1. Introduction

The effect of quantum confinement in semiconductor nanocrystals results in the discreteness of the nanocrystals’ energy spectra and leads to the modification of the nanocrystal’s interaction with both the confined and external elementary excitations [1–3]. This significantly differs the absorption and emission spectra of nanocrystals from the spectra of bulk semiconductors [4–14]. In particular, semiconductor nanocrystals exposed to an electrostatic field exhibit the specific quantum-confined Stark and Franz-Keldysh effects [15–21]. These effects show great promise for useful applications in many fields, including energy harvesting [22,23], near-field sensing [24], and the advancement of nanocrystal-based lasers [25–28].

The effect of electroabsorption has recently been experimentally studied for CdSe quantum dots, nanorods, and nanoplatelets in the form of colloidal solutions [29]. The experiment revealed some intriguing electroabsorption properties of semiconductor nanoplatelets, which are flat nanocrystals with almost zero thickness dispersion (the thickness of a nanoplatelet can be changed discretely, with a step of one monolayer). One of such properties is the pronounced electro-optical response of nanoplatelets compared to the response of quantum dots and nanorods. It was specifically found that the field-induced change in the absorption efficiency of the nanoplatelets is ten times that of the quantum dots. This feature makes nanoplatelets very promising for applications and is due to the relatively weak spatial quantum confinement in the planes of the nanoplatelets. Here we develop a fully quantum-mechanical theory of electroabsorption by semiconductor nanocrystals, which is needed for quantification of these and other experimental findings. We significantly simplify the mathematics by ignoring the Coulomb interaction between the confined charge carriers and assuming the confining potential to be infinite. The developed theory not only describes all the major features of electroabsorption, such as the Stark effect, the Franz-Keldysh effect, and the field-induced broadening of electroabsorption spectra, but also provides guidance toward novel applications.

2. Theoretical formulation

The quantum mechanical treatment of the electroabsorption phenomenon in semiconductor nanocrystals requires knowing the wave functions and energy spectrum of electrons and holes confined by the nanocrystals. We begin this treatment by formulating an analytically solvable model of the electronic subsystem of a semiconductor nanocrystal.

2.1. Electronic subsystem of a semiconductor nanocrystal

In order to analytically calculate the energy spectrum and wave functions of the confined charge carriers, we employ the two-band envelope k · p model [30] and assume that the nanocrystal is in the form of a rectangular parallelepiped l_x × l_y × l_z. A state of the charge carriers is described by a set of three quantum numbers |n〉 = |n_x, n_y, n_z〉, each representing the spatially confined motion along one of the Cartesian axes. The energies and wave functions obey the Shrödinger equation

The form of potential V(r) critically depends on the nanocrystal dimensions and the orientation of the nanocrystal with respect to the external electric field F, which is assumed to be homogenous far away from the nanocrystal. Due to the nonsphericity of the nanocrystal, the electric field inside it, F_int, is inhomogeneous. We first ignore this inhomogeneity by assuming that the external field is simply changed inside the nanocrystal by a constant factor, which is the ratio of permittivities of the nanocrystal and the environment, ε = ε_NC / ε_e, and take the inhomogeneity into account later using the perturbation theory. With this assumption, the electrostatic potential inside the nanocrystal is given by V(r) = qFr= ε, where q = −e for electrons and q = e for holes.

The linearity of potential V (r) allows one to separate variables in the Shrödinger equation. By taking |n〉= |n_x〉|n_y〉|n_z〉 and considering the energy spectrum to be the sum of energies due to the confined motions in three spatial dimensions, $E_{\mathbf{n}}=E_{n_x}+E_{n_y}+E_{n_z}$, we obtain the following one-dimensional Shrödinger equation inside the nanocrystal:

Since the nanocrystal surface is assumed to be impenetrable for electrons and holes, the wave functions at this surface must vanish. The application of this boundary condition at facet v = l_v/2 relates the two integration constants in the general solution as $B_{n_v}=-A_{n_v}\text{Ai}\left[\sigma (l_v/2)\right]/\text{Bi}\left[\sigma (l_v/2)\right]$. Using the normalization condition, 〈n_v|n_v〉 = 1, then gives [31]

Finally, the energy spectrum $E_{n_v}$ can be calculated from the dispersion equation, which follows from the boundary condition at facet v = −l_v/2,

Some algebra shows that in weak fields $E_{n_v}\propto l_v^4$.

The exact electric field inside the nanocrystal can be decomposed into homogeneous and inhomogeneous contributions as F_int(r) = F/ε + δF(r). If the permittivities of the nanocrystal and its environment do not differ too much, then |δF|≪|F|/ε and electrostatic potential δV(r) of the inhomogeneous contribution can be treated as a small perturbation of potential V(r) = qFr/ε. By solving the equation ∇δV = δF and using the stationary perturbation theory, we can write the wave functions and energy spectrum of the nanocrystal in the form

Note that even though δV is determined up to a constant, this constant does not affect the perturbed wave functions and interband transition energies, because it results in the shift of the nanocrystals energy spectrum as a whole.

