Analyzing photoionization cross-sectional modulation in CdSe/CdS core–shell nanodots: Impact of strain and applied electric field

A schematic illustration of a single-centered donor inside a spherical core/shell nanostructure of CdSe/CdS with its potential profile. $r_{1}$ and $r_{2}$ are the outer and the inner radii of the shell region.

The Hamiltonian adopted for on-center impurity confined in a core/shell nanodot, employs spherical coordinates and is formulated within the EMA and parabolic-band approximations as follows:

H = H_{e} + V_{i m p} (\vec{r}, {\vec{r}}_{i m p}) .

(1)

Here,

H_{e}

is the Hamiltonian for a single electron confined in CSQD without impurity and

V_{i m p} (\vec{r}, {\vec{r}}_{i m p})

is the energy generated by the coulombic interaction between the impurity located at

{\vec{r}}_{i m p}

and the electron, defined as

V_{i m p} (\vec{r}, {\vec{r}}_{i m p}) = - \frac{e^{2}}{ε_{r} | \vec{r} - {\vec{r}}_{i m p} |} .

(2)

The

H_{e}

term is given by

H_{e} = - \frac{ℏ^{2}}{2} \nabla \cdot (\frac{1}{m^{*}} \nabla) + V_{c o n f} (\vec{r}) + e \cdot \hat{F} \cdot \vec{z} .

(3)

In this context, $ℏ$ represents the reduced Planck constant, $\vec{r}$ denotes the electron's position vector, and $e \cdot \hat{F} \cdot \vec{z}$ is the axial effective potential energy created by the external applied electric field $\hat{F}$ ⁠.

V_{c o n f} (\vec{r})

denotes the potential of confinement. For an unstrained structure, this potential of confinement is precisely equivalent to the confinement barrier's height:

V_{c o n f} (\vec{r}) = V_{u n s} (r) = E_{c}^{0} (C d S) - E_{c}^{0} (C d S e)

⁠. For the strained structure, the confinement potential denoted as

V_{h y d} (r)

takes on the following shape in the practical case when a strain field is present:

V_{c o n f} (\vec{r}) = V_{h y d} (r) = E_{h y d}^{C d S} - E_{h y d}^{C d S e},

(4)

where

E_{h y d}^{C d S e}

and

E_{h y d}^{C d S}

signify the values of the strained CB for the core and shell materials, respectively, as determined by the following relationship:

{\begin{matrix} E_{h y d}^{C d S e} = E_{c}^{0} (C d S e) + γ_{h y d} (C d S e) \times ε_{h y d}^{C}, \\ E_{h y d}^{C d S} = E_{c}^{0} (C d S) + γ_{h y d} (C d S) \times ε_{h y d}^{S}, \end{matrix}

(5)

where

γ_{h y d}

represents the hydrostatic deformation potentials and

ε_{h y d}

describes the fractional change in volume due to hydrostatic strain, presented in detail in Sec. II B.

In the EMA approximation, and under the modulation of the strain effect and the applied electric field, the single particle Schrödinger equation is given as

H ϕ_{i} (r) = H E_{i},

(6)

where

E_{i}

and

ϕ_{i} (r)

stand for their, respectively, energy levels and envelope wave functions.

Considering the spherical symmetry of the CSQD structure, we may divide wave function

ϕ (r)

into angular and radial components as follows:

ϕ_{i} (r) = R_{n l} (r) Y_{l}^{m} (θ, φ),

(7)

where

R_{n l} (r) = \frac{U_{n l} (r)}{r}

stands for the radial wave function and

Y_{l}^{m} (θ, φ)

represents the spherical harmonics, the radial wave functions conform to the revised Schrödinger's equation in the following structure:

\begin{aligned} [\frac{- ℏ^{2}}{2} \frac{d}{d r} (\frac{1}{m_{C, S}^{*} (r)} \frac{d}{d r}) + \frac{l (l + 1) ℏ^{2}}{2 m_{C, S}^{*} (r) r^{2}} + V_{h y d} (r)] U_{n l} (r) = E_{i} U_{n l} (r) . \end{aligned}

(8)

The symbols n and l are used to designate the principal and angular quantum numbers, respectively.

