Tianyu Dong, Xiaoke Gao, Ke Yin, Chun Xu, and Xikui Ma, "Modified optical response of biased semiconductor nanowires within a nonlocal hydrodynamic framework," J. Opt. Soc. Am. B 37, 3277-3287 (2020)
Semiconductors and their oxides, when properly doped, are potential promising plasmonic material alternatives due to their special properties such as low loss and tunability. The hydrodynamic theory has been applied to describe the nonlocal response of pint-sized nanostructures even when several different kinds of charge carriers are considered, but when an external static magnetic field is presented the interplay between the gyrotropy and nonlocality needs to be considered, which is important and critical for semiconductors. We derive an analytical approach to calculate the optical properties of a plasmonic semiconductor nanowire in an external dc magnetic field within the multi-fluid hydrodynamic framework. The extended nonlocal Mie theory to magnetized multi-fluid plasmas predicts the existence of multiple acoustic and optical longitudinal modes within the multi-fluid hydrodynamic theory and the resonance splitting due to the applied bias magnetic field. We further focus on the nonlocal magneto-plasmonic response of nanowires that consist of thermally excited InSb, and predict the modified Zeeman splitting of the plasmonic extinction resonances due to the interplay between nonlocality and gyrotropy.
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The sources of values can be found in the references. Detailed information is also given in the references of Refs. [26,28].
The conductivity effective mass for the electrons, light and heavy holes is assumed to be equal to that of the density-of-state (DOS) mass, respectively, i.e., $ m_{\text{cond}}^* = {m^*} $. Here, $ {m_0} = {m_e} = 9.1 \times {10^{ - 31}}\; \text{kg} $ is the electron mass. The DOS hole mass is given by [26,39] $ m_h^* = (m_{lh}^{*3/2} + m_{hh}^{*3/2}{)^{2/3}} $. The conductivity effective hole mass is given by [26,40] $ m_{h,\text{cond}}^* = (m_{lh}^{*3/2} + m_{hh}^{*3/2})/(m_{lh}^{*1/2} + m_{hh}^{*1/2}) $.
When the Boltzmann distribution can be used for the electrons, the charge-carrier densities are given by [26,39] $ {n_e} = {n_h} = 2[{k_B}T/(2\pi {\hbar ^2}{)]^{3/2}}{(m_e^*m_h^*)^{3/4}}\exp [ - {E_g}/(2{k_B}T)] $, where $ {E_g} $ is the bandgap and $ \hbar = 1.054 \times {10^{ - 34}}\; \text{J} \cdot \text{s} $ is the reduced Planck’s constant.
Table 2.
Derived Parameters for InSb at Various Temperatureab
200 K
300 K
350 K
400 K
15.68
15.68
15.68
15.68
[]
1.370
6.340
10.344
15.370
[]
0.114
0.507
0.787
1.082
[]
0.251
1.113
1.736
2.420
[]
0.251
1.114
1.739
2.424
[]
1.047
1.529
1.628
1.758
[]
1.099
1.099
1.099
1.099
[]
4.752
4.752
4.627
4.396
[]
4.724
4.724
4.600
4.732
[]
0.093
0.199
0.271
0.366
[]
0.073
0.143
0.183
0.229
[]
0.249
0.559
0.746
0.916
[]
0.298
0.669
0.892
1.090
[]
8.532
10.894
12.142
13.490
[]
7.541
9.236
9.976
10.665
[]
1.568
1.921
2.047
2.133
[]
1.564
1.915
2.041
2.127
For completeness, the high frequency permittivities $ { \epsilon _\infty } $ are included. The parameters $ {\omega _h} $, $ {\gamma _h} $, $ {\beta _h} $, and $ {\omega _{ch}} $ for the two-fluid model are also calculated. Here, $ {\omega _i} = [{e^2}{n_i}/(m_i^*{ \epsilon _0}{)]^{1/2}} $, $ {\gamma _i} = e/(m_{i,\text{cond}}^*{\mu _i}) $, and $ {\beta _i} = (3{k_B}T/m_i^*{)^{1/2}} $ for both the electrons and holes.
