Intrinsic Gap States in Semiconductors With Inverted Band Structure

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C: Physical Processes in Nanomaterials and Nanostructures

Intrinsic Gap States in Semiconductors With Inverted Band Structure: Comparison of SnTe vs PbTe Nanocrystals Roman Vaxenburg, Janice E. Boercker, Danielle L. Woodall, Chase T Ellis, C. Stephen Hellberg, Alexander Lev Efros, and Joseph G. Tischler J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01871 • Publication Date (Web): 09 Apr 2019 Downloaded from http://pubs.acs.org on April 9, 2019

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The Journal of Physical Chemistry

Intrinsic Gap States in Semiconductors with Inverted Band Structure: Comparison of SnTe vs PbTe Nanocrystals Roman Vaxenburg,† Janice E. Boercker,‡ Danielle L. Woodall,¶ Chase T. Ellis,‡ C. Stephen Hellberg,‡ Alexander L. Efros,∗,‡ and Joseph G. Tischler‡ George Mason University, Fairfax VA, USA, Naval Research Laboratory, Washington DC 20375, USA, and Previous Naval Research Laboratory, Washington DC 20375, USA contractor E-mail: [email protected]

Abstract In this contribution we compare size-dependent optical properties of PbTe and SnTe nanocrystals (NCs). We demonstrate that the size dependence of the band edge absorption line and the photoluminescence (PL) of PbTe NCs can be quantitatively described by optical transitions between the lowest quantum confined states of conduction and valence bands. In contrast, the optical properties of SnTe NCs are strongly influenced by intrinsic gap states that arise from the inverted band structure that can be described by introducing a negative energy gap Eg < 0. In principle, these intrinsic gap states could be observed directly in PL and absorption of small size SnTe NCs, where the wave functions of these states are spread over entire NC volume. In our samples, however, these transitions are not observed due to self-doping of SnTe NCs by ∗ To

whom correspondence should be addressed Mason University, Fairfax VA, USA ‡ Naval Research Laboratory, Washington DC 20375, USA ¶ Previous Naval Research Laboratory, Washington DC 20375, USA contractor † George

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holes created from negatively charged Sn vacancies. As a result, the absorption is blue shifted due to filling of the gap states and confined valence band levels by holes (phenomenon known as Moss-Burstein effect) and PL is completely suppressed due to nonradiative Auger recombinations. The size –dependent Moss-Burstein effect in our NCs is quantitatively modeled by assuming a hole concentration of p ≈ 2.2 · 1020 cm−3 which corresponds to several tens of holes per a NC.

Introduction The multi-band effective mass approximation is a powerful tool that can be used to calculate the electronic properties of both bulk and nanostructured semiconductors. These electronic properties include electron/hole transport, 1 optical transitions, 2 and magneto-optical effects. 3 The quantitative description of all direct gap II-VI and III-V compound semiconductors, which have the band edge at the Γ-critical point of the Brillouin zone is conducted within the 8 band (two electron bands and six valence bands) Pidgeon and Brown model introduced in 1966. 4 In order to describe IVVI semiconductors that have a direct band gap at the L- critical point of the Brillouin zone, such as PbS, PbSe and PbTe, the commonly used 4 band model was introduced in 1964 by Dimmock and Wright. 5 Simplification of the model from 8 bands to 4 bands is a result of the reduction in the valence band complexity due to the electron-hole band symmetry at the L-point. All semiconductors within these models are described by a specific set of energy band parameters, which include the bulk energy gap Eg ; the Kane matrix element P that describes coupling between conduction and valence bands; and the remote band contributions to the effective masses of electrons and holes. The quantum confinement levels and the spectra of the inter-band optical transitions of spherical nanocrystals (NCs) for semiconductors with the band edge at the Γ-critical point have been successfully described using the Pidgeon and Brown model or its 6 band modification that describes the level structure of the valence band. 6,7 Similarly, for semiconductors with the band edge at the L-critical point, the energy levels and the optical properties of NCs have been successfully described using a spherical approximation of the 4 band Dimmock and Wright model. 2 These 2


