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Field-Induced Broadening of Electroabsorption Spectra of Semiconductor Nanorods and Nanoplatelets Nikita V. Tepliakov,† Irina O. Ponomareva,† Mikhail Yu. Leonov,† Alexander V. Baranov,† Anatoly V. Fedorov,† and Ivan D. Rukhlenko*,†,‡ †

ITMO University, Saint Petersburg 197101, Russia Monash University, Clayton Campus, Melbourne, Victoria 3800, Australia

‡

ABSTRACT: We theoretically study the broadening of optical absorption spectra of monodisperse ensembles of randomly oriented nanorods and nanoplatelets exposed to a static electric field. It is found that the weaker quantum confinement inside nanoplatelets results in a much stronger field impact on their absorption than on the absorption of nanorods. The most notable manifestation of this impact is that the widths of the absorption lines in the nanoplatelets’ spectra peak as functions of the electric field strength. Both the maximal widths and the corresponding optimal field strengths are found to decrease with the largest dimensions of the nanoplatelets. In sharp contrast to this, the broadening of the nanorods’ absorption grows monotonously and saturates with the field strength. The high tunability of electroabsorption and the possibility to maximize its efficiency, revealed by our study, suggest semiconductor nanoplatelets as a promising material base for the next-generation electro-optic devices.

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If a nanostructure is exposed to an external electric field, its optical absorption is changed due to the quantum-confined Franz-Keldysh effect32 and the quantum-confined Stark effect.33 The Franz-Keldysh effect is due to field-induced interband tunneling of the confined charge carriers caused by the spatial overlap of their wave functions, whereas the Stark effect is due to the field-induced shift of the carrier energies.34−39 An important feature of both effects is that their strengths increase significantly with the nanostructure dimensions.40 Since this increase is accompanied by the reduction of the energy spacing in the nanostructure energy spectrum, the Stark and FranzKeldysh effects are only efficient for a certain range of nanostructure dimensions. The overall impact of the external electric field on the absorption of a semiconductor constitutes the electroabsorption phenomenon. Electroabsorption shows great promise for the advancement of electro-optical sensing. In particular, the pronounced electrooptical response of quantum nanostructures may be used to measure the near fields of plasmonic nanoparticles.41 These fields are resonantly enhanced by factors 10−100 regardless of the nanoparticle shape, making the studies of electroabsorption in strong electric fields topical. Recently, we experimentally studied the effect of electroabsorption in colloidal quantum dots, nanorods, and nanoplatelets made of CdSe.29 This effect was explained using a simple quantum mechanical model, which was further developed in our recent theoretical paper.42 These works

INTRODUCTION It is common knowledge that the absorption and emission spectra of semiconductor nanostructures differ significantly from the spectra of bulk materials due to the quantum size effect.1−6 The main reason for this is the discreteness of energies of elementary excitations residing inside the nanostructures, which shows up as distinct peaks of absorption and photoluminescence.7−9 An effective control over the absorption and emission spectra of semiconductor nanostructures is needed to further advance photonics devices for converting between the energy of solar radiation and electrical energy10,11 and will significantly benefit the performance of nanocrystal-based lasers.12−17 There are many ways of altering the absorption properties of semiconductor nanostructures. First of all, one can vary the widths and positions of the absorption peaks of quantum dots by changing the dot diameters.18,19 Similar modifications occur in the absorption spectrum of semiconductor nanowires upon changing their thickness.20 Certain peaks in this spectrum can also be suppressed dynamically by varying the incidence angle of light exciting the nanowires.21,22 Another way of changing the amplitudes and widths of the absorption lines is through varying the concentration of the confined charge carriers, as it was demonstrated for unbound electrons in the quantum wells.23 The absorption properties of semiconductor nanostructures also vary depending on temperature,24,25 the presence of various elementary excitations residing inside or near the nanostructures,26−28 or the presence of external fields.29−31 The field-based control over the absorption properties is especially attractive for applications due to its dynamical nature. © XXXX American Chemical Society