We assume that the charge for holes is identical to electron charge, i.e., $ {q_{lh}} = {q_{hh}} = {q_h} = e $. For the cyclotron frequencies $ \omega _{ci}^{\text{ref}} = e{B_{\text{ref}}}/m_i^* $, $ {B_{\text{ref}}} = 1\; \text{T} $ is adopted so that the corresponding cyclotron frequencies for various bias magnetic fields $ {B_0} $ can be readily derived.
Tables (2)
Table 1.
Material Properties of InSb at Various Temperaturesa
The sources of values can be found in the references. Detailed information is also given in the references of Refs. [26,28].
The conductivity effective mass for the electrons, light and heavy holes is assumed to be equal to that of the density-of-state (DOS) mass, respectively, i.e., $ m_{\text{cond}}^* = {m^*} $. Here, $ {m_0} = {m_e} = 9.1 \times {10^{ - 31}}\; \text{kg} $ is the electron mass. The DOS hole mass is given by [26,39] $ m_h^* = (m_{lh}^{*3/2} + m_{hh}^{*3/2}{)^{2/3}} $. The conductivity effective hole mass is given by [26,40] $ m_{h,\text{cond}}^* = (m_{lh}^{*3/2} + m_{hh}^{*3/2})/(m_{lh}^{*1/2} + m_{hh}^{*1/2}) $.
When the Boltzmann distribution can be used for the electrons, the charge-carrier densities are given by [26,39] $ {n_e} = {n_h} = 2[{k_B}T/(2\pi {\hbar ^2}{)]^{3/2}}{(m_e^*m_h^*)^{3/4}}\exp [ - {E_g}/(2{k_B}T)] $, where $ {E_g} $ is the bandgap and $ \hbar = 1.054 \times {10^{ - 34}}\; \text{J} \cdot \text{s} $ is the reduced Planck’s constant.
Table 2.
Derived Parameters for InSb at Various Temperatureab
200 K
300 K
350 K
400 K
15.68
15.68
15.68
15.68
[]
1.370
6.340
10.344
15.370
[]
0.114
0.507
0.787
1.082
[]
0.251
1.113
1.736
2.420
[]
0.251
1.114
1.739
2.424
[]
1.047
1.529
1.628
1.758
[]
1.099
1.099
1.099
1.099
[]
4.752
4.752
4.627
4.396
[]
4.724
4.724
4.600
4.732
[]
0.093
0.199
0.271
0.366
[]
0.073
0.143
0.183
0.229
[]
0.249
0.559
0.746
0.916
[]
0.298
0.669
0.892
1.090
[]
8.532
10.894
12.142
13.490
[]
7.541
9.236
9.976
10.665
[]
1.568
1.921
2.047
2.133
[]
1.564
1.915
2.041
2.127
For completeness, the high frequency permittivities $ { \epsilon _\infty } $ are included. The parameters $ {\omega _h} $, $ {\gamma _h} $, $ {\beta _h} $, and $ {\omega _{ch}} $ for the two-fluid model are also calculated. Here, $ {\omega _i} = [{e^2}{n_i}/(m_i^*{ \epsilon _0}{)]^{1/2}} $, $ {\gamma _i} = e/(m_{i,\text{cond}}^*{\mu _i}) $, and $ {\beta _i} = (3{k_B}T/m_i^*{)^{1/2}} $ for both the electrons and holes.
We assume that the charge for holes is identical to electron charge, i.e., $ {q_{lh}} = {q_{hh}} = {q_h} = e $. For the cyclotron frequencies $ \omega _{ci}^{\text{ref}} = e{B_{\text{ref}}}/m_i^* $, $ {B_{\text{ref}}} = 1\; \text{T} $ is adopted so that the corresponding cyclotron frequencies for various bias magnetic fields $ {B_0} $ can be readily derived.
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