^{1,a)}and Thomas G. Pedersen

^{1}

### Abstract

Surface plasmon polaritons (SPPs) and localized surface plasmon (LSP) resonances are not limited to noble metals. Any material with a substantial amount of free carriers will support surface plasma oscillations which, when coupled to an electromagnetic field, will result in surface plasmon polaritons and localized surface plasmon resonances in confined systems. Utilizing a full band structure approach, we analyze the plasmonic properties of several heavily doped semiconductors. We present rigorous quantum mechanical calculations of the plasma frequency, and study in detail its dependence on impurity doping concentration. Results are presented for silicon, germanium, gallium arsenide, zinc oxide, and gallium nitride. For silicon and zinc oxide, the surface plasmon resonance frequency is calculated for a large range of doping concentrations and we study the dispersion of surface plasmon polaritons on thin films. The investigated properties of heavily doped semiconductors hold promises for several interesting applications within plasmonics.

The authors gratefully acknowledge the financial support from the Project “Localized-surface plasmons and silicon thin-film solar cells—PLATOS” financed by the Villum foundation.

I. INTRODUCTION

II. THEORY

III. RESULTS: PLASMA FREQUENCY

A. Zincblende crystals

B. Wurtzite crystals

IV. RESULTS: SEMICONDUCTORPLASMONICS

A. The role of inter-band transitions

B. Plasmonic properties of Si and ZnO

1. Silicon

2. Zinc oxide

V. CONCLUSION

### Key Topics

- Doping
- 92.0
- II-VI semiconductors
- 67.0
- Semiconductors
- 34.0
- Plasmons
- 29.0
- III-V semiconductors
- 26.0

## Figures

Electron and hole concentrations in Si calculated as a function of the relative Fermi energy eV. The carrier concentrations are calculated for eV. The inset is the Si band structure for which we have calculated the carrier densities.

Electron and hole concentrations in Si calculated as a function of the relative Fermi energy eV. The carrier concentrations are calculated for eV. The inset is the Si band structure for which we have calculated the carrier densities.

Plasma frequency of Si calculated as a function of the relative Fermi energy, . The solid green curve is calculated in the effective mass approximation. The solid black curve is a calculation based on the full band structure at room temperature , and the dashed curve is the same, but at 0 K. The inset shows a zoom near the conduction band edge.

Plasma frequency of Si calculated as a function of the relative Fermi energy, . The solid green curve is calculated in the effective mass approximation. The solid black curve is a calculation based on the full band structure at room temperature , and the dashed curve is the same, but at 0 K. The inset shows a zoom near the conduction band edge.

Calculated plasma frequency as a function of the free carrier concentration. Both *n*- and *p*-type dopings are considered. Results are presented for three zincblende semiconductors: Si, Ge, and GaAs.

Calculated plasma frequency as a function of the free carrier concentration. Both *n*- and *p*-type dopings are considered. Results are presented for three zincblende semiconductors: Si, Ge, and GaAs.

Electron and hole concentrations calculated as a function of the relative Fermi energy , for ZnO. The inset display a segment around the Γ-point of the band structure for ZnO used in the calculation of the carrier densities.

Electron and hole concentrations calculated as a function of the relative Fermi energy , for ZnO. The inset display a segment around the Γ-point of the band structure for ZnO used in the calculation of the carrier densities.

Calculated plasma frequency as a function of the relative Fermi energy . Both polarizations *xy* and *z*, perpendicular and parallel to the crystal *c*-axis, respectively, are presented. The green curve is calculated in the effective mass approximation.

Calculated plasma frequency as a function of the relative Fermi energy . Both polarizations *xy* and *z*, perpendicular and parallel to the crystal *c*-axis, respectively, are presented. The green curve is calculated in the effective mass approximation.

Calculated (full band structure) plasma frequency as a functionof the free carrier concentrations. Results are presented for both *n*-and *p*-type dopings. Two wurtzite semiconductors, ZnO and GaN, are considered.

Calculated (full band structure) plasma frequency as a functionof the free carrier concentrations. Results are presented for both *n*-and *p*-type dopings. Two wurtzite semiconductors, ZnO and GaN, are considered.

