Optical Transitions and Photoluminescence in Cylindrical Core/Layer/Shell β-CdS/β-HgS/β-CdS Heterostructure

Abstract

We present the theoretical consideration of the states of charge carriers in the HgS quantizing layer of a cylindrical core/layer/shell β-CdS/β-HgS/β-CdS heterostructure in the effective mass approximation within the framework of a simple two-band model. Various ranges of the geometric dimensions of the sample are considered by implementing the corresponding regimes of dimensional quantization for charge carriers in the layer. The electrostatic interaction between the electron and the hole is taken into account for each case, and, depending on the size of the sample, the corresponding values of the ground state energy of the pair are obtained. We also considered interband optical transitions in the sample and photoluminescence in each case. In each case, taking into account the electrostatic interaction leads to a shift in the threshold frequency of interband absorption and luminescence to the short-wavelength region, and the frequency itself is determined by the geometric dimensions of the sample.

Appendices

Calculation of the Averaged Potential from Eqs. (14) and (16)

In triangular coordinates, we have

$$\left| {{{{\mathbf{r}}}_{c}} - {{{\mathbf{r}}}_{{v}}}} \right| = \sqrt {r_{c}^{2} + r_{{v}}^{2} - 2{{r}_{c}}{{r}_{{v}}}\cos \alpha } .$$

(A.1)

Here, α is the angle between vectors r_e = r_c(x_c, y_c, 0) and ${{{\mathbf{r}}}_{{v}}}({{x}_{{v}}}$, ${{y}_{{v}}}$, 0). Substituting Eqs. (13) and (A.1) into Eq. (16), we obtain the following intermediate result:

$$\left\langle {{{U}_{{e - h}}}({{r}_{{v}}})} \right\rangle = - \frac{{2{{e}^{2}}}}{{\gamma L}}$$

$$ \times \;\left\{ {\int\limits_{{{R}_{1}}}^{{{R}_{2}}} {\left[ {\frac{1}{{{{r}_{c}} + {{r}_{{v}}}}} - \frac{{\cos \frac{{2\pi - {{n}_{c}}({{r}_{c}} - {{R}_{1}})}}{L}}}{{{{r}_{c}} + {{r}_{{v}}}}}} \right]F\left( {\frac{\pi }{2},k} \right)d{{r}_{c}}} } \right\},$$

(A.2)

$${{k}^{2}} = \frac{{4{{r}_{c}}{{r}_{{v}}}}}{{{{{({{r}_{c}} + r)}}^{2}}}}.$$

Here F$\left( {\pi {\text{/}}2,k} \right)$ is an incomplete elliptic integral of the first kind.

Considering now that k < 1, we use the asymptotic expansion of the elliptic integral for small values of the parameter [60] to calculate the integrals in Eq. (A.2). We obtain

$$\left\langle {{{U}_{{e - h}}}({{r}_{{v}}})} \right\rangle = - \frac{{{{e}^{2}}}}{\gamma }\int {\frac{{{{{\left| {{{\Phi }_{{{{n}_{e}}}}}({{r}_{c}})} \right|}}^{2}}}}{{\left| {{{{\mathbf{r}}}_{c}} - {{{\mathbf{r}}}_{{v}}}} \right|}}d{{{\mathbf{r}}}_{c}}} = - \frac{{{{e}^{2}}({{l}_{1}} - {{l}_{2}})}}{{\gamma ({{R}_{2}} - {{R}_{1}})}},$$

$${{l}_{1}} = \pi \left[ {\ln \frac{{{{R}_{2}} + {{r}_{{v}}}}}{{{{R}_{1}} + {{r}_{{v}}}}} + \frac{{{{r}_{{v}}}({{R}_{2}} - {{R}_{1}})}}{{({{R}_{1}} + {{r}_{{v}}})({{R}_{2}} + {{r}_{{v}}})}}} \right]$$

$${{l}_{2}} = \pi \cos \frac{{2\pi {{n}_{c}}{{R}_{1}}}}{L}\left\{ {\cos \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}\left[ {{\text{ci}}\left( {\frac{{2\pi {{n}_{c}}{{r}_{2}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right.} \right.$$

$$\begin{gathered} \left. { - \;{\text{ci}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{1}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right] + \sin \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L} \\ \times \;\left. {\left[ {{\text{si}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{2}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right) - {\text{si}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{1}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right]} \right\} \\ \end{gathered} $$

