Exchange interaction in InAs nanocrystal quantum dots

Superlattices and Microstructures, Vol. 22, No. 4, 1997

Exchange interaction in InAs nanocrystal quantum dots U. Banin†, J. C. Lee, A. A. Guzelian, A. V. Kadavanich, A. P. Alivisatos Department of Chemistry, University of California at Berkeley, and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (Received 1 September 1997) The near band-gap level structure in high-quality colloidal InAs nanocrystal quantum dots within the very strong confinement regime is investigated. Size-selective photoluminescence excitation and fluorescence line narrowing measurements reveal a size-dependent splitting between the absorbing and the emitting states. The splitting is assigned to the confinementenhanced electron–hole exchange interaction. The size dependence of the splitting significantly deviates from the idealized 1/r 3 scaling law for the exchange splitting. A model incorporating a finite barrier which allows for wavefunction leakage is introduced. The model reproduces the observed 1/r 2 dependence of the splitting and good agreement with the experimental data is obtained. The smaller barriers for embedded InAs dots grown by molecular-beam epitaxy, are predicted to result in smaller exchange splitting as compared with colloidal dots with a similar number of atoms. c 1997 Academic Press Limited

Key words: semiconductor nanocrystal, quantum dots, exchange splitting, InAs.

1. Introduction Understanding the nature and properties of the emitting state in semiconductor nanocrystal quantum dots is important both from fundamental and applied viewpoints. The broad color range afforded by tuning semiconductor nanocrystal size is a potentially useful property in light emitting diodes with the nanocrystals incorporated as chromophores [1] as well as in laser applications. Strongly related to realization of such applications, and equally challenging, is the understanding of scaling laws of electronic and optical properties of the nanocrystals as the size gradually changes from the bulk semiconductor into the molecular regime [2, 3]. For evolution of electronic states, such scaling laws may be obtained from the simplified visualization of the electron and hole wavefunctions in the nanocrystal as eigenstates of a particle in an infinite spherical well [4, 5]. In this paper we examine the scaling law for the enhancement of the electron–hole exchange interaction which leads to a size-dependent splitting between the absorbing and emitting states in semiconductor nanocrystals [6–9]. As the exchange interaction is proportional to the electron–hole charge density overlap this splitting is predicted to scale inversely with the spherical box volume [10]. We investigate the near-band-edge emission and absorption level structure and observe effects of the sizedependent exchange interaction in colloidal InAs nanocrystal quantum dots [11]. This is of particular interest .

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† Present address: Department of Physical Chemistry, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel.

0749–6036/97/080559 + 09 $25.00/0 sm970504

c 1997 Academic Press Limited

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as InAs belongs to the same family of tetrahedrally bonded semiconductors as CdSe, which is the prototypical colloidal nanocrystal quantum-dot system in the strong confinement regime (r ≤ a0 ) [3, 6]. InAs is a narrow ˚ very different from the properties band gap semiconductor (E g = 0.418 eV) with a Bohr radius a0 of 340 A, ˚ We study colloidal InAs nanocrystals with radii of the more ionic CdSe system (E g = 1.84 eV, a0 = 55 A). ˚ ˚ ranging between 10 A < r < 35 A, placing us in a very strong confinement regime (r  a0 ). We find that in this regime the leakage of the confined wavefunction outside of the spherical box is substantial. InAs quantum dots can also be prepared via strain-induced growth by molecular beam epitaxy (MBE) [12, 13]. This provides us with a unique opportunity to cross compare the photophysical properties of such samples with colloidal nanocrystals as studied here. These two samples differ in shape—InAs MBE dots are squarepyramidal while the colloidal nanocrystals are close to spherical. The surfaces of the two types of dots are also distinctly different. Colloidal dots are passivated by coordinating organic ligands bound to the surface atoms. The ligands serve to limit the growth of the particle during the synthesis and prevent particle aggregation. By virtue of ligand exchange schemes, they render the particles soluble in a variety of environments. This provides the ability to chemically manipulate colloidal nanocrystals into complex structures such as nanocrystal dimers [14], polymer/nanocrystal composite devices [1, 15], arrays and superlattices [16]. On the other hand, MBEgrown dots are embedded in a host semiconductor matrix. As a result of the large lattice mismatch between the matrix and the dot material, MBE dots are strained. The epitaxial matrix acts as the confining barrier at the surface of MBE dots. While this barrier is substantially lower than the barrier afforded by the organic passivating ligands in colloidal dots, the epitaxy affords more complete electronic passivation of both the cation and anion sites. We will examine the consequences of the effective barrier of the organic ‘shell’ in the colloidal dots on the exchange interaction in these samples, and compare the results with the expectations for MBE dots. Enhancement of the exchange interaction has been reported previously in other nanocrystal quantum dots [6, 9]. In the prototypical semiconductor nanocrystal system CdSe, the observed level fine structure close to the band gap has been assigned by Efros et al. within the ‘dark exciton’ model [7]. The primary conclusions of this study were that the band-gap transition is split by the exchange interaction, the crystal shape asymmetry and the symmetry of the wurtzite lattice into five levels. The lowest level is the emitting state which is characterized by low oscillator strength and consequently long lifetimes, hence the term ‘dark exciton’. In a different theoretical treatment, Franceschetti and Zunger [17] performed pseudopotential calculations which showed a 1/r 3 scaling for the exchange splitting in Si and GaAs quantum dots and a lower power law for exchange splitting in cubical CdSe quantum dots. The structure of the paper is as follows. Section 2 describes the characteristics of the colloidal InAs nanocrystal samples used in this study. The results of size selective spectroscopic measurements are given in Section 3. The observed results are modeled and discussed in Section 4 and compared with expectations for the MBE-grown samples. We conclude in Section 5. .