Calculated plasma frequency as a function of the free carrier concentrations for ZnO. Results are presented for both *n*- and *p*-type dopings. The black curves are full band structure calculations, and the green curves are the result of the effective mass approximation using effective masses of and for electrons and holes, respectively. The magenta curve is the result of the Dirac effective mass model presented in Eq. (14) for eV.

Calculated plasma frequency as a function of the free carrier concentrations for ZnO. Results are presented for both *n*- and *p*-type dopings. The black curves are full band structure calculations, and the green curves are the result of the effective mass approximation using effective masses of and for electrons and holes, respectively. The magenta curve is the result of the Dirac effective mass model presented in Eq. (14) for eV.

Band gap narrowing and electron density as a function of the relative fermi energy calculated for ZnO.

Band gap narrowing and electron density as a function of the relative fermi energy calculated for ZnO.

Real and imaginary part of the interband dielectric constant of ZnO for polarization parallel to the crystal *c*-axis *z*, where band gap narrowing is taken into account in the calculation. Four different *n*-type doping concentrations are considered. A phenomenological broadening of eV has been used in the calculation.

Real and imaginary part of the interband dielectric constant of ZnO for polarization parallel to the crystal *c*-axis *z*, where band gap narrowing is taken into account in the calculation. Four different *n*-type doping concentrations are considered. A phenomenological broadening of eV has been used in the calculation.

(a) Surface plasmon resonance wavelength calculated as a function of the free carrier density in Si. The solid line is for *n*-type doping and the dashed line is for *p*-type doping. (b) Dispersion relation of SPPs on a heavily doped (free carrier density of ) Si film of thickness *t* = 300 nm. The dashed lines are for *p*-type doping and the solid lines are for *n*-type doping. Black curves are the dispersion of long-range modes, green curve are for short-range modes, and magenta lines are the dispersion relation of canonical SPPs, i.e., for . The gray shading is the light cone, which has the dashed black light line as the boundary between propagating and evanescent modes in the surrounding medium. The two horizontal dashed lines are for *p*- and *n*-type doping (the lowest is for *p*-type.) The two insets show the transverse component of the electric field for (1) the long-range mode and (2) the short-range mode.

(a) Surface plasmon resonance wavelength calculated as a function of the free carrier density in Si. The solid line is for *n*-type doping and the dashed line is for *p*-type doping. (b) Dispersion relation of SPPs on a heavily doped (free carrier density of ) Si film of thickness *t* = 300 nm. The dashed lines are for *p*-type doping and the solid lines are for *n*-type doping. Black curves are the dispersion of long-range modes, green curve are for short-range modes, and magenta lines are the dispersion relation of canonical SPPs, i.e., for . The gray shading is the light cone, which has the dashed black light line as the boundary between propagating and evanescent modes in the surrounding medium. The two horizontal dashed lines are for *p*- and *n*-type doping (the lowest is for *p*-type.) The two insets show the transverse component of the electric field for (1) the long-range mode and (2) the short-range mode.

(a) The same as in Fig. 10(a) but for ZnO. Results are shown for both polarizations and *n*- and *p*-type doping. (b) The same as in Fig. 10(b) but for ZnO, a free carrier concentration of cm , and a film thickness of *t* = 100 nm. Results are shown for the polarization parallel to the crystal *c*-axis (*z*) for both *n*- and *p*-type doping.

(a) The same as in Fig. 10(a) but for ZnO. Results are shown for both polarizations and *n*- and *p*-type doping. (b) The same as in Fig. 10(b) but for ZnO, a free carrier concentration of cm , and a film thickness of *t* = 100 nm. Results are shown for the polarization parallel to the crystal *c*-axis (*z*) for both *n*- and *p*-type doping.

## Tables

Key parameters of the band structures of the zincblende crystals, where is the indirect band gap, is the direct band gap, and is the spin-orbit splitting.

Key parameters of the band structures of the zincblende crystals, where is the indirect band gap, is the direct band gap, and is the spin-orbit splitting.

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