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models were also successfully employed to describe the energy spectra and optical properties of nanostructures with more complex shape, as demonstrated by Sercel and Vahala. 8 Although the Pidgeon and Brown model were highly successful describing bulk properties of many of II-VI and III-V compounds, experimental data on bulk HgTe revealed two controversial properties: 1) the absence of an optical gap and 2) a finite electron effective mass. These two properties were not expected to occur simultaneously since within this model the effective mass is generally inversely proportional to the energy gap ∼ P2 /Eg . 9 In order to resolve this controversy, the authors of Ref. 9 formally introduced the so-called negative energy band Eg < 0 into the Pidgeon and Brown Hamiltonian. This modification moves the Γ6 band that typically describes the bottom of the conduction band below the four-fold degenerate Γ8 band that typically describes the top of the valence band. In addition, the inclusion of Eg < 0 in the model flips the curvature of the former light hole band (Γ8 ), making it the electron band, and similarly, the former conduction band Γ6 becomes the hole band. With the inclusion of a negative gap, the effective mass of the light hole in narrow gap semiconductors is ∼ P2 /|Eg |. 4 In addition to the work done on HgTe, the inverted band structure was also formally introduced to describe the composition dependence of the conduction and valence bands of Pb1−x Snx Se alloys. 10 In this material system, as the Sn content is increased in the PbSnSe alloy, the bandgap (Eg > 0) decreases and vanishes (Eg = 0) at a composition of x = 0.19(0.30) at T = 77(300) K. Increasing the Sn content further opens the gap again, but the symmetry of the electron and valence band is reversed (i.e., Eg < 0) as shown in the band diagram in Fig. 1a. Although introduction of the negative Eg appears to be a simple mathematical formalism that describes the inversion of the electron and hole band symmetry in Pb1−x Snx Se and Pb1−x Snx Te alloys, it has serious physical consequences. One of known consequences of the negative Eg is the presence of intrinsic gap states that form at the interface between PbSe (semiconductor with Eg > 0) and SnSe or SnTe (semiconductors with Eg < 0). 11 Using the Dimmock and Wright model, Volkov and Pankratov demonstrated that these intrinsic surface states do not depend on the properties of the interface transition area. 11 The significance of this prediction was not understood until

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2012 when it was demonstrated that the band structure of SnTe corresponds to that of a topological crystalline insulator (TCI). 12 The electronic structure of a topological crystalline insulator is protected not by time reversal symmetry, as in conventional topological insulators, but rather by a symmetry of the bulk crystal. In SnTe, this symmetry is a reflection symmetry about the 110 mirror plane. 13–15 Topological crystalline insulators have protected surface states with an even number of Dirac cones on any surface that does not break the symmetry. In SnTe, the 001, 110, and 111 surfaces are all symmetric with respect to the 110 mirror plane and have protected surface states. Angular-resolved photoemission spectroscopy (ARPES) was used to probe the 001 and 111 surfaces of SnTe, finding the expected Dirac-cone surface states. 16,17 In contrast to SnTe, PbTe is topologically trivial under ambient pressure and has no protected surface states. 12 In genreral, gap states only exist in the vicinity of the surface for bulk materials with Eg < 0, as such, these gap states usually do not affect the optical absorption of bulk semiconductors, and can be observed only in photololuminescence (PL) or surface reflection measurements that mainly probe the surface. In contrast to bulk materials, for small–sized NCs composed of TCI materials, surface states are extended over their entire volume. For example, in InAs NCs these states lead to the addition of optically allowed transitions that can exist within the bulk energy gap, resulting in a strong modification of both absorption and PL of InAs NCs. 18 The electronic properties of another topoplogical insulator HgTe NCs were studied in Ref. 19 In this contribution we investigate the size–dependent optical properties of PbTe and SnTe NCs. We show that in the case of PbTe NCs no gap states are observed and that the optical properties of this material system can be explained through regular quantum confined states. In contrast, we find that the optical properties of SnTe NCs are strongly influenced by surface gap states due to the inversion symmetry of the electron-hole bands (i.e., Eg < 0). However, we demonstrate that in SnTe NCs these effects are experimentally obscured by the well known self-doping of SnTe 20–22 from negatively charged Sn vacancies. 23 As a result the optical transitions undergo a Moss-Burstein effect, where transition energies are blue-shifted due to filling of confined valence