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2. Sample description and experimental set-up The synthesis and characterization of the colloidal InAs nanocrystal samples has been reported previously [11]. Briefly, samples are prepared using the dehalosilyation reaction of InCl3 and As(Si(CH3 )3 )3 at a temperature of 265◦ C with trioctylphishine and trioctylphosphine oxide serving as both solvents and capping agents. Following the initial reaction which produces a wide range of sizes, size-selective precipitation techniques were used to isolate specific size distributions. The samples are discrete particles consisting of a crystalline core with the bulk InAs zincblende crystal structure. Figure 1 presents a high resolution transmission electron microscope (HRTEM) picture of InAs nanocrystals. The high degree of crystalinity is evident. Faceting of the surface is also observed. The homogeneity in size and shape of the samples is demonstrated in the lower frame of Figure 1 which shows a lower resolution transmission electron micrograph of a larger field of InAs .

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˚ in diameter. Bottom: Lower resolution image of a field of InAs Fig. 1. Top: HRTEM images of two InAs nanocrystals, 22 and 50 A nanocrystals. .

nanocrystals. The particles spontaneously close pack on the carbon grid, and because of the presence of the ligand shell on the surface, they do not aggregate. Particle size and size distributions were determined using a HRTEM. We count the number of lattice fringes along the 111 direction of a statistically significant number of particles (∼200 nanocrystals per sample). The mean size obtained by this method was correlated with the band gap obtained from the first excitonic transition observed in absorption, and with the emission spectra. Figure 2 presents the histograms of sizes obtained by this technique for several samples along with their respective absorption and emission spectra. The absorption clearly exhibits excitonic structure shifted considerably from the band gap of bulk InAs at 0.418 eV. The emission consists of a single peak positioned close to the absorption onset. Both absorption and emission shift with size. The colloidal method provides an advantage of affording continuous size tunability over a broad ˚ range which is ideal for examining fundamental quantum confinement effects. Sizes ranging from 20 to 70 A in diameter have been isolated with size distributions on the order of 10%. For the low-temperature optical studies, the nanocrystals were embedded in free-standing polyvinyl butyral (PVB) films at large dilution (optical density of ∼0.3 at the first excitonic feature). Eight different samples spanning the relevant size range were used. Samples were mounted and cooled to 10 K in a continuous flow He cryostat with optical access. Emission and excitation scans were measured using a chopped monochromatized tungsten lamp as the excitation source. The emission was collected at a right-angle configuration, dispersed by a 0.25 m monochromator and detected using a liquid-nitrogen-cooled Ge diode with locking amplification. The typical resolution of emission/excitation scans was 1–2 nm. Emission scans were corrected relative to a calibrated source. Excitation scans were corrected using calibrated Si or Ge diodes placed at the sample position.

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Relative frequency

Absorbance and emission (a.u.)

31 Å

46 Å

53 Å

63 Å

0.8

1.2

1.6

2.0

20

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50

80

Diameter (Å)

Fig. 2. The right panel shows the distribution of diameters of four InAs nanocrystal samples obtained from HRTEM measurements as described in the text. The left panel shows the room-temperature absorption spectra (solid lines), and emission spectra (dashed lines) for the corresponding samples. The extracted mean diameter for each sample is noted. The emission was excited with the 514 nm line of an Ar–ion laser. .

