Rotational Dipole Plasmon Mode in Semiconductor Nanoparticles

Abstract

We analyze a new type of the plasmon mode in nanosize semiconductor crystals. Rotational dipole plasmon resonance, in which only angular degrees of freedom are excited, prevails in the optical spectra of photodoped nanocrystals. Such a collective mode differs basically from surface plasmon resonances in typical photoabsorption spectra of metallic nanoclusters and can be described as an excited state in a finite Fermi system as well as a rotational motion of a quantum liquid. We demonstrate that such dipole oscillations are harmonic, which makes it possible to identify them as a plasmon resonant excitation.

APPENDIX

The collective excited state |Φ_ν〉 in the RPAX (29) is characterized by certain values of total angular momentum L and its projection M. This is ensured by the dependence of amplitudes X and Y on the angular momentum projections in terms of the Clebsch–Gordan coefficient

$$\begin{gathered} {{X}_{{{{n}_{p}}{{l}_{p}}{{m}_{p}}{{\sigma }_{p}},{{n}_{h}}{{l}_{h}}{{m}_{h}}{{\sigma }_{p}}}}} = {{( - 1)}^{{{{m}_{h}}}}}C_{{{{l}_{p}}{{m}_{p}},{{l}_{h}}M - {{m}_{h}}}}^{{LM}}\frac{{{{\delta }_{{{{\sigma }_{p}}{{\sigma }_{h}}}}}}}{{\sqrt 2 }}{{X}_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}}, \\ {{Y}_{{{{n}_{p}}{{l}_{p}}{{m}_{p}}{{\sigma }_{p}}{{n}_{h}}{{l}_{h}}{{m}_{h}}{{\sigma }_{p}}}}} = {{( - 1)}^{{{{m}_{p}}}}}C_{{{{l}_{p}}M - {{m}_{p}}{{l}_{h}}{{m}_{h}}}}^{{LM}}\frac{{{{\delta }_{{{{\sigma }_{p}}{{\sigma }_{h}}}}}}}{{\sqrt 2 }}{{Y}_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}}. \\ \end{gathered} $$

(45)

In the case of dipole excitations considered here, L = |l_p – l_h| = 1, and the dependence of the reduced parts of amplitudes ${{X}_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}}$ and ${{Y}_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}}$ on radial quantum numbers n_p,h and orbital angular momenta l_p,h of single-particle states is determined from the numerical solution of matrix equation (30).

In the two-level 2 × 2 RPAX model, in superposition (29), we take into account only one HOMO–LUMO transition, for which

$${{n}_{p}} = {{n}_{h}} = 1,\quad {{l}_{p}} - 1 = {{l}_{h}} = {{l}_{{\max }}}.$$

Consequently, only one pair of reduced amplitudes X and Y satisfying Eqs. (35) is left, and their values are written in explicit form as relation (37). Coulomb matrix element V in Eqs. (35), (36) can be written in form

$$\begin{gathered} V = \sum\limits_{{{m}_{h}}{{m}_{p}}m_{h}^{'}m_{p}^{'}}^{} {{{{( - 1)}}^{{{{m}_{h}} + m_{h}^{'}}}}C_{{{{l}_{h}} - {{m}_{h}}\,{{l}_{p}}{{m}_{p}}}}^{{10}}} C_{{{{l}_{h}} - m_{h}^{'}\,{{l}_{p}}m_{p}^{'}}}^{{10}} \\ \times \,\int {\psi _{{{{l}_{h}}{{m}_{h}}}}^{*}({\mathbf{r}}){{\psi }_{{{{l}_{p}}{{m}_{p}}}}}({\mathbf{r}}){v}({\mathbf{r}},{\mathbf{r}}{\kern 1pt} ')\psi _{{{{l}_{p}}m_{p}^{'}}}^{*}({\mathbf{r}}{\kern 1pt} '){{\psi }_{{{{l}_{h}}m_{h}^{'}}}}({\mathbf{r}}{\kern 1pt} ')d{\mathbf{r}}d{\mathbf{r}}{\kern 1pt} ',} \\ \end{gathered} $$

(46)

where the result of summation over spin indices has already been included in the form of factor 2. Single-particle wavefunctions are defined as product (7), and we disregard the dependence of their radial components on the angular momentum

$${{P}_{{1,{{l}_{{\max }}}}}}(r) = {{P}_{{1,{{l}_{{\max }}} + 1}}}(r) = {{P}_{1}}(r).$$

Operator ${v}$(r, r') of the Coulomb interaction between particles is represented as the sum of “direct” and exchange components (19), which makes it possible to calculate the corresponding terms separately, V = V_d + V_x.