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Figure 1: Comparison of band structure and optical properties of PbTe and SnTe NCs. a) Diagram of the composition dependence of the energy band structure for Pb1−X SnX Te. b) and c) show TEM images of a typical PbTe NC sample at different magnifications. d) Absorbance (blue curves) and PL (green curves) spectra from PbTe NCs with radii in the range of 4.7 - 8.5 nm, as determined by TEM measurements. e) and f) TEM images of a typical SnTe NC sample at different magnifications. g) Measured absorbance spectra of SnTe NC samples with radii in the range of 6.9 - 9.7 nm, as determined through TEM. band levels with holes. Our results indicate that the broad absorption feature 24–27 and absence of PL in SnTe NCs can be explained as a result of inhomogeneous doping of different NCs, which broadens the absorption feature and leads to high non-radiative recombination rates as a result of Auger processes. Our absorption measurements and modelling indicates that the SnTe NC studied in this work have a hole doping density of approximately ∼ 2.2 · 1020 cm−3 , which corresponds to several tens of holes per NC. In prior work, the broad absorption feature of SnTe NCs has been attributed to plasmonic effects; 25,26 however, in contrast to this, we find that the doping density is too low to support a plasmonic absorption feature in the 0.4-0.7 eV energy range.

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Methods General Considerations. Standard Schlenk-line techniques were used unless otherwise noted. Oleic acid (90%), 1-octadecene (90%), lead oxide (99.999%), tellurium (shot, 99.999%), trioctylphosphine (90%), oleylamine (70%), bis[bis(trimethylsilyl)amino]tin(II), toluene (anhydrous 99.8%), acetonitrile (anhydrous, 99.8%), tetrachloroethylene (anhydrous, =99%), acetone (99.9%), and chloroform (anhydrous, =99%) were all purchase from Sigma Aldrich. Trioctylphosphine (97%) was purchase from Strem Chemicals. All chemicals were used as received except that the 1-octadecene, used for the bis[bis(trimethylsilyl)amino]tin(II) solution, and the oleic acid, used in the SnTe ligand exchange, were dried at 110 o C under vacuum for 2 hours and placed over activated 0.3 nm molecular sieves in a glovebox prior to use. PbTe Nanocrystal Synthesis. PbTe NCs were made using a method developed by Murphy et al. 33 with only minor changes. First, a 0.5 M stock solution of trioctylphosphine telluride (TOPTe) was made by mixing 0.638 g Te with 10 mL of trioctylphosphine (90%) and heating to 150o C under argon for 30 minutes. As the Te dissolved the TOPTe solution turned yellow. Next, 0.225 g PbO, 1.89 g oleic acid, and 4.7 g 1-octadecene were added to a 25 mL three neck round-bottom flask and heated to 110o C under vacuum for 30 minutes, the yellow/orange PbO powder dissolved and the solution went clear, indicating the formation of Pb oleate. The Pb oleate solution was backfilled with argon and heated to the injection temperature, whereupon 1 mL of room temperature, 0.5 M TOPTe solution was injected into the hot Pb oleate solution. Immediately after injection, the solution tuned dark brown and the flask was moved to an oil bath at 110o C for the remainder of the reaction. Reaction time (3-6 minutes) and injection temperature (150-190o C ) were used to change the NC size, with higher temperatures and longer times resulting in larger NCs. At the end of the reaction the flask was moved to a cold water bath and 2 mL of room temperature toluene was injected. Finally, the reaction solution was taken into a glovebox and unreacted precursors and reaction byproducts were removed from the PbTe NCs by adding acetonitrile and toluene and centrifuging. This purification process was repeated before the nanocrystals were stored dried out (evaporation of excess solvents) in a glovebox. Typically reaction yields were ∼25%. 6