3. Results The low-temperature emission and near-band-gap absorption spectra of a representative InAs nanocrystal sample are shown in Fig. 3. Both in emission and absorption we observe large confinement-related blue shifts from the bulk Inas band gap. In absorption an excitonic feature is clearly resolved. Higher-lying transitions are also observed. Using size-selective photoluminescence excitation spectroscopy a complex level structure is resolved consisting of nine states. The level structure in this very strong confinement regime exhibits substantial non-parabolicity. These experiments and a proper theoretical assignment of these high excited states is discussed elsewhere [18]. In emission, a well-defined peak is observed close to the absorption onset with no substantial luminescence at lower energies indicating good passivation of potential deep surface traps which would result in emission shifted to the far red. It is interesting to note in this context, the recent progress in employing the epitaxial growth approach to colloidal nanocrystals. Growth of a high-band-gap CdS shell on CdSe core nanocrystals allowed us to obtain complete electronic passivation with close-to-unity room temperature band-gap emission quantum yields [19]. As these spectra are inhomogeneously broadened by size, shape, and environment, we utilize two sizeselective optical measurements to obtain narrowed emission and absorption spectra (Fig. 3). In fluorescence line narrowing (FLN), resonant excitation at the band gap selects the largest nanocrystals of the ensemble and a narrowed emission spectra of the subset is obtained [6, 20, 21]. We use the position of the excitation window to assign a unique nanocrystal size to each spectrum. In photoluminescence excitation (PLE), narrowed detection of the emission allows us to select a subset of the ensemble and the excitation scan provides information on its absorption [22]. The detection energy is used to assign a size to each of the measured curves. .

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T = 10 K 0.5

PL Luminesence intensity (a.u.)

OD

0.3

Absorbance

0.4

0.2

0.1 FLN

PLE 0

1

1.2

1.4

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Energy (eV) Fig. 3. Ensemble absorbance (OD), and photoluminescence (PL, excitation wavelength 514 nm) along with size-selected FLN (excitation at 1.29 eV denoted by arrow) and PLE (detection at 1.32 eV denoted by arrow) for a representative InAs nanocrystal sample. .

˚ is A set of such size-selected spectra for InAs nanocrystals ranging in diameter between ∼20 and 50 A presented in Fig. 4. The set of FLN and PLE spectra resemble the mirror image relationship between absorption and luminescence well known in molecular photophysics. In both the FLN and the PLE data, a shift between the detection (or excitation) windows and the first peak in the spectrum is observed. The shift is size dependent ˚ diameter particles. A second peak is observed in both cases, and in some rising to a value of 23 meV for 22 A of the FLN spectra, a third peak is resolved as well. Peak positions from FLN and PLE spectra are extracted by non-linear fits using a sequence of Lorentzian lines and a broad background. Figure 5 shows the extracted shift of the first peak and the spacing between the first and second peaks as a function of particle radius. ˚ to We show asymmetric error bars to represent the uncertainty in the radius which is approximately 2.5 A larger radius [23]. The HRTEM method of sizing provides a lower bound to the size as it is insensitive to a non-periodic layer at the nanocrystal surface. .

4. Model and discussion We focus our attention to the observed size-dependent shift. The shifts in FLN and PLE provide evidence for a splitting between the absorbing and emitting states in the nanocrystal. Following excitation to the optically allowed band-gap state, relaxation to a lower-lying state with weaker oscillator strength takes place from which radiative recombination then occurs. The spacing between the excitation (detection) window and the first peak provides a quantitative measure of the splitting between the absorbing and emitting states. In figure 5 it can be seen that the extracted shifts in both FLN and PLE for a given size coincide. The second feature in the FLN and PLE spectra results from coupling to the LO phonon in emission and in absorption respectively.

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FLN int. (a.u)

35 Å

29 Å 31 Å 34 Å

38 Å

PLE int. (a.u)

28 Å

27 Å

44 Å 36 Å

48 Å

48 Å

52 Å

52 Å –0.12

–0.09

–0.06

–0.03

00

0.03

E–Eexc (eV)

0.06

0.09

0.12

E–Edet (eV)

Fig. 4. Size selected FLN (left frame) and PLE (right frame) spectra for a sequence of InAs nanocrystal diameters. The excitation (detection) energy is subtracted for FLN (PLE). .