Received: August 29, 2015 Revised: January 5, 2016

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DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

of electrons and holes, me and mh. We also assume that nanocrystals are made of cubic semiconductor and limit ourselves to the case of strong spatial confinement. A state of an electron or hole in such nanocrystals is characterized by a set of three integer quantum numbers, n = (nx, ny, nz). The respective wave function is the product of wave functions describing one-dimensional motions along the three spatial axes

were inspired by the studies of others, demonstrating unique electronic properites of CdSe nanoplatelets, such as giant oscillator strength.43−45 Our experiments showed that the change in the absorption of the nanoplatelets induced by the field is 10 times that of the quantum dots and revealed that this change unusually decreases with the nanoplatelet thickness. These features were qualitatively explained by applying the developed theory to a pair of the lowest-energy interband transitions in the three kinds of nanocrystals. While confirming that the nanoplatelets are the most promising for applications due to their pronounced electro-optical response, the theory also allowed us to elucidate the selection rules for one-photon optical transitions in individual nanocrystals. The experiments were conducted with ensembles of almost monodisperse nanocrystals, which were randomly oriented in space. The random nanocrystal orientation leads to the fieldinduced broadening of the electroabsorption spectrum. This broadening is a result of the averaging of the absorption spectra of individual nanocrystals, which are exposed to the Stark and Franz-Keldysh effects of different strengths due to their different orientations relative to the electric field. In this work, we thoroughly examine the impact of the field-induced broadening on electroabsorption of CdSe nanorods and nanoplatelets by taking into account the random orientation of the nanocrystals in the ensemble. Cubic nanocrystals, also known as quantum dots, are excluded from this study due to their more symmetric shape and the resulting weaker broadening of their electroabsorption spectrum. The results of our study may prove useful for engineering electroabsorption in various kinds of semiconductor−nanocrystal-based photonics devices.

Ψn(F , r) = ψn (x , Fx)ψn (y , Fy)ψn (z , Fz) x

z

(2)

which are expressed through the Airy functions of the first and second kinds as ψn (ξ , Fξ) = A nξ Ai(ξ , Fξ) + Bnξ Bi(ξ , Fξ) ξ

(3)

Here the constants Anξ and Bnξ are found from the standard boundary and normalization conditions.40,42 The total energy of an electron or hole is a sum of energies Enξ(Fξ) associated with its confined motions in the three spatial dimensions, ξ = x, y, z, i.e. En(F) = Enx(Fx) + Eny(Fy) + Enz(Fz)

(4)

These energies are solutions of the dispersion equation following from the boundary conditions, which require the wave functions to vanish at the nanocrystal surface. The energy spectrum of electron−hole pairs inside a nanocrystal is then given by En, m(F) = En(F) + Em(F) + Eg

(5)

where Eg is the semiconductor bandgap and the sets of quantum numbers n = (nx, ny, nz) and m = (mx, my, mz) represent the states of electrons and holes. Excitation of a nanocrystal with photons of energy ℏω generates electron−hole pairs at a rate40

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THEORETICAL MODEL Electroabsorption of a Nanocrystal Ensemble. Consider a monodisperse ensemble of arbitrarily oriented semiconductor nanocrystals in the form of rectangular parallelepipeds. Following the electroabsorption model developed in our previous paper,40,42 we shall assume that the nanocrystals have impenetrable surfaces and denote the lengths of their three edges as Lx, Ly, and Lz. When such nanocrystals are immersed in an external electric field, it creates a lineal potential which decays along the field direction inside the nanocrystals as V (r) = ±e Fr

y

W (ω , F ) ∝

1 (ℏω)2

∑ m, n

γn, m Jm2 , n (F) [ℏω − En, m(F)]2 + γm2 , n

(6)

where the summation is evaluated over all the quantum states of the nanocrystal, γn,m is the full dephasing rate of interband transition m → n, and the overlap integral is defined as Jm, n (F) =

(1)

∫ Ψn(F, r)Ψm(F, r) dr

(7)

We can now calculate the absorption coefficient of the ensemble by averaging W(ω,F) over all the possible nanocrystal orientations with respect to the internal field, i.e.

where the plus and minus signs refer to electrons and holes, respectively, −e is the charge of a free electron, and F = (Fx, Fy, Fz) is the electric field inside the nanocrystals. Strictly speaking, if the external field is homogeneous, then F is not due to the nonsphericity of the nanocrystals. We shall, however, neglect the effects of field penetration inside the nanocrystals and assume the internal electric field homogeneous. This assumption will not affect the overall picture of electroabsorption but will enable us to study its important general features qualitatively. It also allows us to ignore the effects of a uniform layer of surface ligands, which is often formed on colloidal nanocrystals. These include the screening of an external field and additional anisotropy of electroabsorption if the ligands are chiral and may affect the spectral line widths. In order to analytically calculate the spectrum and wave function of charge carriers confined by the nanocrystals, we employ a two-band model of semiconductor.46 This model allows for the interaction of charge carriers with the periodic potential of the nanocrystal lattice through the effective masses

K (ω , F ) ∝

ℏω 4π

∫ W (ω , F ) d Ω

(8)

where the integration is performed over a solid angle of 4π, dΩ = sin ϑ dϑ dφ, F = (F sin ϑ cos φ, F sin ϑ sin φ, F cos ϑ), and F = |F|.