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SnTe Nanocrystal Synthesis. SnTe NCs were made using a method created by Kovalenko et al. 24 with very minor modifications. First, a stock solution of 0.73 M trioctylphosphine telluride (TOPTe) was made by mixing 0.931 g Te with 10 mL of trioctylphosphine (97%) and heating to 150o C under argon for 30 minutes. Note: we have tried this reaction with the 97 % and the 90% pure trioctylphosphine, and found that more spherical, less-jagged, nanocrystals are formed when the 97% purity was used. Next, 14 mL of oleylamine was placed in a 100 mL three neck round-bottom flask and heated to 110o C for 1 hour under vacuum. Then the flask was back filled with argon and 1 mL of the yellow 0.73 M TOPTe solution was injected and the whole solution was heated to the injection temperature. Meanwhile, in a glovebox, 0.18 g of bis[bis(trimethylsilyl)amino]tin(II) was mixed with the dried 1-octadecene and loaded into a syringe. This orange/yellow, room temperature, bis[bis(trimethylsilyl)amino]tin(II) solution was then injected into the hot TOPTe solution, at which point the mixture immediately turned a dark brown color. The reaction was allowed to continue for the allotted reaction time and then was cooled using a liquid nitrogen bath. Once near room temperature, 3 mL of dried oleic acid was added which caused the solution temperature to rise 5-10o C before it cooled back to room temperature. The SnTe NCs were then removed from unreacted precursors and byproducts by adding either the nonsolvent/solvent pair acetone/chloroform or ethanol/chloroform and centrifuging. The SnTe NC precipitate was then dried, washed with ethanol, and stored dried (excess solvents evaporated) in a glovebox. Reaction time (1.5-10 minutes) and injection temperature (150-180o C) were used to change the NC size, with higher temperatures and longer times resulting in larger NCs. Typical reaction yields were 20%. Nanocrystal Diameter Determination. Transmission electron microscopy (TEM) was used to measure the NC diameters. Small volumes (less than 500 µL) of dilute NC solutions in chloroform were drop casted onto lacey carbon TEM grids to make the TEM samples. A JEOL JEM-2200FS TEM operating at 200kV was used to obtain bright-field images to measure the NC diameters, at least 100 NCs per sample were measured and averaged, by hand, using the software Image J. The CCD camera, on which the images were recorded, was calibrated with a gold lattice magnification

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standard. Absorbance Spectra. The NCs were suspended in tetrachloroethylene to measure the absorbance. The absorbance spectra were obtained at room temperature using a Perkin-Elmer Lambda 750 spectrometer equipped with deuterium and tungsten lamps as well as PbS and PMT detectors. Through a combination of these optical elements the spectrometer is capable of operating at energies between 6.5 and 0.41 eV. For the SnTe NCs with absorbance features at energies lower than 0.41 eV, attenuance spectra of NC films were acquired instead of absorbance spectra of NCs suspended in tetrachloroethylene. The NCs were drop cast on double side polished Si wafers with a resistivity larger 1000Ωcm. The attenuance of these films was measured using a Bruker VERTEX 80v Fourier-transformed infrared (FTIR) spectrometer equipped with a NIR/MIR KBr beamsplitter, MIR globar source, and DLaTGS detector. A portion of the Si wafer not coated with the SnTe film was used as a reference. The films were evacuated to 1.6 Torr inside the sample chamber and measured at room temperature. Photoluminescence Spectra. A Bruker Vertex 80v FTIR equipped with a UV/vis/NIR CaF2 beamsplitter and a Hamamatsu PbS/Si detector was used to collect the room temperature PbTe NC PL spectra. The samples were suspended in tetrachloroethylene and excited with 153 mW from a 660 nm diode laser perpendicular to the collection axis. While we found that the SnTe NCs did not exhibit PL over the combined operating spectral range of the CaF2 beamsplitter and PbS/Si detector (4.96 to 0.38 eV), we also performed PL measurements at lower energies with a combination of a NIR/MIR KBr beamsplitter and an Infrared Associates InSb/MCT detector (0.38 to 0.07 eV). The samples were suspended in tetrachloroethylene and excited in the same way as the PbTe NCs samples.