B. Strain effect for the one shell model

The theory of elasticity models the behavior of elastic solids subjected to stress. One of the foundations of this theory is Hooke's law, which describes the relationship between stress and strain in an elastic material. In the case of an isotropic material, the general formulation of this law—depicting the relationship between the elements of the strain tensor

ε_{i j}

and the elements of the stress tensor

σ_{i j}

—is as follows:^22,23

ε_{i j} = \frac{1}{E} [(1 + ν) σ_{i j} - ν T r (σ) δ_{i j}],

(9)

where E is Young’s modulus and

ν

is the Poisson ratio.

We derive the deformation expression for a spherically CSQD in the context of a single shell. Here, the displacement, expressed in the conventional spherical coordinates, assumes the following form:

u = (u_{r} (r), 0, 0) .

(10)

We need to impose the following continuum elasticity model (CEM) for the radial displacement at the interfaces between the core and shell materials:

σ_{r r}^{C} (r_{1}) = σ_{r r}^{S} (r_{1}),

(11)

u_{r}^{C} (r_{1}) - u_{r}^{S} (r_{1}) = ε_{1} r_{1},

(12)

where

ε_{1} = \frac{a_{υ} (C d S e) - a_{υ} (C d S)}{a_{υ} (C d S e)}

is the relative lattice mismatch.

Equation (11) describes the interface's mechanical equilibrium, while Eq. (12) is true for shrink-fit brought on by lattice mismatch. It establishes a link between the crystalline structure of the materials and a continuum approach.

In the limit of identical elastic constants [ν (Core) = ν (Shell); E (Core) = E (Shell)], the strain tensor component expressions that are non-zero are as follows:

for core material : {\begin{matrix} ε_{r r}^{C} = ε_{θ θ}^{C} = ε_{φ φ}^{C} = \frac{2}{3} ε_{1} \frac{1 - 2 ν_{C}}{1 - ν_{C}} (1 - \frac{r_{1}^{3}}{r_{2}^{3}}), \\ ε_{h y d}^{C} = ε_{r r}^{C} + ε_{θ θ}^{C} + ε_{φ φ}^{C} = 2 ε_{1} \frac{1 - 2 ν_{C}}{1 - ν_{C}} (1 - \frac{r_{1}^{3}}{r_{2}^{3}}), \end{matrix}

(13)

for shell material : {\begin{matrix} ε_{r r}^{S} = \frac{2}{3} ε_{1} \frac{1 - 2 ν_{C}}{1 - ν_{C}} \frac{r_{1}^{3}}{r_{2}^{3}} (\frac{r_{2}^{3}}{r^{3}} \frac{1 + ν_{C}}{1 - 2 ν_{C}} - 1) ≃ \frac{2}{3} ε_{1} \frac{1 + ν_{C}}{1 - ν_{C}} \frac{r_{1}^{3}}{r^{3}}, \\ ε_{θ θ}^{S} = ε_{φ φ}^{S} = \frac{- 2}{3} ε_{1} \frac{1 - 2 ν_{C}}{1 - ν_{C}} \frac{r_{1}^{3}}{r_{2}^{3}} (\frac{1}{2} \frac{r_{2}^{3}}{r^{3}} \frac{1 + ν_{C}}{1 - 2 ν_{C}} + 1) ≃ \frac{- 1}{3} ε_{1} \frac{1 + ν_{C}}{1 - ν_{C}} \frac{r_{1}^{3}}{r^{3}}, \\ ε_{h y d}^{C} = ε_{r r}^{C} + ε_{θ θ}^{C} + ε_{φ φ}^{C} = - 2 ε_{1} \frac{1 - 2 ν_{C}}{1 - ν_{C}} \frac{r_{1}^{3}}{r_{2}^{3}} . \end{matrix}

(14)

C. Photoionization cross section of a donor impurity

The light absorption process can be described as an optical transition probability that takes place from the initial state to a final state one assisted by a photon. In the dipole approximation, the PCS for a hydro genic impurity describing the transitions from the impurity ground state to a final sub-band state under an external optical excitation are based on Fermi's golden rule derived from time-dependent perturbation theory can be written as¹⁵

σ (ℏ ω) = [{(\frac{ξ_{e f f}}{ξ_{0}})}^{2} \frac{n_{r}}{ε_{r}}] \frac{4 π^{2}}{3} β_{F S} ℏ ω \sum_{κ} | M_{f i} |^{2} δ (E_{f} - E_{i} - ℏ ω),

(15)

where

n_{r}

is the refractive index of the material,

β_{F S}

is the fine structure constant,

ℏ ω

is the incident photon energy, and

E_{f}

and

E_{i}

are the energies of the final and initial states, respectively.