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RESULTS AND DISCUSSION Absorption Spectra of Nanorods and Nanoplatelets. To study the field-induced broadening in the electroabsorption spectrum described by eq 8, we first focus on two kinds of nanocrystals: nanorods of square cross section, with Lx = Ly = 3 nm and Lz = 20 nm, and square nanoplatelets of dimensions Lx = Ly = 20 nm and Lz = 2 nm. To specify material parameters, we assume that the nanocrystals are made of CdSe, which is characterized by the effective masses me = 0.11 m0 and mh = B

DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C 0.44 m0, where m0 is the mass of a free electron. To keep the generality of our conclusions and simplify further analysis, we ignore the permanent dipole moment of the first excited state of CdSe nanocrystals.47 The interaction of this moment with an electric field can contribute to the anisotropy of electroabsorption and will be considered in a separate publication elsewhere. Finally, we focus on the electric-dipole-allowed fundamental interband transition between the ground states of electrons and holes, n = m = (1,1,1), and take γn,m = 5 meV.48 The solid curves in Figure 1 are the fundamental peaks of the normalized electroabsorption spectra of nanorods and nano-

Figure 2. (a) Field dependencies of the FWHMs of the fundamental absorption peaks of 3 × 3 × 20 nm3 nanorods and three ensembles of square nanoplatelets of dimensions 16 × 16 × 2, 20 × 20 × 2, and 24 × 24 × 2 nm3. (b) Optimal field strength F0 and maximal FWHM P0 versus the lateral size of 2 nm thick nanoplatelets. Material of all nanocrystals is CdSe.

of a larger and a smaller nanoplatelets in the same figure. The maximal electric field in Figure 2a is about 10 times greater than that used in our earlier experimental work29 (note the difference in the employed definitions of the electric field strength). The monotonous spectral broadening of the first excitonic peak, seen from the figure to occur in fields below 20 kV/cm, agrees well with the experiment. The optimal field strength and the peak FWHM are plotted as functions of L in Figure 2b. Owing to the presence of the maximum of the FWHM, the field-induced broadening of the fundamental absorption peak of the nanoplatelets can be more than a 4-fold for electric fields as weak as 80 kV/cm. Two-Dimensional Averaging of the Absorption. The observed features of the electroabsorption spectra can be explained by considering auxiliary spectra of nanoplatelets and nanorods obtained via two separate averagings of the electron− hole pair generation rate over the polar and azimuthal angles. If a nanoplatelet and nanorod are in the xy-plane, as shown in Figures 3a and 3b, and the electric field vector is parallel to the y-axis, the auxiliary spectra are

Figure 1. Fundamental absorption peak of (a) 3 × 3 × 20 nm3 CdSe nanorods and (b) 20 × 20 × 2 nm3 CdSe nanoplatelets.

platelets for two electric field strengths. For reference, the dashed spectra show the same peaks in the absence of the field. One can see that the peak of the nanorods’ spectrum broadens at half-maximum from 10 to almost 15 meV when the field strength is increased from 0 to 50 kV/cm. The bottom part of the peak broadens further without affecting its full width at halfmaximum (FWHM) when the field strengthens up to 200 kV/ cm. At the same time, the peak is seen to turn asymmetric due to the extension of its left wing. A more pronounced broadening occurs in the absorption spectrum of nanoplatelets at low field intensities: the FWHM of the fundamental peak rises from about 10 to more than 30 meV as the field is increased from 0 to 45 kV/cm. This broadening is due to an additional absorption line arising at the low-frequency side of the fundamental peak. As the field strength grows further, the additional spectral line weakens and the fundamental peak becomes asymmetric, much like in the case of nanorods. The electric-field dependence of the FWHMs of the considered absorption peaks is plotted in Figure 2a. One can see that the width of the fundamental peak of the nanorods saturates with the field strength whereas that of the nanoplatelets exhibits a pronounced maximum. The position of the maximum, F0, and the peak value of the FWHM, P0, both grow with the reduction of the nanoplatelets’ largest dimensions Lx = Ly = L. This is illustrated by the examples