Experimental Results In order to provide a clear picture of the band symmetry inversion and the existence of surface gap states, we studied PbTe and SnTe nanocrystals as a function of size. PbTe nanocrystals of

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different sizes were produced following the synthesis methods developed by Murphy et. al. 33 (for details see Methods). Transmission electron microscopy (TEM) images for a typical PbTe sample are shown in Fig. 1b and 1c. Similarly, SnTe NCs were made using the solution phase synthesis developed by Kovalenko et al. 24 The nanocrystal size for both NC systems is varied by adjusting the injection temperature and reaction time (for details see Methods). Figures 1e and 1f show TEM images of a typical SnTe sample. The room temperature absorption spectra of PbTe NCs varying in radius from 2.4 nm to 4.3 nm are shown in Fig. 1d. The sharp absorption features are consistent with the narrow size distribution of NCs, determined via TEM (shown in the labels in Fig. 1d). In the same figure, we also show the room temperature PL spectra which shows the high quality of the nanocrystals. The PL emission lines are only slightly shifted with respect to the absorption peak of the lowest interband transition, indicating that the observed PL is from band-edge transitions and not from defect or surface states. Similarly, Fig. 1g shows room temperature absorption spectra for SnTe NCs size. In contrast to PbTe NCs, the absorption spectra only reveal one broad absorption feature, and no PL signal was observed within our signal to noise limits (not shown here). Futher details on the optical measurements are provided in the Methods section. To model these experimental results we calculated the energy levels of electrons and holes using the single-particle, multi-band, effective mass approximation introduced by Dimmock and Wright, 5 which can also be used for negative energy gap semiconductors. Combining the electron and hole levels with optical selection rules, we calculated the energy of allowed optical transitions, neglecting electron-hole Coulomb interactions due to the large dielectric constants of these materials.

Theoretical Analyses The band structure of PbTe and SnTe at the L-point of the Brillouin is highly anisotropic for both electrons and holes 2,28–32 and their energy levels are described by the model of Dimmock and

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Wright. 5 In this model, the Hamiltonian for electrons and holes is given by: 5 

Eg 2

   Hˆ DW =     

+

h¯ 2 kt2 2mt−

+

h¯ 2 kz2 2m− l

h¯ m0 Pt (kx − iky )

h¯ m0 Pt (kx + iky )

− mh¯0 Pl kz

0 Eg 2

0

h¯ m0 Pl kz

+

h¯ 2 kt2 2mt−

+

h¯ 2 kx2 2m− l

h¯ m0 Pl kz

h¯ m0 Pt (kx − iky )

h¯ m0 Pt (kx + iky )

− mh¯0 Pl kz

−

Eg 2

h2 k2

h¯ 2 k2

¯ t z − 2m + − 2m+ t



0

l

0

−

Eg 2

h2 k2

h¯ 2 k

        2

(1)

¯ t z − 2m + − 2m+ t

l

− + + where m0 is the mass of the free electron; m− l (ml ) and mt (mt ) are the energy band parameters

that describe the remote band contribution to the electron (hole) effective masses that are longitudinal and transverse to the four axes defined by Γ point and all four L valleys, respectively; and Pl and Pt are the direct longitudinal and transverse Kane matrix elements, respectively. In Eq.(Eq. (1)) the z axis is selected along the valley direction, while x and y can be selected arbitrary, resulting in the three projections of momentum operator kz,x,y and kt2 = kx2 + ky2 . The anisotropic energy parameters of several lead chalcogenides and SnTe semiconductors are given in Table. 1. Table 1: The energy band parameters describing dispersion of electrons and holes within Dimmock and Wright model. 5 The parameters for SnTe were linearly extrapolated from the Pb1−x Snx Te parameters for various compositions given in Hews et al. [ 29] and Adler et al. [ 30]. Parameter Eg (T = 300) (eV) m0 /mt− m0 /m− ` m0 /mt+ m0 /m+ ` 2Pt2 /m0 (eV) 2P`2 /m0 (eV)

PbS 2,31,32 0.41 1.9 3.7 2.7 3.7 3.0 1.6

PbSe 2,31,32 0.28 4.3 3.1 8.7 3.3 3.0 1.7

PbTe 2,31,32 0.31 11.6 1.2 10 0.7 5.6 0.52

SnTe 29,30,32 – 0.19 9.12 1.2 5.36 0.65 5.6 0.52

Numerical solutions of this Hamiltonian for PbTe and SnTe require a numerical approach, which we address later in this paper. Nevertheless initially we consider a formation of the gap states in SnTe NCs within the Kang and Wise approximate model, 2 which neglects band anisotropy and assumes an infinitely high potential barrier surrounds the NCs. This approach allows an analytical description of the electron and hole energy spectra and provides better insight into the problem.