2.2. Electroabsorption by a semiconductor nanocrystal

Consider electroabsorption of a semiconductor nanocrystal upon an interband transition between a hole state m and an electron state n, which is induced by light of intensity I and frequency ω. The transition frequency is given by

2.3. Electroabsorption by an ensemble of semiconductor nanocrystals

The absorption coefficient of a monodisperse ensemble of N equally oriented nanocrystals is given by K_nm = N(ħω/I)W_nm. The absorption coefficient of a monodisperse ensemble with randomly oriented nanocrystals can be found by the averaging

If the nanocrystal dimensions in the ensemble are distributed about their mean values 〈l_v〉 (v = x,y,z), and the distribution is characterized by the probability density G(〈l_x〉, 〈l_y〉, 〈l_z〉, l_x, l_y, l_z), then the inhomogeneous size broadening of the electroabsorption spectrum of the ensemble is described by the absorption coefficient

The three dimensions of nanocuboids are usually independent of one another and characterized by the same distributions, which results in the following factorization of the probability density: G(〈l_x〉, 〈l_y〉, 〈l_z〉, l_x, l_y, l_z) = g (〈l_x〉, l_x) g (〈l_y〉, l_y) g (〈l_z〉, l_z). The size distribution of the nanocrystal edges is often described by the lognormal density function

3. Results and discussion

The developed theory describes all the significant features of electroabsorption of semiconductor nanocrystals, including the Stark effect, the Franz-Keldysh effect, and the field-induced broadening of electroabsorption spectra of nanocrystal ensembles.

The Stark effect is the field-induced shifting and splitting of absorption peaks in the nanocrystal spectrum. When a nanocrystal is exposed to a dc electric field, the size-quantized energies of its electrons and holes shift, as illustrated in Fig. 1(a), changing the frequencies of interband transitions. The change of the transition energies is described by Eqs. (5), (6), and (7). If the field is sufficiently strong, then the energies of some transitions are smaller than the band gap of bulk semiconductor, which makes the nanocrystal to absorb light at otherwise forbidden frequencies. The developed theory also predicts that the strength of the Stark effect critically depends on the dimensions of the nanocrystal and its orientation with respect to the electric field. In particular, the red shifts of the absorption peaks are maximal when the largest nanocrystal dimension is aligned with the field. The theory also describes the lifting of the degeneracy of the nanocrystal states and splitting of the respective peaks in the absorption spectrum due to the symmetry breaking of the confining potential by the electric field.

The Franz-Keldysh effect is the suppression of interband transitions that were dipole allowed without the field and the emergence of the forbidden transitions in the absorption spectrum of the nanocrystal. Both features are described by the overlap integral 〈〈n|m〉〉 in Eq. (8). In the absence of the electric field, the confined electrons and holes are described by the same set of wave functions, which means that only electric dipole transitions between the states with the same quantum numbers are allowed, i.e. 〈n|m〉 = δ_nm. The electric field shifts electrons and holes to the opposite sides of the nanocrystal, making the overlap integral 〈〈n|m〉〉 decrease with the field strength and reducing the rates of the dipole-allowed transitions. At the same time, the overlap integral of the originally orthogonal wave functions of electrons and holes becomes nonzero, which leads to the appearance of additional absorption peaks in the nanocrystal spectrum. The variation of the overlap integral is illustrated by Fig. 1(b). Like the Stark effect, the Franz-Keldysh effect is the most pronounced when the electric field is directed along the largest dimension of the nanocrystal, because the spatial separation of electrons and holes in a given field grows with the reduction of the quantum confinement.

The field-induced broadening of electroabsorption spectrum is a result of random orientation of the nanocrystals in space. This type of broadening comes from the averaging of the absorption spectra of individual nanocrystals, which differ due to the strengths of the Stark and Franz-Keldysh effects being dependent on the nanocrystal orientation relative to the external field. Indeed, nanocrystals that have their largest dimensions aligned to the electric field exhibit the strongest Stark and Franz-Keldysh effects, featuring spectra that are the most modified by the field. On the other hand, the spectra of nanocrystals with the smallest dimensions parallel to the field are almost unchanged by the field. As a result of spectra averaging, described by Eq. (10), each line in the absorption spectrum of the nanocrystal acquires a more or less pronounced low-frequency wing, like those shown in Fig. 1(c). The nonmonotonic dependence of the absorption peak on the field strength, evidenced by the figure, is another clear manifestation of the Franz-Keldysh effect. Also noteworthy is that the field-induced broadening is the strongest in ensembles of nanorods and nanoplatelets, whose three dimensions are significantly different, and is absent in the electroabsorption spectra of spherical nanocrystals.

 

Fig. 1 (a) Stark shift of electronic energies $E_{n_x}$, (b) overlap integral 〈n_x|m_x〉, and (c) in-homogeneously broadened absorption peak (111) → (112) of randomly oriented 2 × 20 × 20 nm³ CdSe nanoplatelets. In (a) and (b) l_x = 20 nm. In all panels δV = 0, the field strength is given inside the nanoplatelets, and the effective masses of electrons and holes are 0.11m₀ and 0.44m₀, where m₀ is the free-electron mass.

Download Full Size | PDF