#
THz Mixing with High-T_{C} Hot Electron Bolometers: A Performance Modeling Assessment for Y-Ba-Cu-O Devices

^{1}

^{2}

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## Abstract

**:**

_{C}superconducting YBa

_{2}Cu

_{3}O

_{7}

_{–x}(YBCO) oxide nano-constrictions are promising THz mixers, due to their expected wide bandwidth, large mixing gain, and low intrinsic noise. The challenge for YBCO resides, however, in the chemical reactivity of the material and the related aging effects. In this paper, we model and simulate the frequency dependent performance of YBCO HEBs operating as THz mixers. We recall first the main hypotheses of our hot spot model taking into account both the RF frequency effects in the YBCO superconducting transition and the nano-constriction impedance at THz frequencies. The predicted performance up to 4 THz is given in terms of double sideband noise temperature T

_{DSB}and conversion gain G. At 2.5 THz for instance, T

_{DSB}≅ 1000 K and G ≅ − 6 dB could be achieved at 12.5 μW local oscillator power. We then consider a standoff target detection scheme and examine the feasibility with YBCO devices. For instance, detection at 3 m through cotton cloth in passive imaging mode could be readily achieved in moderate humidity conditions with 10 K resolution.

## 1. Introduction

_{C}(T

_{C}is the superconducting critical temperature) and high-T

_{C}HEBs, Nb and YBa

_{2}Cu

_{3}O

_{7}

_{–x}(with x lower than 0.2, and called YBCO hereafter), respectively. The concept then fruitfully evolved [2], rather in favor of low T

_{C}devices to start with, mainly Nb and NbN, exhibiting the diffusion cooled and phonon cooled processes, respectively [3]. In fact, the research on HEBs was mainly driven by immediate applications to THz radio astronomy as mixers for heterodyne reception [4,5]. Besides, YBCO was attractive in several respects, e.g., operating temperature with light cryogenics, or fast response, due to its very short electron-phonon relaxation time τ

_{ep}, in the ps range [6,7,8]. Early YBCO HEB mixers were demonstrated at mm-wave [9,10,11,12] or THz [9,10,13,14,15] frequencies. Difficulties were encountered, however, which mainly arose from YBCO chemical reactivity to atmospheric water and carbon dioxide and to the fabrication process as well, which compromised the durability of good performances for frequency mixing operation [16]. They also arose from the superconducting film to substrate phonon escape time τ

_{esc}, much longer for YBCO [17] than for Nb, for instance [18], with a detrimental effect on the instantaneous bandwidth [11,12,13,19]. More recently, MgB

_{2}has met a sustained interest as a compromise between NbN and YBCO, in terms of operating temperature, but also in terms of instantaneous bandwidth [20,21].

_{e}interacts with the phonon reservoir at temperature T

_{p}; this interaction is governed by the above-mentioned electron-phonon relaxation time τ

_{ep}. The phonons release their energy to the device substrate with the above-mentioned escape time τ

_{esc}[22]. If the electron-electron interaction time τ

_{ee}is much shorter than τ

_{ep}, (which is the case for Nb, NbN or YBCO), hot electron bolometric action occurs, such that T

_{e}> T

_{p}, provided that τ

_{esc}<< τ

_{ep}. This latter condition is related to the film-substrate interface quality, which is mainly correlated with the device technological elaboration process. As a further refinement of this model, a three thermal reservoir model can be considered, including the substrate to the cold finger thermal resistance [23]; although 0D, this latter model also discusses the dimensions of the device constriction to achieve good sensitivity HEBs. As shown in Figure 1b, the constriction is defined by its length L, width w and thickness θ.

_{e}(x) with respect to the constriction length coordinate (0 < x < L). First developed for low T

_{C}HEBs [24], this so-called “hot spot” modeling was further applied to high T

_{C}devices [25]. The YBCO HEB model then evolved to include more realistic descriptions, such as the phonon temperature spatial dependence T

_{p}(x) [26] and the operating radio frequency (RF) influence on the shape of the resistance vs. temperature superconducting transition [27].

## 2. Models and Methods

#### 2.1. Describing the Superconducting Transition at THz Frequencies

_{e}of the YBCO constriction at angular frequency ω = 2πf, we describe the superconductor by a two-fluid (2F) model. The concentrations of the superconducting and normal charge carriers are n

_{S}and n

_{N}, respectively, with n

_{S}+ n

_{N}= n, the total free electron concentration [28]. Defining the two-fluid critical temperature T

_{C}

^{2F}, the dependence of n

_{N}/n = f

_{N}(t

_{r}) as a function of reduced temperature t

_{r}= T

_{e}/T

_{C}

^{2F}was chosen as f

_{N}(t

_{r}) = t

_{r}

^{4}. This choice minimizes the electron system free energy, according to the Gorter-Casimir 2F model [28]. Hence the real and imaginary parts of the complex conductivity σ = σ

_{1}− jσ

_{2}, assuming Ohm’s law is still valid in an impurity scattering description [29]:

_{N}is the normal state conductivity, r = ωτ

_{S}is the impurity scattering parameter (for scattering time τ

_{S}), μ

_{0}is the permeability of a vacuum and λ

_{0}is the London’s penetration depth at t

_{r}= 0.