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Figure 2: Comparison of the energy levels of a) PbTe and b)SnTe NCs as a function of the radii for the isotropic infinite barrier model. 2 Solid and dashed curves indicate energy levels with parity π = +1 and π = −1, respectively. Energy levels with different angular momentum j are represented by different colors. Optically allowed transitions are indicated by black solid vertical arrows.

Kang-Wise Approximation Kang and Wise described the the optical spectra of spherical NCs made of lead chalcogenide semiconductors, within a spherical approximation. The isotropic band edge parameters are selected 11



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according the the following rule: αc,v 1 1 2 + ± and ≡ ±= ± m0 m 3mt 3m`

2 1 P2 = Pt2 + P`2 , 3 3

(2)

where c and v denote the conduction and valence bands, respectively. In this approximation, each electron and hole state is characterized by a total angular momentum j = 1/2, 3/2, 5/2, .... Also, the conservation of angular momentum allows for the separation of the angular and radial motion, which results in two subsets of radial wavefunction solutions. One of these subsets is characterized by a parity of π = +1 while the other one has parity of π = −1. This parity number that was introduced by Kang and Wise, 2 is particularly useful since optical transitions are only allowed when they involve a transition of +1 to -1 and viceversa (i.e., |∆π| = +1). For the case of an infinite potential barrier the radial wavefunction vanishes at the NC surface, resulting in analytical equations that determine the energy E of the quantum confined levels: 2

A(k1 , k2 ) j j−1/2 (k1 a) j j+1/2 (k2 a) − j j−1/2 (k2 a) j j+1/2 (k1 a) = 0 , for π = +1 , A(k1 , k2 ) j j+1/2 (k1 a) j j−1/2 (k2 a) − j j+1/2 (k2 a) j j−1/2 (k1 a) = 0 , for π = −1 .

(3)

Here A(k1 , k2 ) = (k1 /k2 )[Eg + αc (¯hk2 )2 /m0 − 2E]/[Eg + αc (¯hk1 )2 /m0 − 2E] are the energy dependent coefficients, jn (x) are the spherical Bessel functions, and the squares of k1 and k2 variables are connected with the energy levels E via the following relationship: 2 h¯ 2 k1,2 b =− ± 2m0 2

r

b2 −c , 4

(4)

where b = [(Eg /2 − E)αv + (Eg /2 + E)αc + 2P2 /m0 ]/(αc αv ), c = (Eg2 /4 − E 2 )/(αc αv ) and the energy E is Eq. (Eq. (4)) is calculated from the middle of the energy gap. To calculate the electron and hole levels, it is necessary to find the roots for eq. (Eq. (2)) as a function of E for both parities. First inspection of Eq. (Eq. (4)) shows that in the case when c < 0 (when the energy is in the conduction or valence band, |E| > |Eg |/2) one of the "quasi-momenta" k1 or k2 is real and the

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other is imaginary. In this case we have regular quantum confined electron and hole levels whose energies are affected by exponentially small corrections coming from the coupled band. In the case when c > 0 (when the energy lies within the bulk band gap, |E| < |Eg |/2) both k1,2 become 2 are negative imaginary. This case is realized only when Eg < 0 in SnTe NCs where both k1,2