The “direct” part of operator V(r, r') is determined from the multipole expansion in spherical harmonics (2). After integration with respect to angular variables [59],

$$\begin{gathered} \langle {{Y}_{{{{l}_{p}}{{m}_{p}}}}}({\mathbf{n}}){\text{|}}{{Y}_{{lm}}}({\mathbf{n}}){\text{|}}{{Y}_{{{{l}_{h}}\,{{m}_{h}}}}}({\mathbf{n}})\rangle \\ = {{( - 1)}^{{ - {{m}_{h}}}}}\sqrt {\frac{{(2{{l}_{h}} + 1)(2{{l}_{p}} + 1)}}{{4\pi (2l + 1)}}} C_{{{{l}_{h}} - {{m}_{h}}\,{{l}_{p}}{{m}_{p}}}}^{{lm}}C_{{{{l}_{h}}\,0{{l}_{p}}0}}^{{l0}}, \\ \end{gathered} $$

(47)

$$C_{{{{l}_{h}}0{{l}_{p}}0}}^{{10}} = {{( - 1)}^{{{{l}_{h}}}}}\sqrt {\frac{{3({{l}_{h}} + 1)}}{{(2{{l}_{h}} + 1)(2{{l}_{p}} + 1)}}} ,$$

(48)

and subsequent summation over m_{p, h}, we obtain

$$\begin{gathered} {{V}_{d}} = \frac{{2{{e}^{2}}({{l}_{{\max }}} + 1)}}{3}\int {{{P}_{1}}{{{(r)}}^{2}}{{P}_{1}}{{{(r{\kern 1pt} ')}}^{2}}} \\ \times \left( {\frac{{r_{ < }^{l}}}{{{{\varepsilon }_{1}}r_{ > }^{{l + 1}}}} + \frac{{({{\varepsilon }_{1}} - {{\varepsilon }_{2}})(l + 1){{{({{r}_{1}}{{r}_{2}})}}^{l}}}}{{{{\varepsilon }_{1}}(l{{\varepsilon }_{1}} + (l + 1){{\varepsilon }_{2}}){{R}^{{2l + 1}}}}}} \right)drdr{\kern 1pt} '. \\ \end{gathered} $$

(49)

The corresponding contribution 2ΔV_d/${{\hbar }^{2}}$ to Ω² on the right-hand side of Eq. (36) with account for relations (14), (25), (22), and (20) exactly coincides with term K_C/M in formula (24) for $\omega _{p}^{2}$ in the hydrodynamic model.

Analogously, the exchange part of the matrix element can be written in form

$$\begin{gathered} {{V}_{x}} = - 2{{\left( {\frac{1}{{9\pi }}} \right)}^{{1/3}}}\frac{{{{e}^{2}}}}{{{{\varepsilon }_{1}}}}\sum\limits_{{{m}_{h}}{{m}_{p}}m_{h}^{'}m_{p}^{'}}^{} {C_{{{{l}_{h}} - {{m}_{h}},{{l}_{p}}{{m}_{p}}}}^{{10}}} C_{{{{l}_{h}} - m_{h}^{'},{{l}_{p}}m_{p}^{'}}}^{{10}}{{( - 1)}^{{{{m}_{h}} + m_{h}^{'}}}} \\ \times \int {\frac{{\psi _{{{{l}_{h}}{{m}_{h}}}}^{*}({\mathbf{r}}){{\psi }_{{{{l}_{p}}{{m}_{p}}}}}({\mathbf{r}})\psi _{{{{l}_{p}}m_{p}^{'}}}^{*}({\mathbf{r}}){{\psi }_{{{{l}_{h}}m_{h}^{'}}}}({\mathbf{r}})}}{{{{\rho }_{0}}{{{(r)}}^{{2/3}}}}}} d{\mathbf{r}}, \\ \end{gathered} $$