_{C0}, as discussed in Reference [27], with

_{N}is a normalization coefficient and ΔT

_{G}is the critical temperature standard deviation, to be adjusted to fit the experimental transition.

_{1}

^{2F}(ω → 0) and the Gaussian function, so that

^{2F}(ω) in the so-obtained DC transition, and then fit a new Fermi-Dirac-like transition with frequency dependent parameters (mid-transition temperature, transition width and minimum resistivity, as detailed in Section 3.1) [27].

#### 2.2. HEB Master Equations

_{C}= L × w × θ (Figure 1b), the thermal power balance is governed by the heat diffusion equations for the electron and phonon baths, represented by Equation (4a,b), respectively, as follows:

_{e}and κ

_{p}are the corresponding thermal conductivities, C

_{e}and C

_{p}the unit volume specific heats (evaluated at temperatures T

_{ec}and T

_{pc}, respectively). Integers n and m characterize the thermal exchanges between electrons and phonons and phonons and substrate (at reference temperature T

_{0}), respectively. The source term in Equation (4a) represents the Joule power locally injected in a constriction slice of width δx. This power results from both DC and THz currents flowing along the constriction of cross-sectional area S

_{C}= w × θ. More specifically, after having dropped the higher order terms:

_{0}is the DC bias current, I

_{LO}(local oscillator) and I

_{S}(signal) are the THz current amplitudes at angular frequencies ω

_{LO}and ω

_{S}, respectively. The resistivity functions are deduced as discussed in Section 2.1. It therefore appears that (i) the current components are constant over S

_{C}(see below, hypothesis 2) and (ii) the knowledge of T

_{e}(x) is required; this implies an iterative resolution procedure for the coupled Equation (4a,b).

- These equations obviously describe a phonon cooling mechanism for the electrons in YBCO, as opposed to Nb HEBs, where cooling by electron diffusion to the metal contacts is the dominant process [3,31]. In fact, using YBCO data (cf. Section 3.2, Table 1), we evaluate the electron diffusion length l
_{e}≈ π(τ_{ep}κ_{e}/C_{e})^{1/2}≅ 25 nm, a significantly smaller value than the constriction length (100 to 400 nm) considered here, so that phonon cooling will prevail [22]. - It also appears from Equation (5) that the DC and THz current densities are assumed to be constant across the area S
_{C}, a point discussed at length in Reference [32]. Due to the resistivity vs. T_{e}(x) dependence, DC and THz powers are non-uniformly absorbed along the constriction, as noticed in Reference [33]. - The constriction ends (at coordinates x = 0 and x = L) are at reference temperature T
_{0}, which is the cryostat/cryogenerator cold finger temperature (gold antenna contacts). So that: T_{e}(0) = T_{p}(0) = T_{e}(L) = T_{p}(L) = T_{0}. - The solutions T
_{e}(x) and T_{p}(x) follow the geometrical symmetry of the constriction with respect to its center: T_{e}(x) = T_{e}(L − x) and T_{p}(x) = T_{p}(L − x). Consequently, the equation solving will be performed in a half-constriction, e.g., x ∈ [L/2, L]. - The dissipated power in the constriction tends to raise T
_{e}(x) and T_{p}(x), which will reach their maximum values at the constriction center, so that: T_{e}^{max}= T_{e}(L/2), and T_{p}^{max}= T_{p}(L/2), with dT_{e}(L/2)/dx = dT_{p}(L/2)/dx = 0. - The energy exchange exponent n = 3 was taken for YBCO, as extracted from electron-phonon interaction time measurements (see, e.g., Reference [1]). The YBCO to substrate phonon mismatch index m = 3 resulted from the number of available phonon modes, proportional to T
^{3}(Debye’s model). - For numerical computation convenience, the resistive superconductive transition has been approximated by a Fermi-Dirac function of the form ρ = ρ
_{0}[1 + exp((T_{C}− T_{e})/ΔT)]^{−1}, to fit the variation given by Equation (3). - Due to thermal effects, T
_{C}is sensitive to the constriction DC bias current I_{DC}; this effect has been taken into account by writing T_{C}(I_{DC}) = T_{C}(0)[1 − (I_{DC}/(J^{ref}S_{C}))^{2/3}], where J^{ref}is a reference current density. This point is mentioned in Reference [25] and discussed in Reference [34], in the context of YBCO thin film electrical transport vs. microstructure relationship. - Because we can have access to the constriction RF impedance, it was possible to handle the effective power dissipated in the constriction as α × P
_{LO}, with α = α_{imp}× α’, where α_{imp}is the impedance matching factor between the constriction and the antenna, and α’ represents all the losses from other origins (focusing optics, diplexer, etc.).