M_{f i} = ⟨ ϕ_{f} | e . r | ϕ_{i} ⟩

is the matrix element of dipole moment. The Dirac δ-function can be represented as a narrow Lorentzian function as follows:¹⁵

\begin{aligned} δ (E_{f} - E_{i} - ℏ ω) \to \frac{Γ}{π [{(E_{f} - E_{i} - ℏ ω)}^{2} - Γ^{2}]} = \frac{Γ}{π [{(E_{b} - ℏ ω)}^{2} - Γ^{2}]} . \end{aligned}

(16)

We take z-polarization, for example, to discuss, Therefore, the matrix element can be written as

\begin{aligned} M_{f i} = M_{z} & = \int {R_{f} (r) Y_{f}^{0} (θ, φ)}^{*} \times r \cos θ \times {R_{i} (r) Y_{i}^{0} (θ, φ)} \\ \times r^{2} \sin θ d r d θ d φ . \end{aligned}

(17)

Under these conditions, the PCS can be rewritten as

σ (ℏ ω) = [{(\frac{ξ_{e f f}}{ξ_{0}})}^{2} \frac{n_{r}}{ε_{r}}] \frac{4 π^{2}}{3} β_{F S} ℏ ω M_{z}^{2} \frac{Γ}{π [{(E_{b} - ℏ ω)}^{2} - Γ^{2}]} .

(18)

III. RESULTS AND DISCUSSION

We utilized the physical parameters listed in Table I in our numerical computations concerning electron and donor impurity states along with their associated characteristics, confined within type-II CdSe/CdS spherical CSQD.

TABLE I.

List of parameters for CdSe/CdS CSQD.²⁴

	Core material	Shell material
	CdSe	CdS
$m^{*} / m_{0}$	0.15	0.22
$E_{c}^{0} (e V)$	−4.26	−3.93
a_υ (nm)	0.605	0.582
$ε_{1}$	–	0.0380
γ_hyd(eV)	−2	−2.54
ν	0.408	0.410
$ε_{r}$	10.6	8.9

	Core material	Shell material
	CdSe	CdS
$m^{*} / m_{0}$	0.15	0.22
$E_{c}^{0} (e V)$	−4.26	−3.93
a_υ (nm)	0.605	0.582
$ε_{1}$	–	0.0380
γ_hyd(eV)	−2	−2.54
ν	0.408	0.410
$ε_{r}$	10.6	8.9

Figure 2 displays the EPD distributions in the x-projection of the normalized wave function for electron confinement within type-II CdSe/CdS CSQD. These distributions correspond to the four lowest states and are presented for three distinct shell widths. These projections are provided under the condition of a constant core radius, denoted as $r_{1}$ ⁠, without considering the impact of strain. It is worth noting that the symmetry of the ground state remains unaffected by the shell's width. Specifically, the EPD for this state consistently reaches its maximum and maintains concentricity with the core, regardless of the shell's width. As we transition to the first excited state $(ϕ_{2})$ ⁠, a distinct antinode becomes apparent. In contrast to the ground state, the behavior of the EPD in the third excited state $(ϕ_{4})$ is notably different. The EPD exhibits minimal values at the core of the structure and achieves its maximum values within a region of the barrier material. Additionally, the electronic cloud of the second excited state $(ϕ_{3})$ shows dual maxima (degeneracy) that symmetrically form within the core and then extend toward the boundary of the barrier.

FIG. 2.

The 2D-projection EPD $| ϕ_{i} |^{2}$ for the four electronic low-lowest states (i = 1, 2, 3, and 4) in x-projection for three different shell widths. Each row corresponds to a different shell width $w_{s}$ ⁠. Red color represents the maximum positive value, while blue is associated to the minimum value.