(50)

where factor 2 appears again as a result of summation over the spin indices. The sum of the product of two spherical harmonics in $\psi _{{{{l}_{h}}{{m}_{h}}}}^{*}$(r)${{\psi }_{{{{l}_{p}}{{m}_{p}}}}}$(r) and the Clebsch–Gordan coefficient can be transformed to [59]

$$\begin{gathered} \sum\limits_{{{m}_{h}},{{m}_{p}}}^{} {C_{{{{l}_{h}} - {{m}_{h}},{{l}_{p}}{{m}_{p}}}}^{{10}}} {{( - 1)}^{{{{m}_{h}}}}}{{Y}_{{{{l}_{p}},{{m}_{p}}}}}({\mathbf{n}})Y_{{{{l}_{h}}, - {{m}_{h}}}}^{*}({\mathbf{n}}) \\ = \sqrt {\frac{{(2{{l}_{p}} + 1)(2{{l}_{h}} + 1)}}{{12\pi }}} C_{{{{l}_{h}}0,{{l}_{p}}p}}^{{10}}{{Y}_{{1,0}}}({\mathbf{n}}). \\ \end{gathered} $$

(51)

After the calculation of the angular part, expression (50) becomes

$${{V}_{x}} = - {{\left( {\frac{1}{{9\pi }}} \right)}^{{1/3}}}\frac{{{{e}^{2}}}}{{{{\varepsilon }_{1}}}}\frac{{2({{l}_{{\min }}} + 1)}}{{4\pi }}\int\limits_0^\infty {\frac{{P_{1}^{4}(r)}}{{{{r}^{2}}{{\rho }_{0}}{{{(r)}}^{{2/3}}}}}dr.} $$

(52)

Therefore, with account for relations (14), (25), (22), and (20), exchange contribution 2ΔV_x/${{\hbar }^{2}}$ to Ω² exactly coincides with exchange term K_x/M in relation (24).

Expressions (45) for RPAX amplitudes were also used in the calculation of transition density $\rho _{{{\text{tr}}}}^{{(\nu )}}$(r) = 〈Φ_ν|δρ(r)|Φ₀〉. After the summation over spin indices, formula (41) is transformed to

$$\begin{gathered} \rho _{{{\text{tr}}}}^{{(\nu )}}({\mathbf{r}}) = \sqrt 2 \sum\limits_{{{n}_{p}},{{l}_{p}},{{m}_{p}},{{n}_{h}},{{l}_{h}},{{m}_{h}}}^{} {{{{( - 1)}}^{{{{m}_{h}}}}}} \\ \, \times (X_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}^{\nu }C_{{{{l}_{p}}{{m}_{p}},{{l}_{h}} - {{m}_{h}}}}^{{10}}{{\psi }_{{{{l}_{h}}{{m}_{h}}}}}({\mathbf{r}})\psi _{{{{l}_{p}}{{m}_{p}}}}^{*}({\mathbf{r}}) \\ \, + Y_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}^{\nu }C_{{{{l}_{p}} - {{m}_{p}}\,{{l}_{h}}{{m}_{h}}}}^{{10}}\psi _{{{{l}_{h}}{{m}_{h}}}}^{*}({\mathbf{r}}){{\psi }_{{{{l}_{p}}{{m}_{p}}}}}({\mathbf{r}})). \\ \end{gathered} $$

(53)

The summation over m_p and m_h is performed using expression (51) and leads to final result

$$\begin{gathered} \rho _{{{\text{tr}}}}^{{(\nu )}}({\mathbf{r}}) = \frac{1}{{\sqrt {6\pi } {{r}^{2}}}}\sum\limits_{{{n}_{p}},{{l}_{p}},{{n}_{h}},{{l}_{h}}}^{} {{{P}_{{{{n}_{p}}{{l}_{p}}}}}(r){{P}_{{{{n}_{h}}{{l}_{h}}}}}(r)} \\ \, \times (X_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}^{\nu } + Y_{{{{n}_{p}}{{l}_{p}},{{n}_{h}}{{l}_{h}}}}^{\nu })\sqrt {(2{{l}_{h}} + 1)(2{{l}_{p}} + 1)} \\ \, \times C_{{{{l}_{h}}0{{l}_{p}}0}}^{{l0}}{{Y}_{{l0}}}({\mathbf{n}}), \\ \end{gathered} $$

(54)

which with account for relation (48), gives exactly expressions (42) and (43).