(A.3)

$$ + \;\pi \sin \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}\left\{ {\sin \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}\left[ {{\text{ci}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{1}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right.} \right.$$

$$\left. { - \;{\text{ci}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{2}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right] + \cos \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}$$

$$\left. { \times \;\left[ {{\text{si}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{2}}}}{L} - \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right) - {\text{si}}\left( {\frac{{2\pi {{n}_{c}}{{R}_{1}}}}{L} + \frac{{2\pi {{n}_{c}}{{r}_{{v}}}}}{L}} \right)} \right]} \right\}.$$

Here, si(x) and ci(x) are the integral sine and cosine, respectively. In dimensionless units for potential (A.3), we obtain

$$\begin{gathered} {{V}_{{e - h}}}({{\rho }_{{v}}},{{\rho }_{1}},{{\rho }_{2}}) = \frac{{\left\langle {{{U}_{{e - h}}}({{r}_{{v}}},{{R}_{1}},{{R}_{2}})} \right\rangle }}{{{{E}_{{{\text{et}}}}}}} \\ = - \frac{{{{e}^{2}}[{{l}_{1}}({{\rho }_{{v}}},{{\rho }_{1}}{{\rho }_{2}}) - {{l}_{2}}({{\rho }_{{v}}},{{\rho }_{1}}{{\rho }_{2}})]}}{{\gamma {{E}_{{{\text{et}}}}}({{\rho }_{2}} - {{\rho }_{1}})}}. \\ \end{gathered} $$

(A.4)

The graphs of functions (A.4) are presented in Fig. 3.

The dependence of potential (A.4) on the radius of the hole is practically linear. Therefore, the bulky expressions (A.3) and (A.4) can be extrapolated with great accuracy by a simple linear function

$${{V}_{{e - h}}}({{\rho }_{{v}}}) \approx - \left| a \right| + b{{\rho }_{{v}}}.$$

(A.5)

Table 7 shows the values of parameters a and b for various values of the layer thickness.

Solution of Equation (18)

By introducing the variable from Eq. (18), we obtain the following equation:

$$\begin{gathered} \xi = \left( {{{r}_{{{v} - \frac{E}{F}}}}} \right){{\left( {\frac{{2{{\mu }_{h}}F}}{{{{\hbar }^{2}}}}} \right)}^{{1/\beta }}},\quad (E = {{E}_{{e - h}}} + \left| V \right|), \\ \frac{{{{d}^{2}}\chi (\xi )}}{{d{{\xi }^{2}}}} - \xi \chi (\xi ) = 0. \\ \end{gathered} $$

(B.1)

The solutions of this equation are given, as is known [61], by a linear combination of the Airy functions of the first Ai(ξ) and the second (Bi(ξ)) kind,

$$\chi (\xi ) = {{C}_{1}}\operatorname{Ai} (\xi ) + {{C}_{2}}\operatorname{Bi} (\xi ).$$

(B.2)

Here, C₁ and C₂ are normalization constants. Given the boundary conditions (18), we have for the wave functions (B.2)

$$\begin{gathered} \chi ({{\xi }_{1}}) = \chi ({{\xi }_{2}}) = 0, \\ {{\xi }_{{1,2}}} = \left( {{{R}_{{1,2}}} - \frac{E}{F}} \right){{\left( {\frac{{2{{\mu }_{h}}F}}{{{{\hbar }^{2}}}}} \right)}^{{1\beta }}}. \\ \end{gathered} $$

(B.3)