Figure 3 shows the combined effects of strain and axial electric field on the spatial distribution of EPD. It can be seen that, in the absence of an applied electric field, it becomes apparent that the strain effect exerts negligible influence on the EPD distribution of the four lowest electronic state wave functions, as this strain remains symmetrical and uniform throughout the lattice mismatch. However, in the scenario where the strain effect is present and an axial electric field is applied along the (Oz) axis at an intensity of 100 kV/cm, we observe the EPD of the ground state moving in the opposite direction to the applied field, while the EPDs of the excited states undergo significant spatial distortion and deformation as a result of the applied electric field.

FIG. 3.

The 2D-projection EPD $| ϕ_{i} |^{2}$ for the four electronic low-lowest states (i = 1, 2, 3, and 4) in x-projection without impurity. 1st row: without strain effect and F = 0 kV/cm, 2nd row: strain effect and F = 0 kV/cm, and 3rd row: strain effect and F = 100 kV/cm. The inner and outer radii of the core/shell are fixed. Red color represents the maximum positive value, while blue is associated to the minimum value.

Figure 4 illustrates the percentage of overall hydrostatic deformation in both the core and shell, varying with the shell width $w_{s}$ ⁠. Notably, the deformation in the core surpasses that in the shell, exhibiting an increase with growing shell width attributed to strain-induced lattice distortions. Furthermore, thin layers exhibit heightened sensitivity to hydrostatic deformation, while for thick shells, the deformation percentage stabilizes at 2%. In a prior investigation, Park et al. presented the strain-induced piezoelectric polarization concerning the diameter of the CdSe QD in a strained CdSe/CdS CSQD.²⁵

FIG. 4.

The percentage of strain in the core (blue line) and shell (orange line) in a spherical core/shell nanostructure of CdSe/CdS as a function of shell width $w_{s}$ ⁠, with a fixed core radius $r_{1} = 4 nm$ ⁠.

The fluctuation of the impurity's binding energy with and without the strain effect for the four lowest-energy levels in the CSQD nanostructure, with a fixed core radius of $r_{1} = 4 nm$ ⁠, is depicted in Fig. 5 as a function of shell width $w_{s}$ ⁠. Looking at the figure, it is evident that in the case of thin shells $(w_{s} \leq 2 nm)$ ⁠, the binding energy $E_{b}^{4}$ of the 3rd excited state exceeds 100 meV, whereas that of the ground state, $E_{b}^{1}$ is higher than that of the 1st and 2nd excited states. For small shell radii, tunneling effect improves with larger barrier areas, whereas the binding energies drop quickly with increasing $r_{2}$ ⁠. The binding energies continue to decrease, albeit only slightly, once the barrier width $w_{s}$ approaches 1.5 nm, i.e., $r_{2} = 5.5 nm$ ⁠, with the exception of $E_{b}^{4}$ which continues its rapid decay. We also note that the strain effect had no impact on $E_{b}$ of various states with small shell widths $(w_{s} \sim 1 nm)$ ⁠. Nonetheless, once $w_{s}$ surpasses 1 nm, the strain effect gradually becomes apparent, leading to a minor increase in $E_{b}$ for all states, except for the last excited state, where it significantly boosts $E_{b}^{4}$ ⁠. The increase in binding energy in the presence of the deformation effect is in reasonable agreement with the experimental result presented by Peng et al.²⁶

FIG. 5.

Binding energy $E_{b}^{i}$ of four low-lying states as a function of shell width $w_{s}$ with and without the strain effect.

Figure 6 displays the binding energy levels $E_{b}^{i}$ as a function of the CdS shell width $(w_{s})$ ranging from 0.4 to 8.4 nm. This plot includes two sets of lines: one representing the data with an applied electric field F = 100 kV/cm (shown as dashed lines) and the other without an applied electric field (represented by solid lines). These data pertain to a spherical CdSe/CdS CSQD, with the CdSe core radius $(r_{1})$ held constant at 4 nm. This illustration demonstrates that the electric field has a minimal impact on the binding energies of the first and second excited states (⁠ $E_{b}^{2}$ and $E_{b}^{3}$ ⁠). While the electric field does reduce the binding energies of these states, it does so to a limited extent, and this subtle reduction becomes noticeable when the shell width surpasses 2 nm. In the ground state, the impact of the electric field on $E_{b}^{1}$ is more significant compared to the two prior states, and this effect remains constant throughout the entire range of shell widths $w_{s}$ ⁠. Furthermore, it is worth highlighting that the 3rd excited state exhibits the highest sensitivity to the electric field. With increasing electric field strength, there is a corresponding decrease in binding energy, and the impact of the electric field becomes increasingly pronounced as the shell width nears 2 nm or when the outer radius of the CSQD surpasses 6 nm.

