Influence of Polytypism on the Electronic Structure of CdSe/CdS and CdSe/CdSe Core/Shell Nanocrystals

We address theoretically differences and similarities on the electronic structure of CdSe/CdS dot-in-dot nanocrystals (NCs) with wurtzite/wurtzite (WZ/WZ), zincblende/zinc-blende (ZB/ZB) and polytype WZ/ZB crystalline phases, as they are currently being synthesized and used in opto-electronic devices. We show that the electronic structure of polytypic CdSe/CdS NCs closely resembles that of WZ or ZB NCs with regard to quantum confinement and strain. This results in similar single-exciton wave functions, with the (s-like) ground state showing quasi-type-II behavior and excited (p-like) states showing fully type-II behavior. The main differences arise in the nature and magnitude of built-in electric fields. We predict these fields are stronger in polytypes than in pure WZ or ZB NCs due to the sharp spontaneous polarization mismatch between the cubic core and the hexagonal shell lattices. Polarization in NCs is currently believed to be screened by several surface effects. In polytypical structures, however, the polarization mismatch at the interface may create effective charges which are sufficiently far from the outer surface to be quenched. To make a definitive assessment on this controversial issue, we propose experiments in polytypic NCs where both core and shell are made of CdSe, albeit with ZB and WZ structure, respectively. In such a case, band offsets are small, strain is absent and our calculations predict pyroelectricity should become the driving force, inducing transitions from type-I to type-II excitons with increasing core or shell size.


Introduction
Colloidal NCs with a CdSe core embedded in a thick CdS shell have taken a leading position in the research of semiconductor nanostructures for opto-electronic applications owing to their high luminescence quantum yield, narrow emission linewidth and reduced blinking. 1 In bulk CdSe and CdS crystals, WZ phase is more stable than ZB one. However, the energy difference is smaller than 1.5 meV/atom. 2 As a consequence, in NCs -where the surface to volume ratio is high-one can grow either phase through appropriate choice of synthetic temperature or ligands. [3][4][5][6][7] In the last years, this has made possible the synthesis of CdSe/CdS NCs with pure WZ phase, [8][9][10][11][12] pure ZB phase 5,[13][14][15] or a polytypic form where the CdSe core has ZB phase and the CdS shell WZ one. [16][17][18] Several spectroscopic studies have analyzed the near-band-edge luminescence of such systems. In all instances, a pronounced dependence of the emission energy on the core and shell size was observed, consistent with a quasi-type-II exciton ground state, where holes are localized inside the core and electrons delocalized over both core and shell. The study of radiative lifetimes, however, reveals quantitative differences between WZ, ZB and ZB/WZ structures. For core radius ∼ 1.5 nm and thick shells (∼ 5.5 nm), reported lifetimes in pure WZ NCs range between 100 and 200 ns, 19 in pure ZB NCs around 50 ns 14 and in polytypic NCs around 150 ns. 17,20 The question arises if such a variation reflects fundamental differences in the electronic structure of the three systems, making excitons in WZ and polytypic NCs longer lived than in ZB ones.
Theoretical studies have used atomistic and continuum models to provide detailed descriptions of the band structure of both ZB 21-23 and WZ 19,24 CdSe/CdS NCs. By contrast, the band structure of polytypic CdSe/CdS NCs remains largely unexplored. Attempts in this direction are essentially limited to simplified effective mass models neglecting strain, permanent dipoles intrinsic to the WZ shell and Coulomb effect on the excitonic wave function. 16,17 For improved understanding, in the present work we calculate the electronic structure and single exciton states in polytype CdSe/CdS NCs using a multi-band k·p method that ac-counts for the above effects. We then compare the results with those of pure WZ and pure ZB heterocrystals, calculated on equal footing, explaining the similarities and analyzing the differences.
Our calculations show that the electronic structure of polytypic NCs resembles that of ZB and WZ NCs. Lattice mismatch strain is comparable in all cases, which results in hydrostatic strain stimulating electron delocalization to a similar extent. The main difference is found in the magnitude and nature of built-in electric fields. These are very weak in ZB NCs because of the cubic lattice symmetry, but reach values of 5-25 mV/nm inside the core of WZ and polytypic NCs. The fields arise from the competition between piezo and pyroelectricity, the latter being particularly important in polytypic NCs due to the sharp spontaneous polarization mismatch between core and shell lattices. As a result, the core of polytypic NCs -in spite of its ZB phase-experiences stronger fields than those of WZ NCs.
The presence of alloyed ZB interlayers, which are often found in polytypic NCs, 20 barely changes this picture.
Because the origin of internal fields observed in CdSe NCs is a controversial issue, [25][26][27][28][29][30][31] we propose a system which should confirm the presence of spontaneous polarization induced dipoles in core/shell NCs. Namely, polytypic NCs made of CdSe only, albeit with different phase in the core (ZB) and shell (WZ). This turns out to a be strain-free system, hence lacking piezoelectricity, but with sharp polarization mismatch at the core/shell interface.
Because band offsets between polytypic forms are low, spontaneous polarization should have a strong influence, and exciton lifetimes should be very sensitive to the core size.