Hence, we obtain the following expression to determine the energy spectrum:

$$\frac{{\operatorname{Ai} [{{{(1.222)}}^{{1\beta }}}({{\rho }_{1}}{{b}^{{1\beta }}} - \varepsilon {{b}^{{ - 2\beta }}})]}}{{\operatorname{Ai} [{{{(1.222)}}^{{1/\beta }}}({{\rho }_{2}}{{b}^{{1/\beta }}} - \varepsilon {{b}^{{ - 2\beta }}})]}}$$

$$ = \frac{{\operatorname{Bi} [{{{(1.222)}}^{{1\beta }}}({{\rho }_{1}}{{b}^{{1\beta }}} - \varepsilon {{b}^{{ - 2\beta }}})]}}{{\operatorname{Bi} [{{{(1.222)}}^{{1/\beta }}}({{\rho }_{2}}{{b}^{{1/\beta }}} - \varepsilon {{b}^{{ - 2\beta }}})]}},$$

(B.4)

$${{\rho }_{{1,2}}} = \frac{{{{R}_{{1,2}}}}}{{{{a}_{{{\text{et}}}}}}},\quad \varepsilon = \frac{E}{{{{E}_{{{\text{et}}}}}}}.$$

The solutions to this transcendental equation are the sought energy.

Calculation of Averaged Potential (31)

As we repeatedly noted, the ground state of charge carriers in various modes is of interest in this paper. Thus, we explicitly have for Eq. (31)

$$\begin{gathered} \bar {U}({{{\mathbf{r}}}_{e}},{{{\mathbf{r}}}_{{v}}};{{z}_{e}},{{z}_{{v}}}) \equiv U({{z}_{e}},{{z}_{{v}}}) \\ = - \frac{{4{{e}^{2}}}}{{\gamma {{L}^{2}}}}\int {\frac{{{{{\sin }}^{2}}\frac{{\pi ({{r}_{e}} - {{R}_{1}})}}{L}{{{\sin }}^{2}}\frac{{\pi ({{r}_{{v}}} - {{R}_{1}})}}{L}}}{{{{r}_{e}}{{r}_{{v}}}\sqrt {{{{({{{\mathbf{r}}}_{e}} - {{{\mathbf{r}}}_{{v}}})}}^{2}} + {{{({{z}_{e}} - {{z}_{{v}}})}}^{2}}} }}d{{{\mathbf{r}}}_{e}}d{{{\mathbf{r}}}_{{v}}}.} \\ \end{gathered} $$

(C.1)

Considering the motion in the (x, y) plane in the triangular coordinates (A.1), we obtain the following intermediate result

$$\bar {U}({{{\mathbf{r}}}_{e}},{{{\mathbf{r}}}_{{v}}};{{z}_{e}},{{z}_{{v}}}) \equiv U({{z}_{e}},{{z}_{{v}}})$$

$$\begin{gathered} = - \frac{{4\pi {{e}^{2}}}}{{\gamma {{L}^{2}}}}\int {\frac{{{{{\sin }}^{2}}\frac{{\pi ({{r}_{e}} - {{R}_{1}})}}{L}{{{\sin }}^{2}}\frac{{\pi ({{r}_{{v}}} - {{R}_{1}})}}{L}}}{{\sqrt {{{{({{{\mathbf{r}}}_{e}} - {{{\mathbf{r}}}_{{v}}})}}^{2}} + {{{({{z}_{e}} - {{z}_{{v}}})}}^{2}}} }}} \\ \times \;F\left( {\frac{\pi }{2};K} \right)d{{r}_{e}}d{{r}_{{v}}}, \\ \end{gathered} $$

(C.2)

$${{K}^{2}}\frac{{4{{r}_{e}}{{r}_{{v}}}}}{{{{{({{r}_{e}} - {{r}_{{v}}})}}^{2}} + {{{({{z}_{e}} + {{z}_{{v}}})}}^{2}}}}.$$

Here, $F\left( {\frac{\pi }{2}:K} \right)$ is an incomplete elliptic integral of the first kind. Now, using the asymptotic behavior of the function $F\left( {\frac{\pi }{2}:K} \right)$ for small values of the argument, we have, instead of (C.2),

$$U({{z}_{e}},{{z}_{{v}}}) \equiv U(t)$$

$$ = - \frac{{4\pi {{e}^{2}}{{x}_{1}}}}{{\gamma {{R}_{1}}}}\int\limits_0^1 {\int\limits_0^1 {\frac{{{{{\sin }}^{2}}\pi {{x}_{e}}{{{\sin }}^{2}}\pi {{x}_{{v}}}}}{{\sqrt {{{{({{x}_{e}} + {{x}_{{v}}} + 2{{x}_{1}})}}^{2}} + {{t}^{2}}} }}} } {\kern 1pt} {\kern 1pt} d{{x}_{e}}d{{x}_{{v}}}$$