FIG. 6.

Binding energy $E_{b}^{i}$ of four low-lying states as a function of shell width $w_{s}$ without an applied electric field and with F = 100 kV/cm considering the strain effect. The radius of the core material $r_{1}$ is fixed at 4 nm.

Figure 7 illustrates the variation of the binding energy $E_{b}^{i}$ for the four lowest-energy states concerning the electric field strength F. The nanosystem size is fixed, with a constant shell width of $w_{s} = 8 nm$ ⁠. For all states, when we neglect the strain, the values of binding energies are lower than when the strain effect of taken into account, while retaining the same fluctuation pattern. The strain effect is less important in the ground state $E_{b}^{1}$ and in the first excited state $E_{b}^{2}$ ⁠, and more relevant for the other two excited states. For low electric field values (F ≤ 100 kV/cm), as F increases from 0 to 100 kV/cm, binding energies decrease monotonically. At high electric field strengths, the binding energies of the first two states decrease rapidly after critical F values, and these critical values change with the presence or absence of the strain effect. The energies of the other two excited states vary non-monotonically at high F intensities.

FIG. 7.

Binding energy $E_{b}^{i}$ of four low-lying states as a function of applied electric field F for strained and unstrained cases and fixed size in $(r_{1}, r_{2}) = (4, 12) nm$ ⁠.

Figure 8 illustrates the PCS spectrum of the on-center donor as a function of the photon field, polarized along z axis, for two distinct shell width values: 8 nm represented by solid lines and 16 nm indicated by dashed lines. In our study, we are examining the inter-band transitions involving the ground-donor state generated by the on-center impurity near the CB and the four lowest electron states produced by the quasi-free electrons within the CB. As we arrange the electronic energy levels sequentially, starting from state 1^e (ground state) and progressing to state 4^e (3rd excited state), we observe that the transition 1^imp–4^e shows a PCS peak shifted toward higher energy (blue-shift). This trend persists with transitions like 1^imp–3^e, 1^imp–2^e, and so on, ultimately leading to transition 1^imp–1^e, which displays a PCS peak in the low-energy range (red-shift). After scrutinizing the amplitudes of the PCS peaks, it becomes evident that the 1^imp–3^e transition shows the greatest amplitude, while the 1^imp–1^e transition showcases the smallest amplitude. As the shell width expands from 8 to 16 nm, one can notice an augmentation in the amplitude of peaks across various transitions, with no impact on the resonance positions of these peaks. This phenomenon arises because, with the widening of the barrier, there is a growing overlap between the ground and excited state wave functions due to their localization within the same region. Consequently, this leads to an amplified of the magnitude of the PCS peak.

FIG. 8.

PCS related to the transitions between the donor ground state and the four lowest electronic states, as a function of the energy of the incident photons at F = 0 kV/cm and $r_{1} = 4 nm$ ⁠, for two values of shell width $w_{s}$ ⁠, taking into account the strain effect.

The PCS coefficient as a function of incident photon energy for the same transitions, as in Fig. 8, for the size configuration $(r_{1}, r_{2}) = (4, 12) nm$ and without an applied electric field is shown in Fig. 9. In this figure, we study the correction made to PCS when considering the strain effect. For this purpose, we analyze two cases. Unstrained, presented by solid lines, and strained, presented by dashed lines. When comparing the strained and unstrained systems, one can observe that the PCS resonance peak shifting to higher energies (or experiencing a blue-shift) when considering the strain effect, in addition to the amplitudes of the peaks of the various transitions narrowing due to the strain effect. The influence of strain is particularly significant for transitions 1^imp–3^e and 1^imp–4^e in comparison to the other transitions, and this behavior corresponds with the variation in binding energy caused by the strain effect previously discussed in Fig. 5.