Methods
The excitonic electron and hole states are calculated with Hamiltonians of the form: where j = e, h stands for electron or hole, H kin j is the kinetic energy term, V j the singleparticle confinement potential and V e−h j is Coulomb attraction exerted upon carrier j by the other carrier. The single-particle confinement can be split into several terms: where V conf j the confining potential defined by the band offsets between bulk CdSe and CdS,

Results and discussion
We compare four different models of CdSe/CdS dot-in-dots. The first one is a WZ/WZ core/shell NC with sharp interface, see Fig. 1(a). The second one is a ZB/WZ (polytypic) NC also with sharp interface, Fig. 1(b). The third one a ZB/WZ NC with a diffusion interlayer in between the core and the shell, Fig. 1(c). The interlayer, which accounts for the alloying often observed in polytypic dots, 20 has ZB structure and CdSe1 − xS x composition, with x varying exponentially between the core and shell interfaces. The last model is a ZB/ZB NC, also with sharp interface, Fig. 1(d). holds because strain turns out to be very similar regardless of the crystal phase. For pure WZ and pure ZB NCs, the lattice mismatch between CdSe core and CdS shell is known to be 3.9% and 4.3%, respectively. For the polytypic structure, we find it to be also ∼ 3.9%.
After relaxation, the equilibrium strain remains similar in all structures. Even for the alloyed NC, strain in the center of the core is as compressive as that of sharp interface NCs, since it is ultimately set by the thick and stiff CdS shell rather than by the alloyed interlayer (see Supporting Information).
Dashed lines in Fig. 1(f-g) also reveal the presence of significant piezoelectric fields in polytypic NCs. These are very weak in ZB NCs owing to the high lattice symmetry, see  R is the core radius and H the shell thickness. In (c), we include a 2-nm-thick alloyed interlayer with exponential change of composition between core and shell. (e-h) Corresponding single-particle potential profiles along c axis. Solid lines are used for complete single-particle potential V j , dashed lines for potential excluding spontaneous polarization, and dotted ones for unstrained band offset, V conf j . Different colors are used for conduction band (blue) and valence A-band (red). In (e-h), the core has R = 1.5 nm radius and the shell H = 5 nm.
In addition to piezoelectric fields, one can consider the influence of pyroelectric fields.
The latter originate in the spontaneous polarization arising from the intrinsic deviation of WZ crystals from their ideal geometry. This effect had been suggested as the source of permanent dipoles observed in WZ CdSe NCs, 25,30 although the polarization strength is currently believed to be quenched by several surface effects, including screening by passivating ligands, surface reconstruction and relaxation, 27 non-stochiometric cation termination of the NCs, 28 and migration of mobile surface charges. 29,31 In polytypical structures, however, the polarization mismatch at the heterostructure interface may create effective charges which are sufficiently far from the outer surface to be quenched, as is actually the case of piezoelectric fields. 36,37 Thus, we next include V sp j in our calculations. We neglect the polarization mismatch at the outer interface, which is presumably quenched, but consider that between the core and shell materials. The resulting band profile is represented with solid lines in Fig. 1(e-h).
As can be seen, pyroelectricity compensates and reverses the sign of piezoelectric fields inside the core. For WZ NCs, the field switches from 14 mV/nm (dashed line) to −6 mV/nm (solid line). The opposite sign of pyro and piezoelectric contributions is due to core lattice being compressed. 38 On the other hand, the magnitude of the pyroelectric field is proportional to the polarization mismatch at the interface. For WZ heterocrystals, P CdSe sp = −0.42 µC/cm 2 and P CdS sp = −0.93 µC/cm 2 . 30 For polytypic NCs, in turn, the influence of pyroelectricity is even stronger because the polarization mismatch is maximized, as the ZB core leads to P sp = 0 µC/cm 2 . Consequently, the field felt in the core switches from 13 mV/nm (dashed lines) to −23 mV/nm (solid lines).
The abovementioned fields are strong enough to open venues for exciton wave function modulation via internal fields, as we recently suggested for pure WZ NCs. 19 To illustrate this point, in Fig. 2 we plot the squared electron-hole overlap (proportional to radiative lifetimes) for the NCs under study, excluding (a) and including (b) pyroelectric fields. In general, increasing the core size leads to larger overlap, as expected for quasi-type-II systems. 35 Yet, for large enough core radius (R > 2 − 2.5 nm) the increasing magnitude of internal dipoles may revert this trend. This can be seen for WZ/WZ dots when considering piezoelectric fields only -panel (a)-or for alloyed polytype NCs -panels (a) and (b)-. The exact core size where internal fields overcome quantum confinement effects strongly depends on the conduction band deformation potential parameter, whose value is not precisely determined for hexagonal materials (see Supporting Information). In any case, we stress that this behavior is shared by pure WZ and polytypic NCs, but it is absent in pure ZB NCs where internal fields are negligible. 21 Altogether, the electron-hole overlaps calculated in Fig. 2 reveal moderate differences between different crystal phases. For R = 1.5 nm core, we estimate values between 0.4 and 0.5 in all cases. Only for alloyed NCs it is slightly larger (0.6), but this is related to the increase in effective volume. We then conclude that the different exciton lifetimes reported in experiments for WZ, ZB and polytype NCs 14,17,19,20 are not a consequence of the band structure, but more likely reflect the difficulty of determining exactly the lifetime in NCs with multi-exponential photoluminescence decay. 39 To investigate the influence of the band profiles of Fig. 1 on the wave functions, in Fig. 3 we compare the electron and hole charge densities for a few low lying single-exciton states in the four prototypical CdSe/CdS NCs we consider. The carrier localization in polytypic structures is similar to that in pure phase structures. In particular, the exciton ground state (left column, s-like orbitals) shows a quasi-type-II behavior, with the hole well localized inside the core and the electron partially leaking into the shell. A similar picture is also found in the case of alloyed interface.
