We found that the variation of dielectric permittivity of metals with temperature is responsible for the change in response of an optical antenna with temperature. Metals behave as non-perfect conductors at optical frequencies and the optical properties in this regime are also temperature dependent. As an example, in Fig. 1a, we show how both the real and imaginary parts of the dielectric constant of nickel change with temperature. Consequently, these changes modify the penetration depth within the resonant structures and vary the resonance of the dipole.
The electric field incident on an optical antenna may have a spectral composition E(ω). This spectral distribution generates a current given by Ohm’s law: J(ω) = σ(ω)E(ω). In a bolometric device, this current dissipates as Joule heat and produces a change in temperature of the device that can be sensed externally. If the transducer is a MIM tunnel junction, the current through the junction will produce the signal. In both cases, electric conductivity, σ, is the key factor to understand the response of the device. From the Drude model, this parameter is related to the electric permittivity, 𝜖(ω) = 𝜖
′(ω) + i𝜖
″(ω), by the following equation:
$$ \sigma(\omega)=\omega \epsilon_{0} \left( \epsilon^{\prime\prime}(\omega) - i \left( \epsilon^{\prime}(\omega) -1 \right) \right) , $$
(1)
where 𝜖
0 is the dielectric constant of vacuum. As far as the dielectric constant is dependent on temperature, the conductivity varies with temperature. Then, generated currents become temperature-dependent.
For MIM antenna-coupled devices, besides conductivity, other temperature-dependent parameters are also involved in the transduction [22], specially when temperature spans on several hundreds degrees. Temperature changes the response of the MIM diode because of changes in the Fermi level produced by the increase of mobility [23, 24]. MIM junctions rectify the currents flowing through them [7]. MIM junctions are typically placed at the feed point of the antenna as a thin insulation layer. In this case, the rectified current density arriving to the junction could be given as the component across the junction, J
z
:
$$ J_{z}= \frac{1}{S} {\int}_{S} \sigma E_{z} ds , $$
(2)
where the average is made across the transverse section of the junction, S, and we only consider the z component of the electric field at the junction, E
z
. The signal delivered by the device is obtained from the current excited by optical radiation, J
z
, considering the characteristics of the junction (biasing, temperature, geometry, material parameters, and charge carrier’s dynamics) [7].
Previous work has demonstrated that one of the simplest ways to detect incoming radiation by using an antenna is by incorporating a bolometer into it [8, 9, 25]. This antenna-coupled transductor works by increasing its temperature through Joule heating generated by the induced current in the antenna; this will give a signal that is proportional to the incoming radiation. By using a bolometer coupled to an antenna, it is possible to make faster detectors since the antenna would be used to couple the radiation which is usually done by the surface of the bolometer. The total power absorbed by the resonant element changes the temperature of the device. To simplify the device even further, it is possible to distribute the bolometric effect along the whole antenna structure. The response of optical antennas working as distributed bolometers is sensed as a voltage variation across the detector [9]. In this case, the dissipated power is given as
$$ Q= {\int}_{v} \mathbf{J}^{\ast} \mathbf{E} dv= {\int}_{v} \sigma(T,\omega) | E(\omega) |^{2} dv , $$
(3)
where the integration is carried out within the whole volume of the antenna, v.
We have illustrated this dependence with temperature and wavelength analyzing a nickel dipole antenna perpendicularly oriented with respect to a load line (see Fig. 2). The dimensions of the dipole are 2.5 × 0.2 × 0.05 μm
3 (length × width × thickness). This geometry is optimized to properly cover a spectral range between 8 and 12 μm. The lead lines have the same thickness of 50 nm and a width of 300 nm. The dipole is placed on a SiO 2 layer of 1.2 μm in thickness. The substrate is modeled as a Si wafer. This element works as a MIM rectifier if both arms of the dipole are electrically isolated by a thin oxide layer. If the element is fabricated in a single deposition step, the antenna can be considered as a distributed bolometer [9]. Figure 3a shows the variation of the density current through the feed point of this dipole antenna configuration, applicable to MIM antenna-coupled detectors, J
z
(Eq. (2)). Figure 3b represents the total dissipated power, Q (Eq. (3)), valid for distributed bolometers. These maps are obtained with COMSOL Multiphysics and express the dependence of these magnitudes with temperature and wavelength.
