function has penetrated into the barrier material (me;GaAs¼ 0:067m0), leading to an increased ef-fective mass [8]. This reasoning is also supported by the result shown in Fig. 3, where a systematic increase of diamagnetic shift with the increasing emission energy can be seen. Such an increasing trend implicates that the electron gradually loses confinement as the dot size re-duces [8], which will push the electron level toward the wetting-layer continuum, resulting in a more extended electron wave function penetrating into the barrier mate-rial. On the other hand, the hole wave function remains well localized in the QDs. This explains why the estimated lh=le 0:53 for our QDs is smaller than the reported ratio of0:75 for typical InðGaÞAs=GaAs self-assembled QDs [15,19].
We would like to emphasize that the systematic differ-ences in diamagnetic shift could be observed only when the confined electron and hole wave functions exhibit a large difference in their lateral extents. Small InAs QDs inves-tigated here just meet this requirement. For larger In(Ga) As QDs, since le and lh are expected to be similar, the
diamagnetic response of all exciton complexes will be nearly identical [13,15], so that the magnetic response of interparticle Coulomb energies of such strongly con-fined few-particle systems becomes unable to resolve experimentally.
In summary, the diamagnetic responses of different ex-citon complexes confined in smallInAs=GaAs QDs have been investigated. Systematic differences between the dia-magnetic shifts of neutral excitons, biexcitons, and trions were observed. Our experimental results provide a general guideline of how the interparticle Coulomb interactions
affect the overall diamagnetism of such strongly confined few-particle systems. Whenever the diamagnetic shift was determined experimentally, both the contributions from the single-particle energies and interparticle Coulomb energies can be deduced quantitatively. Most importantly, we found that the magnetic response of interparticle Coulomb energy is even more sensitive to the wave function extent (scales as l3), as compared with the single-particle energies (l2) and the direct Coulomb energies (l1). Therefore, the measured magnetic response of Coulomb interactions can thus be a sensitive probe to the asymmetry of the electron and hole wave function extents. The experimental ap-proach and theoretical arguments reported here can also be applied to other QD systems, providing a way to take a closer look at the detailed electronic structures of various quantum confined systems.
This work was supported in part by the program of MOE-ATU and the National Science Council of Taiwan under Grant No. NSC-96-2112-M-009-014. S. J. C. ac-knowledges support from the National Center Theoretical Science of Taiwan.
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γ + ∆γ µ
γµ
∆
FIG. 5 (color online). A plot of measured to ðXþ Þ3=2
for both XX and Xþ.
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