The quasi-type-II behavior and the mild influence of interface alloying on the singleexciton ground state are consistent with photoluminescence experiments in ZB/WZ NCs. 18 They also agree with, and hence largely validate, simpler effective mass models previously used to describe such systems. 16,17 Note however that such models missed the internal fields arising in polytypic NCs. For typical core sizes, R = 1.5 nm, the potential drop inside the core is still moderate and so is its effect on the carrier localization. Yet, the influence becomes visible in the alloyed NCs, where the smooth confinement facilitates electron-hole charge separation. For larger cores, one should obviously expect an increasingly important role of built-in fields.
While the ground state of CdSe/CdS NCs has been extensively studied by luminiscence techniques, much less is known about excited states, which can be addressed via differential transmission spectra. 40 To shed light on such states, in the mid and right columns of Fig. 3 we plot the first and second excited states (p z and p x /p y orbitals). Interestingly, for all crystal structures the excited states show a clear type-II character, which is markedly different from the ground state. While the hole remains in the core owing to the large band-offset, the electron mostly localizes in the shell, placing its node in the core region. This should render such states optically dark. Note also that p z orbitals are found to be very sensitive to built-in fields in the shell even for ZB NCs.
We have shown so far that the main difference between polytypic and pure phase NCs is the build up of significant pyroelectricity in the core. Still, its effect is generally masked by piezoelectricity and strong quantum confinement. To reduce such competing effects, we next invesigate CdSe-only polytype NCs. It is known that CdSe dots with ZB phase become unstable with increasing size. 4,5 In what follows we consider a colloidal CdSe dot with ZB phase, which at a certain volume starts growing in WZ phase, thus forming a fully CdSe yet polytypic core/shell NC. The resulting band profile is shown in Fig. 4(a) should be present. Consequently, internal fields of ∼ 15 mV/nm build up in the core (solid line).
The band profile described above makes the electronic structure of CdSe polytype NCs very sensitive to the geometry. In Fig. 4(b) we show the excitonic electron and hole charge densities in a NC with thin (left column) and thick (right column) shell. When the shell is thin, electron and hole are largely localized inside the core, because Coulomb interaction and the strong confinement of the shell allow the hole to overcome the valence band offset.
By contrast, when the shell becomes thicker the hole moves towards the shell. The effect of varying core radius and shell thickness on the electron-hole overlap is summarized in Fig. 4(c).
A qualitative different behavior is observed when including -solid lines-or excluding -dashed lines-spontaneous polarization. We thus propose experiments using polytypic CdSe/CdSe NCs to verify the presence of pyroelectric fields in core/shell structures involving WZ phases.
A first possibility is to synthesize small core NCs (R = 1.5 nm) and let the shell thickness increase. If pyroelectricity is present, it should enhance exciton lifetimes, much like in quasitype-II NCs. A second possibility is to keep thin shells (H = 1 nm) and increase the core size. Again, if pyroelectrictiy is present, it should rapidly increase exciton lifetimes.

Conclusions
We have studied the influence of polytypism in ZB/WZ core/shell NCs using k·p theory.
In CdSe/CdS heterocrystals, we find quantum confinement and strain are comparable to those of pure WZ and pure ZB NCs. Internal fields are formed inside the ZB core due to piezoelectricity and pyroelectricity imposed by the WZ shell. In this regard, polytypic structures resemble WZ NCs rather than ZB ones. Interface alloying softens the confinement potential, but leaves strain and built-in fields essentially unaltered. As a result, low-energy excitons of polytypic NCs have quantitatively similar properties to their WZ counterparts, which is consistent with experiments. The main difference is the strength of pyroelectricity, which we predict to be stronger than in WZ heterocrystals.
To test unambiguously the presence of pyroelectricity in core/shell structures, we propose the study of polytypical CdSe/CdSe NCs. These are strain-free objects with low band offsets. In these conditions, pyroelectricity emerges as a leading factor determining the electronic structure. Should experiments confirm our predictions, polytypic CdSe/CdSe NCs -with high sensitivity of the exciton wave function to core and shell dimensions-could become advantageous structures for opto-electronic applications where tuning of carrier-carrier interactions is required, such as lasing, non-linear optics, photodetectors and photovoltaic cells. 35