(C.3)

$$ = - \frac{{4{{\pi }^{2}}{{e}^{2}}{{x}_{1}}}}{{\gamma {{R}_{1}}}}{{l}_{0}}({{x}_{1}},t).$$

In this integral, we replace functions sin²πx_e and sin²π${{x}_{{v}}}$ by their average values in the integration region x_e, ${{x}_{{v}}}$ ∈ [0; 1]: that is, 〈sin²${{x}_{{e,{v}}}}$〉 = $\frac{1}{2}$. For the corresponding integral, we obtain

$${{I}_{1}}({{x}_{1}},t) = \frac{1}{4}\int\limits_0^1 {\int\limits_0^1 {\frac{{d{{x}_{e}}d{{x}_{{v}}}}}{{\sqrt {{{{({{x}_{e}} + {{x}_{{v}}} + 2{{x}_{1}})}}^{2}} + {{t}^{2}}} }}} } $$

$$ = \frac{1}{4}\{ 2{{x}_{1}}\ln [2{{x}_{1}} + \sqrt {{{{(2{{x}_{1}})}}^{2}} + {{t}^{2}}} ] - \sqrt {{{{(2{{x}_{1}})}}^{2}} + {{t}^{2}}} \} $$

$$\begin{gathered} + \;\frac{1}{4}\{ 2{{x}_{1}} + 2 + \ln [2{{x}_{1}} + 2 + \sqrt {{{{(2{{x}_{1}} + 2)}}^{2}} + {{t}^{2}}} ] \\ - \;\sqrt {{{{(2{{x}^{1}} + 2)}}^{2}} + {{t}^{2}}} \} \\ \end{gathered} $$

(C.4)

$$ - \;\frac{1}{2}\{ 2{{x}_{1}} + 1 + \ln [2{{x}_{1}} + 1 + \sqrt {{{{(2{{x}_{1}} + 2)}}^{2}} + {{t}^{2}}} ]$$

$$ - \;\sqrt {{{{(2{{x}_{1}} + 1)}}^{2}} + {{t}^{2}}} \} .$$

Figure 4 shows the graphs of functions I₀(x₁, t) and I₁(x₁, t) for various geometric sizes of the sample.

It is seen that the results of approximate analytical and accurate numerical calculations completely coincide. In further calculations, we use the analytical expression (C.4). Since we are interested in the lowest, that is, the ground energy state of the e–h pair or the energy values near the minimum potential (C.1), we expand the function

$$U({{z}_{e}} - {{z}_{{v}}}) \equiv U(t) = \frac{{4\pi {{e}^{2}}}}{{\gamma {{R}_{1}}}}{{x}_{1}}{{I}_{1}}({{x}_{1}},t)$$

(C.5)

into a series about the minimum point of t₀ = 0. As a result, we obtain

$$U(t) \simeq {{x}_{1}}[\alpha ({{x}_{1}}) + \beta ({{x}_{1}}){{t}^{2}}].$$

(C.6)

Here,

$$\alpha ({{x}_{1}}) = \frac{1}{2}\{ {{x}_{1}}\ln [4({{x}_{{1 + 1}}})] + {{x}_{1}}\ln [4({{x}_{1}} + 1)]$$

$$ - \;\ln [2(2{{x}_{1}} + 1)] - 2{{x}_{1}}\ln [2(2{{x}_{1}} + 1)]\} ,$$

$$\beta ({{x}_{1}}) = \frac{1}{{8(1 + 2{{x}_{1}})}} - \frac{1}{{32{{x}_{1}}}} - \frac{1}{{32(1 + {{x}_{1}})}}.$$

Table 8 shows the values of parameters α and β for various geometric dimensions of the sample.