Ⅲ. Electrically reconfigurable difference-frequency-generation from polaritonic metasurface
3.2 Design of Multiple Quantum Well and Polaritonic Metasurface
The MQWs with energy state differences of 193 meV (6.4 μm), 117 meV (10.6 μm), and 76 meV (16.3 μm) should be designed for DFG. We designed the MQWs by self-consistent Schrodinger-Poisson solver [40] which calculated below three equation iteratively
− ℏ 2 2𝑚∗
𝑑𝜑𝑛
𝑑𝑧2+ 𝑉(𝑧)𝜓𝑛(𝑧) = 𝐸𝑛𝜓𝑛(𝑧) 𝑛(𝑧) = |𝜓𝑛(𝑧)|2𝑓(𝐸 − 𝐸𝑓)
𝜕2
𝜕𝑧2𝑉(𝑧) = −𝑒𝑛(𝑧) 𝜀0𝜀𝑟 .
(3.1)
m* is effective mass, V(z) is the electric potential, Ef is the fermi energy, φn(z) is the n-state wave function, and n(z) is the charge density. V(z), φn(z), and n(z) are calculated by self-consistent Schrodinger-Poisson solver and Figure 3.1 shows the results of the MQWs for DFG.
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Figure 3.2.1 Calculation results of the MQWs conduction band diagrams by voltage. Eij and zij (i, j=1, 2, 3) are the intersubband transition energy and the dipole element from i to j. The MQWs have asymmetric structure which the first well width is much larger than the second well. (a)-(c) The external bias is applied to the MQWs which value is (a) -2 V, (b) 0 V, and (c) +2 V respectively.
The conduction band is tilted when the external bias is applied to the MQWs by quantum-confined Stark effect. Assume that the electric field is biased along the z direction of MQWs, the applied force can be written as
𝑭 = 𝐹𝒛,
(3.2)
and the perturbing Hamiltonian term is
𝐻 = 𝐻0+ 𝑒𝐸𝑧.
(3.3)
By the first order perturbation theory, the first order correction to the eigenenergy can be expressed as
Δ𝐸(1) = ⟨𝜓1|𝑒𝐹𝑧|𝜓1⟩
(3.4)
and it is zero because of symmetry. However, the second order correction can be expressed as
Δ𝐸(2)= 1 24𝜋2(15
𝜋2− 1)𝑚∗𝑒2𝐹2𝑊4 ℏ
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(3.5)
where W is the width of the well. Therefore, the energy states of first well (Figure 3.1), which has larger well width, are changed larger than the second well.
The designed MQWs structures were composed of In0.52Al0.48As and In0.53Ga0.47As which are Ⅲ-Ⅴ compound semiconductor double quantum wells. This MQW structures were deposited on semi- insulating InP substrate (Epi ready wafer), which has similar lattice constant with InAlAs and InGaAs, by using molecular beam epitaxy (MBE). The table 3.1 shows each thickness of MQWs.
Table 3.2.1 MQWs growth table for electrically reconfigurable DFG metasurface
Layer Material Thickness (nm) Doping Level (cm-3) Role
1 InGaAs 300
Etch stop layer
2 InAlAs 100
3 InGaAs 20 1.00E+18 Contact layer
4 InGaAs 10
5 InAlAs 5
Start of 16 repeat periods - - -
6S1 InAlAs 5
MQWs active layer
7S1 InGaAs 9.8 1.60E+18
8S1 InAlAs 1.2
9S1 InGaAs 2.8
10S1 InAlAs 5
End of repeat periods
11 InAlAs 5
12 InGaAs 10
13 InGaAs 20 1.00E+18 Contact layer
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On the InP substrate, there are two etch stop layers with 300 nm-thickness InGaAs and 100 nm- thickness InAlAs. Above the etch stop layers, 20 nm-thickness InGaAs layer with n doped was deposited for contact layer. The 10 nm-thickness InGaAs layer and 5 nm-thickness InAlAs layer were grown on the contact layer as buffer layers. The total 400nm thickness of designed active MQW are deposited on the buffer layers, and it consists of 16 iterations of MQWs which is composed of In0.52Al0.48As and In0.53Ga0.47As. Each layer of a unit MQWs is 5/9.8/1.2/2.8/5 (nm), where the bold numbers mean thickness of In0.53Ga0.47As. The first well is n doped with 1.6 × 1018 cm-3.