and corresponding k1,2 are imaginary, indicating that the energy level is localized within the bulk bandgap. Calculated energy levels for PbTe and SnTe as a function of nanocrystal size within the Kang and Wise approximation are shown in Fig. 2a and 2b, respectively. The optically allowed transitions between electron and hole levels that meet the selection rule criteria |∆π| = 1 are indicated by the black vertical arrows. As can be seen in Fig. 2a, the confined energy levels of PbTe for all NC sizes are always outside the bulk energy gap, where the latter is indicated by horizontal black lines. In contrast, for SnTe NCs with large radii (a ≥ 10 nm; 1/a2 < 0.01) some of the energy levels exist within the bulk energy gap, as shown in Fig. 2b. These states within the bandgap are the so called gap states, which converge to a single level at the surface of the bulk semiconductor. These gap states are a consequence of the negative energy gap (Eg < 0). In large NCs they usually do not contribute to absorption and PL due to the small overlap of their wave functions with wave functions of intrinsic confined levels of a NC. However, in small size NCs the wave functions of gap states spread over the entire NC volume and as a result, they strongly contribute to the NC absorption and PL. In SnTe NCs the gap state levels cross the band edges of the bulk conduction and valence bands at a ≈ 10 nm, as one can see in Fig.2a. As mentioned previously, for large SnTe NCs the gap states converge to single level at the surface of the bulk semiconductor. To estimate this energy level, we observe that in the case of Eg < 0 both k1 and k2 become imaginary for large NCs. In such a case, the spherical Bessel function in Eq. (Eq. (2)) can be replaced by its asymptotic form: jn (ix) ≈ in ex /(2x), for any n, when x = k1,2 and a → ∞. This leads to the equations that determine the position of the discrete

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gap state at the SnTe flat surface: k1 = k2

Eg 2 Eg 2

h¯ 2 k2

− E + 2m−1 −E +

h¯ 2 k22 2m−

.

(5)

The Kang and Wise model provides only a qualitative description of the electronic and optical properties of PbTe and SnTe nanocrystals since both material systems are highly anisotropic (see Table I). To quantitatively model our NC system, we include the effect of anisotropy by employing the Dimmock and Wright Hamiltonian and incorporating a finite barrier at the NC surface, as described below.

Quantum Confined Levels Described by the Dimmock and Wright Model with a Finite Barrier Height The anisotropy of the Dimmock and Wright Hamiltonian, Hˆ DW , described in Eq. (Eq. (1)) does not allow for the separation of the the radial and the angular variables in the wave functions, as done within the Kang and Wise spherical model. 2 Thus, to determine the wave functions and quantum confined levels of electrons and holes we used the numerical diagonalization procedure from Ref. 28. We propose a solution of the Dimmock and Wright Hamiltonian as

Hˆ DW F = EF ,

(6)

  (1)    ∑ cn`m bn`m (r, θ , φ ) F n`m 1      (2)     F2   ∑ cn`m bn`m (r, θ , φ )    n`m , F= =  (3) F     3   ∑ cn`m bn`m (r, θ , φ )    n`m    (4) F4 ∑ c b (r, θ , φ )

(7)

where the wave function, F is given by:

n`m

n`m n`m

14


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which is an expansion over the set of basis of orthogonal functions:

bn`m (r, θ , φ ) = fn (r)Y`m (θ , φ ).

The radial basis functions in Eq. (Eq. (8)) are defined as fn (r) =

(8)

p

2/R sin(nπr/R)/r with n =

1, 2, 3, ... and Y`m (θ , φ ) are spherical harmonics. The radial functions are orthogonal to each other RR 0

fn0 (r) fn (r)r2 dr = δn0 n . In contrast with Ref. 28, to incorporate the finite barrier potential, we

consider R to be the effective radius of the NC , which is defined as R = a + h, where a is the radius of the real NC and h is the thickness of the barrier. One can see that with the used radial wave function definition fn (R) = 0 an efficient integration cutoff is accomplished. We model the barrier surrounding the NC as a wide-gap semiconductor that is also described by the Dimmock and Wright Hamiltonian, with the same contribution of the remote bands to the electron and hole effective mass and the same Kane matrix element as in the studied NCs. The only parameter that changes across the NC/barrier interface is the energy gap Eg :

Eg (r) =

   E

Bulk

  Em

0

Intrinsic Gap States in Semiconductors With Inverted Band Structure

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