This coupling is the well-known Frohlich interaction which is polar in nature. The peak position corresponds closely to the bulk InAs LO phonon frequency. We proceed to assign the observed splitting between absorbing and emitting states. The effective mass approach to the description of the electronic structure in the strong confinement regime, is to calculate the electron and hole states separately as levels in a spherical potential well. This is justified by the large confinement energies. In this approximation, the band-gap transition is degenerate. In a separate publication we discuss in detail the level structure of the higher excited states of these samples extracted from the absorption and from PLE measurements, and introduce the assignment of the states within the framework of an 8-band k · p model [18] based on the multi-band description of narrow-gap semiconductor states by Pidgeon and Brown [24]. The notation we use to identify the valence (conduction) band states is nl F where n is the level number, l is the minimum orbital angular momentum contributing to a particular level and F is the total angular momentum [25]. In this description, the lowest energy conduction band state is the 1S1/2 level, the highest valence band state designated as 1S3/2 is derived primarily from the heavy-hole and light-hole bands. The lowest transition, 1S3/2 1S1/2 , is eight-fold degenerate [18, 25]. The 8-band k · p model provides a good ˚ and yields assignment for the fit to the dependence of the band gap with size down to a radius of about 22 A, observed excited states for all sizes. The degeneracy of the band-gap transition can be lifted by electron-hole exchange interaction, shape asymmetry, strain, or by the level of lattice symmetry [7]. InAs nanocrystals have a cubic lattice structure, are not strained, and are close to spherical in shape. This leaves us with the effect of the electron–hole exchange interaction dominantly, providing a simplification compared with the case of CdSe nanocrystals. The confinement leads to enhanced exchange interaction between electron and hole wavefunctions. This results in the appearance of a splitting between a low-lying five-fold degenerate state with total angular momentum 2 (optically dark), and a higher three-fold degenerate allowed transition with total angular momentum 1. The splitting between the two levels is size dependent. Assuming an infinite spherical well, following a previous derivation in [7], the expected shift due to the exchange interaction is given by the following .

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35

30

Energy shift (meV)

25

20

15

10

5 0 10

15

20

25

Radius (Å) Fig. 5. The splitting between emitting and absorbing states from the PLE (empty circles) and the FLN spectra (filled circles), is plotted along with the best fit using the model described in the text for the enhanced electron–hole exchange splitting (solid line). Also shown is the calculated exchange splitting assuming an infinite barrier (dashed line). All the parameters aside from the barrier height are identical in both cases. Empty and filled triangles denote the LO phonon frequency extracted from PLE and FLN respectively. Bars represent the error in sizing. .


3 ˚ in InAs), a is the nanocrystal expression: 1exc = aa0 h¯ ωST χ(β), a0 is the exciton Bohr radius (340 A radius, h¯ ωST is the bulk exchange splitting value (to our knowledge, not known for bulk InAs), and χ(β) is a size-independent dimensionless function related to electron–hole charge density overlap with β = m lh /m hh , the light- to heavy-hole mass ratio. This treatment predicts a cubic dependence for the exchange splitting with size. This is a direct result of the proportional dependence of the exchange splitting on the electron–hole charge-density overlap which scales as the inverse of the volume for the infinite well case. This is not dependent on the level of approximation used in describing the highest valence band and lowest conduction band states and holds also for the simple case when single valence and conduction bands are considered. However, as shown by the dashed line in Fig. 5, this grossly overestimates the experimental dependence which scales approximately as 1/r 1.9 . A modification of the idealized scaling law for the exchange interaction is required. As the effective masses in InAs, primarily in the conduction band, are very light we expect substantial leakage of the electron wavefunction outside of the ‘box’ in the very strong confinement regime. We use a more realistic finite barrier for the spherical well. This finite barrier results in substantial penetration of the electron probability of presence outside the box and introduces a size-dependent overlap term into the exchange interaction parameter. The electron–hole charge density overlap was calculated within the 8-band k · p model using a finite barrier of 2 eV for both the conduction and valence-band states with the same set of bulk parameters that was used in the analysis of the band-gap and excited states [18]. The 2 eV barriers are used because the optical gap of the polymer matrix is 4.5 eV. The remaining parameters used are: energy gap E g = 0.418 eV, spin–orbit .