We calculated the permittivity of the above MQWs after designed and fabricated the MQWs for DFG.
Each permittivity of MQWs for the out of plane (𝜀⊥) and in-plane (𝜀∥) of electric field is expressed as
𝜀⊥(𝜔) ≈ 𝜀∞(𝜔) +𝑒2𝑁𝑒
𝜀0ℏ { 𝑧122
(𝜔21− 𝜔) − 𝑖𝛾21+ 𝑧132
(𝜔31− 𝜔) − 𝑖𝛾31} 𝜀∥(𝜔) ≈ 𝜀∞(𝜔) + 𝑖 𝑁𝑒𝑒2𝜏𝑑
𝜀0𝜔𝑚∗(1 − 𝑖𝜔𝜏𝑑)
(3.6)
by Lorentz oscillator model. At the equation (3.6), ε∞ is the permittivity of the undoped MQWS and τd
is a Drude relaxation time. Figure 3.2.2 shows the permittivity for the out of plane by applied voltage.
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Figure 3.2.2 Calculation results of the permittivity of MQWs by Lorentz oscillator model (equation 3.6). (a) The real part of the permittivity and (b) imaginary part of the permittivity are changed by applied voltage. The permittivity depends on the IST energy and the IST energy is changed by QCSE. Therefore, the permittivity is also changed by external bias voltage.
Near the 6 μm and 10.6 μm, the values of imaginary part are high due to the IST. When the applied voltage is increased, the two graphs are shifted to longer wavelength (red shift). On the other hand, the graphs are shifted to shorter wavelength (blue shift), when the applied voltage is decreased. After calculated the permittivity of MQWs, we can also calculate the absorption coefficient of the MQWs and the Figure 3.2.3 shows the result.
Figure 3.2.3 Calculation result of the MQWs’ absorption coefficient by external bias voltage. In the case of the values of wavelength are about 6 μm and 10.6 μm, large and narrow absorption is calculated. It is because IST is occurred at this wavelength. When the external bias voltage is 0 V, the absorption coefficient is 10,012 cm-1.
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At the 10.6 μm wavelength, the absorption coefficient is near 10000 cm-1. It is much higher than other nonlinear material due to high dipole momentum value.
Also, we calculated the nonlinear second order susceptibility by equation (2.26) from chapter 2. In equation (2.26), there are complex terms. Therefore, not only magnitude of nonlinear second order susceptibility but also phase of nonlinear second order susceptibility is also change by external bias voltage.
Figure 3.2.4 The magnitude (a) and phase (b) of the second order susceptibility by external bias voltage on the MQWs. At the equation of the second order susceptibility, the complex terms make the change of phase. When the external bias voltage is increased, the second order susceptibility of MQWs is also increased.
Figure 3.2.4 shows the results of calculated second order susceptibility by the external bias voltage.
The magnitude of χ(2), MQW (Figure 3.2.4 (a)) is 380 nm/V at 6.5 μm wavelength and 0 V. The peak value is shifted to longer wavelength when the external bias voltage is increased. When the external bias voltage is 2.5 V, the magnitude value is the highest and it is 655 nm/V at 7.1 μm. The difference of phase (Figure 3.2.4 (b)) is 286.5 ˚ at 6.5 μm wavelength. The second order susceptibility of MQWs depends on the external bias voltage, so we can design the electrically reconfigurable nonlinear metasurface.
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To design the electrically reconfigurable DFG polaritonic metasurface, metal-insulator-metal (MIM) nanocavity structure is adopted. MIM structure is consist of bottom metal with Au, 400 nm-thickness of MQWs for insulator, and top metal with Au. This structure makes absorb strongly and confined nearfields to the metal. Using the MIM structure, we designed and optimized ‘T’ shape of meta-atom for DFG polaritonic metasurface like Figure 3.2.5. Fabrication of metasurface is detailed in the chapter 3.4 and the dimension of meta-atom is detailed in the caption of Figure 3.2.5
Figure 3.2.5 Schematic of the DFG polaritonic meta-atom which is metal-insulator-metal (MIM) structure for high absorption and nearfields confinement. The bottom metal is composed of gold (Au), platinum (Pt), and chrome (Cr) and they are deposited on Si substrate. The dimension values of meta-atom are t (thickness of MQWs) 400 nm, x (top metal width) 1.2 μm, px (x axis period of meta-atom) 2 μm, and py (y axis period of meta-atom) 0.9 μm.
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