Same metal MIM junctions and bolometers require biasing to operate. In our case, this bias voltage, V
b
, is also the heating source. The value of the temperature reached by the nanoantenna as a function of the bias voltage is shown in Fig. 1b. Due to the small thermal inertia of the device associated with its tiny size, these changes can be made in very short time, allowing a high modulation frequency in the order of kHz [5]. From previous calculations, temperature drops two orders of magnitude when moving less than 1 μm away from an optical antenna on a SiO 2 insulation layer (see Fig. 2 in reference [5]). This means that temperature changes are confined to the close vicinity of the resonant elements.
Response of MIM junctions is dependent on temperature, geometry of the junction, and material parameters in a more complicated way than bolometers [23, 24]. Therefore, for simplicity, we focus our attention on distributed bolometer optical antennas. It is known that bolometers produce a signal proportional to the optical irradiance which is dissipated as heat by the resonant structure [25]. The change in resistivity vs. temperature (bolometric effect) is sensed through a biasing circuit. If we know the temperature distribution, it is possible to obtain the signal produced by the optical radiation impinging on the antenna. A simple phenomenological model can be given to obtain the temperature profile along the lead line [9]:
$$ T(x)- T^{*} = \frac{Q_{b}}{2\kappa} \left\lbrace \begin{array}{ll} -x^{2} - \frac{w^{2}}{4}+ wL_{r} & \text{ if } |x|\leq \frac{w}{2} \\ -w|x|+wL_{r} & \text{ if } \frac{w}{2} < |x| \leq L_{r} \end{array}\right., $$
(4)
where w is the width of the dipole, L
r
is the length along the load line to reach a location where the temperature is the operational temperature of the device, T
∗, κ is the thermal conductivity of the metal, and Q
b
is the averaged absorbed power per volume unit, i.e., a power density. We obtain Q
b
from the total dissipated power of the antenna as Q
b
= Q/v, where Q is given by Eq. (3) and v is the volume of the antenna. The temperature of operation of the device, T
∗, is set by the biasing voltage (or current). This bias voltage (or current) must remain stable during measurement at the given temperature. Moreover, at each operational temperature, T
∗ should be stable with respect to the variation in temperature caused by optical irradiance.
The bolometric signal can be given as follows:
$$ {\Delta} V = I_{b} {\Delta} R = \frac{V_{b}}{R} {\Delta} R, $$
(5)
where ΔR describes the change in resistance due to the temperature profile, T(x), produced by the optical power absorbed by the dipole antenna, R is the resistance at the operational temperature T
∗, and I
b
and V
b
are the bias current and bias voltage, respectively. The equations for ΔR and R are as follows:
$$\begin{array}{@{}rcl@{}} {\Delta} R & = & \frac{\rho_{0} \alpha}{S} {\int}_{-L_{r}}^{L_{r}} T(x)dx \end{array} $$
(6)
$$\begin{array}{@{}rcl@{}} R & = &\frac{\rho_{0} L_{r}}{S} [1+ \alpha(T^{*}-T_{0})] \end{array} $$
(7)
where T
0 is the temperature at which the resistivity of the material is ρ
0, α is the coefficient of resistance with temperature (TCR), S is the transverse area of the load line, and L
r
is the length from the dipole to the location where temperature reaches T
∗. The integration means that each portion of the lead line contributes in series to the change in resistance. In this equation, we consider that the material of the load lines and the dipole is the same, which is the case for distributed bolometers fabricated with a single material deposition.