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splitting 1 = 0.38 eV, gamma parameters γ = 8.9 and γ1 = 19.7 [26], and E p = 21.6. We find that the overlap integral is reduced compared with the infinite-well case. The reduction is most substantial in small diameters in which a significant fraction of the probability of presence for the electron leaks out. The scaling of the overlap is approximately 1/r 2 , much lower then the 1/r 3 scaling for the infinite well case and in good agreement with the observed scaling of the experimental splitting. A similar effect of reduction of electron– hole charge-density overlap is observed in simpler models for the conduction and valence band states. For example, using a single-band model for the electron with a finite barrier we find considerable leakage of its probability of presence and consequently the scaling of the overlap integral is reduced compared with the infinite well case. We next use the 8-band model overlaps to calculate the exchange splitting. This amounts to multiplying the overlaps by a constant and very good agreement with the experimental dependence of the red shift with size is observed (see solid line in Fig. 5). The best fit is obtained using a value of 0.0025 meV for the electron–hole bulk exchange coefficient h¯ ωST . The small value for the exchange splitting is consistent with the very large ˚ which makes the electron–hole overlap term small in the bulk. The obtained Bohr radius in InAs (a0 = 340 A) value is approximately 50 times smaller than that reported for bulk CdSe [27], and about 10 times smaller than the value for bulk GaAs [28]. Equivalent PLE and resonant fluorescent measurements for large MBE dots (12 nm base, 6 nm height, with ∼10 000 atoms) exhibit a progression of coupled LO phonons, similar to that observed here for colloidal dots but with stronger coupling [29, 30]. Exchange-induced splitting has not been reported in these MBEgrown dots, and the excitation wavelength dependence did not exhibit a shift in the peak positions. In the MBE samples, the shape and strain effects are also important factors in determining the energy-level structure. These were recently invoked to model observed large shifts between absorption and emission in In(Ga)As-Ga(Al)As self-assembled quantum dots [31]. In light of our present study, it is interesting to estimate and compare the expected contribution of the exchange-related splitting in such samples with our results for colloidal dots. As the shapes of the two-dot samples differ, we use the number of atoms as a common basis for the comparison. We perform this estimation ˚ and for for the large MBE dot mentioned above, which is equivalent to a spherical dot with a radius of 41 A, a small MBE dot (base length 6 nm and height 6 nm, ∼2500 atoms) with an equivalent spherical radius of ˚ [13]. For this illustrative calculation we take the conduction band barrier height for the MBE dot as ∼25 A 0.93 eV, and the valence band barrier height as 0.17 eV [32]. The values calculated for the exchange-related splitting are 1.3 meV for the large dot and 3.3 meV for the small dot. The small splitting may explain why the shift was not observed in the experiments on large MBE dots. The values are smaller than the splitting calculated using the 2 eV barriers (1.6 meV, and 5.3 meV for the large and small MBE dots respectively). ˚ This comparison provides the following We observe a splitting of 5 meV for colloidal dots with r = 25 A. insight; the smaller barriers in MBE dots are expected to reduce the enhancement of the exchange splitting substantially as a result of the wavefunction leakage. In the colloidal dots on the other hand, the organic ligand ‘shell’, along with the polymer matrix provide a large barrier resulting in significant confinement enhanced exchange splitting. .

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5. Conclusions The band-gap degeneracy in InAs nanocrystals is lifted by electron–hole exchange interaction. This is the source of a size-dependent shift between the absorbing and emitting states measured in size-selective FLN and PLE. The 1/r 3 scaling law of the splitting expected within a model using an infinite barrier at the surface overestimates the observed shift. Instead, a model incorporating a more realistic finite barrier quantitatively reproduces the observed ∼1/r 2 size dependence, and a value of 0.0025 meV is extracted for the exchange splitting in bulk InAs. As the electron mass in InAs is light, even with the high barrier afforded by the organic ligand passivants, substantial probability of presence leaks outside of the particle and the scaling of

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the enhancement of the electron–hole overlap is reduced. This effect is expected to be even more significant in MBE grown dots where barrier heights are substantially smaller. An enhanced exchange interaction in three-dimensionally quantum confined nanocrystals is a general phenomena that has profound effects on the nature of the emitting state. Acknowledgements—U.B. thanks the Rothschild and Fulbright foundations for support. We thank Dr W. Jaskolski for assistance in the overlap calculations and Dr Al. L. Efros for helpful discussions. This work was supported by the U.S. National Science Foundation under contract number DMR-9505302.

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