K ϑ(ω , F ) ∝

ℏω 2π

∫ W (ω , F ) d ϑ

(9)

where F = (0, F sin ϑ, F cos ϑ), and Kφ(ω , F ) ∝

ℏω 2π

∫ W (ω , F ) d φ

(10)

where F = (F cos φ, F sin φ, 0). The orientation-averaged absorption spectra Kϑ(ω,F) and Kφ(ω,F) are shown in Figure 3c for a 20 × 20 × 2 nm3 nanoplatelet and F = 45 kV/cm. Since the nanoplatelet is square, its rotation about the z-axis results in an almost symmetric spectral line. When this nanoplatelet is rotated about C

DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 3. Two ways of one-dimensional spatial averaging of electroabsorption spectrum of (a) nanoplatelets and (b) nanorods of square cross sections and (c, d) the resulting electroabsorption spectra. In (c) F = 45 kV/cm, Lx = Ly = 20 nm, Lz = 2 nm, and in (d) F = 50 kV/cm, Lx = Ly = 3 nm, Lz = 20 nm. The material of the nanocrystals is CdSe.

the x-axis, its absorption peak becomes highly asymmetric, much stronger, and notably broader. These differences come from the size-dependent shift of the electron and hole energy states due to the Stark effect and from the reduction of the interband transition rate due to the Franz-Keldysh effect. Both effects are minimal when the electric field vector is normal to the nanoplatelet’s surface and maximal when this vector is parallel to the surface. As a result, the variations of the Stark shift and the transition rate with the nanoplatelet orientation are larger for averaging over ϑ than over φ. Hence, the highfrequency absorption peak of Kϑ(ω,F) predominantly comes from the nanoplatelets oriented perpendicular to the field, whereas the low-frequency peak is mainly due to the nanoplatelets parallel to the field. The situation is different for nanorods, for which the Stark and Franz-Keldysh effects are strong just in one dimension. The field impact on the fundamental absorption peak in this case is relatively small when the electric field is oriented perpendicular to the nanorod’s axis and is maximal when the nanorod is parallel to the field. The relative contribution of the latter to the peak broadening is therefore reduced as compared to the case of nanoplatelets, resulting in the less pronounced additional peak in the electroabsorption spectrum. The orientationaveraged spectra for a 3 × 3 × 20 nm3 nanorod and F = 50 kV/cm are shown in Figure 3d. The comparison of Figures 3c and 3d allows one to explain the origin of the pronounced maximum of the FWHM in Figure 2a for nanoplatelets and the absence of thereof for nanorods. The width of the broadened absorption peak of the nanoplatelets in Figure 1b is determined by two competing effects, developing upon the spatial averaging of the absorption line of a single nanocrystal: the Stark shift of the line, which increases the width of the peak, and the Franz-Keldysh suppression of the line, which lowers the contribution of the Stark shift to the peak’s broadening. When the field is relatively weak, the Stark effect prevails and the absorption peak broadens monotonously. At certain field strength, the broadening is

counterbalanced by the line suppression and the FWHM of the peak reaches its maximum. After that the peak begins to narrow down due to steep line suppression, and its FWHM saturates when the shifted line stops contributing to the FWHM. The FWHM of the nanorods’ absorption peak in Figure 1a is also contributed by the considered two effects. However, owing to the existence of one large dimension instead of two, the contribution from the nanorods oriented along the field is smaller than the contribution from the nanorods oriented perpendicular to the field, and the FWHM monotonously grows without exhibiting a maximum. The size dependencies of the maximal FWHM and the optimal field strength shown in Figure 2 can be explained by noting that the Franz-Keldysh effect weakens more rapidly with the reduction of the nanoplatelet dimensions Lx and Ly than the Stark effect does. The much steeper weakening of the FrantzKeldysh effect is easy to understand by considering the electric field potential V̂ ξ = ±eFξξ (ξ = x, y, z) as a small perturbation of the electronic subsystem of the nanocrystal. Since the matrix element of this perturbation scales in proportion to the nanocrystal dimension Lξ, the second-order correction to the confinement energies |Enξ(Fξ) − Enξ(0)|, which determines the strength of the Stark effect, scales as L4ξ. In the same order of th e perturbation potential, the overlap integral ∫ ψnξ(ξ,Fξ)ψmξ(ξ,Fξ) dξ scales in proportion to the squared ratio of the perturbation matrix element to the transition 2 6 energy, that is, like (Lξ/L−2 ξ ) = Lξ. Since the strength of the Frantz-Keldysh effect is determined by the overlap integral squared, it scales with the nanocrystal dimension like L12 ξ . As consequence of this difference in the weakening rates of the two effects, in smaller nanocrystals, the additional absorption peak shifts to lower frequencies before it stops contributing to the FWHM, and the FWHM peaks at a stronger electric field. For example, F0 and P0 increase from 30 kV/cm and 23 meV in 24 × 24 × 2 nm3 nanoplatelets to 85 kV/cm and 43 meV in 16 × 16 × 2 nm3 nanoplatelets. D

DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Differential Absorption Spectra. It is common to characterize electroabsorption in semiconductor nanocrystals using the differential absorption spectrum K (ω , F ) − K (ω , 0) ΔK = K K (ω , 0)

(11) 29

which is often measured in experiments. The normalized absorption spectra and differential absorption spectra of 20 × 20 × 2 nm3 nanoplatelets are compared in Figure 4. The

Figure 5. (a) Normalized differential absorption spectrum of 20 × 20 × 2 nm3 CdSe nanoplatelets exposed to weak electric fields, similar to the ones that were used in our earlier experimental work.29 (b) ΔK/K at the first excitonic peak of energy E = 2827 meV plotted as a function of electric field strength.

Figure 4. Normalized electroabsorption spectrum K(ω,F) and normalized differential absorption spectrum ΔK/K of 20 × 20 × 2 nm3 CdSe nanoplatelets exposed to different electric fields.

differential spectra are additionally normalized by subtracting the background value of ΔK/K taken far away from the interband resonance. One can see that the zero of a normalized differential spectrum corresponds to the main spectral line of the fundamental absorption peak, coming from the nanoplatelets perpendicular to the field, whereas the maximum of the spectrum indicates the position of the low-frequency line, produced by the nanoplatelets oriented along the field. This clear distinction prevents experimentalists from misinterpreting the additional peak in the absorption spectrum as a separate interband transition in the nanoplatelets. It should also be noted that the positive and negative lobes in the normalized differential spectra are going through maximum with the electric field strength. This typical feature of inhomogeneously broadened electroabsorption spectra of randomly oriented nanoplatelets was not observed previously due to the relative weakness of the electric fields used in experiments. Figure 5a shows the differential absorption spectrum calculated for weaker electric fields, which are similar to the fields that were used in our previous experimental work.29 In good agreement with the experiment, the lobes of the differential spectrum monotonously grow with the field strength rather than pass through maxima. The negative lobe in the real spectrumrepresenting the fundamental interband transitionis notably suppressed due to the presence of higher energy absorption peaks, which are neglected in the present analysis. Figure 5b shows ΔK/K, taken at the energy of the fundamental absorption peak, as a function of the electric field strength. This dependence is also in good agreement with the experimental data. Nonfundamental Transitions. We repeated the same analysis for other dipole-allowed interband transitions, such as (111) → (113) and (113) → (111), and a few dipole-forbidden transitions (which become allowed in the presence of the electric field), including (111) → (112) and (112) → (111), in the ensembles of randomly oriented nanorods and nanoplatelets. All the general features of electroabsorption upon the

fundamental transition were found to be peculiar to electroabsorption upon other transitions, though emerging at stronger electric fields. In particular, the absorption spectrum of nanoplatelets features an additional absorption peak, absent in the spectrum of nanorods, and the FWHMs of the spectral lines peak for a certain field strength, which decreases with the size of the nanoplatelets. Furthermore, even the shapes of the absorption peaks upon forbidden transitions are similar to those in Figures 1 and 3. This may appear somewhat strange at the first glance, given that the rates of forbidden transitions grow with the electric field strength, which results in the additional peak being stronger than the main one. The shape similarity still preserves since the electric field blue-shifts the additional peak and brings it to the high-energy side of the main peak. These results lead us to the conclusion that the field-induced broadening of electroabsorption spectra is predominantly controlled by the geometry of the nanocrystals in the ensemble.

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CONCLUSION As a concluding remark, we would like to emphasize the possibility of flexible engineering the electroabsorption spectrum of semiconductor nanocrystals. As we have seen, the strengths of the Stark and Franz-Keldysh effects are dictated by the largest dimensions of the nanocrystals. On the other hand, the positions of the absorption peaks are controlled by the smallest nanocrystal dimensions: the diameter of nanorods and the thickness of nanoplatelets. Therefore, by changing the size and shape of nanocrystals in a monodisperse ensemble, one can vary the field-induced broadening and positions of the absorption peaks as required for a particular application.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (I.D.R.). E

DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by Grant 14.B25.31.0002 and Government Assignment No.3.17.2014/K of the Ministry of Education and Science of the Russian Federation.

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DOI: 10.1021/acs.jpcc.5b08424 J. Phys. Chem. C XXXX, XXX, XXX−XXX