Using the previous assumptions and the analytic solution for the temperature profile (see Eq. (4)), it is possible to obtain an expression for the bolometric signal, V
s
:
$$ V_{s}={\Delta} V = V_{b} Q_{b} \frac{\alpha}{\kappa} \frac{w}{4L_{r}} \left( {L_{r}^{2}} - \frac{w^{2}}{12} \right) = V_{b} Q_{b} \gamma . $$
(8)
This signal, V
s
, is sensed as a variation with respect to the bias voltage, V
b
. In Eq. (8), we distinguish three main contributions: the bias voltage, V
b
, that sets the temperature of operation, T
∗; the absorbed optical irradiance given by Q
b
; and finally, a parameter, γ, that summarizes the geometrical and material characteristics. If we consider w≪L
r
then γ = (α/κ)(L
r
w/4). The bolometric signal, V
s
, obtained from the device is plotted in Fig. 4a, as a function of wavelength and temperature.
Noise Evaluation
To operate the device, the antenna is heated to a certain operational temperature, T
∗, that is larger than the ambient temperature, T
0. In these conditions, the identification of noise sources is very important to evaluate the operational mode and the capabilities of this device as a spectrometer.
When the bolometric effect is the transduction mechanism, the noise of the element can be modeled by considering thermal noise, temperature noise, and Johnson noise. These contributions depend on the geometry of the detector, the properties of the material (electric and thermal conductivities), and its temperature [26]. Johnson noise acts as a voltage source:
$$ V_{\text{{\small Johnson}}}=\sqrt{4 k_{B} T^{*}R{\Delta} f} , $$
(9)
where k
B
is the Boltzman constant, Δf is the bandwidth of the detection system, and R is the resistance at T
∗ given by Eq. (7). Thermal noise is produced by the heat exchange between the device at a temperature T
∗, and its surroundings at T
0. This term is given as a noise equivalent power, NEP:
$$ \text{NEP}_{\text{{\small therm}}} = \sqrt{\frac{8k_{B} A_{d} \sigma_{\text{SB}} {\Delta} f ({T^{*}}^{5}+{T_{0}^{5}})}{\epsilon}} , $$
(10)
where σ
SB is the Stefan-Boltzman constant, A
d
is the detector area, and 𝜖 is the emissivity of the device. The last contribution that we consider is due to the temperature fluctuations in the device. This contribution can be modeled as a power fluctuation:
$$ \sqrt{< {\Phi}_{\text{{\small temp}}}>^{2} }= \sqrt{4 k_{B} K {T^{\ast}}^{2}} , $$
(11)
where K is the thermal conductance of the device.
The previous noise sources combine in quadrature and produce a variation of the signal voltage, V
s
. Johnson voltage can be directly included within this quadrature addition. For thermal noise and temperature noise, we use Eq. (8) to compute the noise voltage when the power given by Eqs. (10) and (11) are injected in the device. These contributions to noise are plotted in Fig. 5 where we can check that temperature noise as the most predominant source.
Once noise is evaluated, we estimate a signal-to-noise ratio (SNR) for the dipole antenna. The map of Fig. 4b shows SNR as a function of temperature and wavelength. Additionally, our results for the signal and noise of the device establish a condition for the stability of the bias voltage source. From Fig. 1b, we have evaluated that temperature changes at a maximum rate of 3K/mV at the highest bias voltage level (when temperature is around 1000 K). Conversely, at low temperature (close to 293 K), the variation of the bias voltage per degree reaches its maximum value of about 30 mV/K. If the bias voltage varies from 0 to around 1.5 V, we may assure a stability in this value in the order of 0.1 mV for a variation in temperature lower than 1K at the highest range in V
b
. This stability requirement can be relaxed if temperature is closer to room temperature. These requirements are easily fulfilled by a laboratory bias source. It is key that the measurement of the signal, V
s
, is done with synchronous lock-in techniques to sense the variation of the voltage, V
s
, with respect to the bias, V
b
, allowing the acquisition of very low signals. Figure 4a shows signal values in the order of tenths of microvolts that are easily detected with a lock-in measurement method.