FIG. 9.

PCS related to the transitions between the donor ground state and the four lowest electronic states, as a function of the energy of the incident photons at F = 0 kV/cm and fixed size, for unstrained (solid lines) and strained (solid lines) structures.

We conclude our study by examining the PCS spectrum as a function of incident photon energy in Fig. 10, covering the same transitions as shown in Fig. 8, with the size configuration $(r_{1}, r_{2}) = (4, 12) nm$ ⁠. We consider the presence of strain effect under two conditions: without an applied electric field (solid lines) and with an electric field of F = 100 kV/cm. As an initial observation, the 1^imp–1^e transition is minimally influenced by the electric field, with little effect observed on both amplitude and PCS shifting, while increasing the electric field results in a considerable increase in the PCS amplitudes of both transitions 1^imp–2^e and 1^imp–3^e. The impact of F on the PCS of transition 1^imp–4^e is distinct: increasing F significantly reduces the PCS amplitude and shifts the resonance peak to a lower energy (red-shift). These behaviors align well with $E_{b}^{i}$ results depicted in Fig. 7. Specifically, within the low electric field intensity range, an increase in F leads to a decrease in $E_{b}^{i}$ for all transitions, indicating a blue-shift in PCS peaks. Simultaneously, the enhancement of the electric field strengthens the z-polarization, resulting in an increase in peak PCS intensity. Furthermore, while the increase in the electric field strengthens the z-polarization, this implies that the electron's EPD is pushed toward the extremity of the CSQD, leading to a decrease in the Coulombic interaction and resulting in a blue-shift of the PCS. We can observe a similar behavior in the impact of the axial electric field on the PCS spectrum for a core/shell structure of cylindrical geometry, with an on-center and off-center donor, as shown in Ref. 27.

FIG. 10.

https://doi.org/10.1103/PhysRevB.46.15578

PCS related to the transitions between the shallow donor ground state and the lowest states of the electron under the strain effect, as a function of the energy of the incident photons for two values of the applied electric field.

IV. CONCLUSION

In conclusion, we studied the binding energy and PCS of a donor impurity located in the core of a CdSe/CdS CSQD. Within the EMA framework, we utilized the FEM approach. For transitions between the impurity ground-state 1^imp and the four low-lying electron states in the CB, we examined the binding energy $(E_{b})$ and PCS taking into account the size of the QDs, the strain effect, and the strength of the electric field.

In summary, the main conclusions are as follows:

With the presence of strain effect and a thicker CSQD shell, a significant alteration occurs in the binding energy of the donor impurity. In thinner shells, electrons migrate into the shell region, leading to a decrease in the binding energy of the four low-lying states as the width increases. Furthermore, as the shell width rises, strain magnifies this effect on binding energy.
Raising the electric field strength does not notably impact the binding energy of the initial three states $E_{b}^{i = 1, 2, 3}$ ⁠, but it does lead to a substantial reduction in the binding energy of the 4th state $E_{b}^{4}$ ⁠.
Augmenting the shell width $w_{s}$ substantially enhances the amplitude of the PCS peaks without causing a significant shift in the resonance zone. In contrast, accounting for the strain effect leads to a blue-shift in the peaks, causing them to shift toward higher energies for all transitions.
The PCS spectrum exhibits a strong sensitivity to the electric field's intensity, and the extent of its impact differs for each transition.

With this study, we hope to demonstrate how careful examination of the donor impurity's optical and electrical properties along with consideration of the strain effect, applied electric field, and QD size might possibly lead to new developments in experimental research and the construction of innovative devices.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

A. Ed-Dahmouny: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Jaouane: Data curation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Fakkahi: Formal analysis (equal); Investigation (equal). K. El-Bakkari: Formal analysis (equal); Investigation (equal); Methodology (equal). R. Arraoui: Formal analysis (equal); Investigation (equal). H. Azmi: Formal analysis (equal); Investigation (equal). A. Sali: Supervision (equal); Validation (equal). N. Es-Sbai: Supervision (equal); Validation (equal).

DATA AVAILABILITY

All the files with tables, figures, and codes are available. The corresponding author will provide all the files in case they are requested.

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