#### 2.3. Solving HEB Master Equations

_{C}, κ

_{e}, C

_{e}, n, τ

_{ep}, α’, S

_{C}, κ

_{p}, C

_{p}, m, τ

_{esc}. Formally, the fixed (input) variables are P

_{LO}, I

_{0}and T

_{0}, because in a measurement procedure, these are the parameters controlled by the operator. The resolution of the equations should therefore be performed on the (output) variables T

_{e}(x) and T

_{p}(x), and their second derivatives d

^{2}T

_{e}/dx

^{2}and d

^{2}T

_{p}/dx

^{2}as well.

_{e}(x) and T

_{p}(x) are deduced at each point provided that we define T

_{e}

^{max}and T

_{p}

^{max}, where the relation T

_{e}

^{max}≥ T

_{p}

^{max}will help us to converge. The basic resolution approach is therefore to test (T

_{e}

^{max}, T

_{p}

^{max}) pairs until the value found at the extremity of the constriction is: T

_{e}(L) = T

_{p}(L) = T

_{0}, which represents our criterion of success for the calculation.

_{e}

^{max}and I

_{0}, such that T

_{e}

^{max}is now a fixed parameter and I

_{0}a computational variable. This “trick” facilitates the calculations, because, on the one hand, the T

_{p}

^{max}solution search is carried out with a fixed upper bound (T

_{e}

^{max}is known). On the other hand, difficult cases where different constriction resistance (R) values can be associated with the same I

_{0}value (because I

_{0}is surjective towards R), can be simplified: In fact, T

_{e}

^{max}is strongly correlated with the resistance (because T

_{e}

^{max}is bijective with R in all our calculations).

_{LO}(and not P

_{LO}) is the THz input, P

_{LO}being deduced at the end of the calculation. Therefore, it is not possible to perform a calculation that will accurately give a result ascribed to a given P

_{LO}value. What we obtained by our method was a “cloud of points” where each of these points is a configuration of the HEB operating at a certain power P

_{LO}, at a certain bias current I

_{0}and having a certain resistance R (as well as all the other characteristics that we wish to deduce).

_{LO}values we wished to investigate (e.g., P

_{LO}∈ [0, 50 μW]). With these new series of points, we were able to reuse the processing tools for the calculation of the conversion gain and the noise temperature, as further commented in Section 2.4.

_{LO}is homogeneous along the constriction, the RF heat dissipation αP

_{LO}is then deduced from the Z

_{RF}= R

_{RF}+ jX

_{RF}THz impedance. The result is then fed back into Equation (5), according to:

_{imp}× α’ (hypothesis 10 above) is a function of Z

_{RF}, through the impedance matching coefficient

_{a}is the antenna impedance. The left hand term is the regular impedance matching expression, whereas the right hand term is a correction ratio between the RF power dissipated in the constriction and the total input power.

#### 2.4. HEB Mixer Performance

#### 2.4.1. General Considerations and Conversion Gain

_{DC}, and (ii) the contribution originating from the RF power (P

_{LO}and P

_{S}). To do this, we first define two parameters K

_{DC}and K

_{RF}as:

_{S}<< P

_{LO}.

_{DC}can be deduced from the device I−V relationship at given P

_{LO}, whereas K

_{RF}cannot be deduced directly. Indeed, the bolometer electrothermal feedback implies that a P

_{LO}variation introduces a variation of R, and thus a change of the dissipated P

_{DC}. This feedback, which involves power variations ΔP

_{DC}and ΔP

_{LO}, can be taken into account in K

_{RF}from the resistance variation

_{L})/(incident RF signal power) = P

_{L}/P

_{S}can be deduced from the small signal IF current $\tilde{I}$, as (see, e.g., Reference [24]):

_{0}= R − ΔR and I

_{0}is the constriction current when P

_{S}= 0.

^{in}= T

^{out}/G.

#### 2.4.2. Noise Temperature Contributions

_{Jn}in series with the bolometer, such that $\langle {v}_{\mathrm{Jn}}^{2}\rangle =4{\mathrm{k}}_{\mathrm{B}}{T}_{0}{R}_{0}\Delta f$, where Δf is the output signal bandwidth. In this context, v

_{Jn}introduces variations ΔR

_{Jn}and ${\tilde{I}}_{\mathrm{Jn}}$, hence the noise power ${P}_{\mathrm{Jn}}^{\mathrm{out}}={\mathrm{k}}_{\mathrm{B}}{T}_{\mathrm{Jn}}^{\mathrm{out}}\Delta f$ dissipated in the mixer load resistance R

_{L}. Using the conversion gain expression (Equation (13)), one readily obtains the dual sideband (DSB) expression of the mixer input Johnson noise:

_{ee}and is expressed as follows [24]:

_{L}. In the same manner as previously, one obtains the DSB expression of the mixer input thermal fluctuation noise:

_{IF}, hence the approximate expression T

_{IF}/2G for the DSB input noise temperature. Besides, we have considered here T

_{IF}as independent of the working frequency and equal to the cooling temperature T

_{0}of the HEB [38].

_{BB}= T

_{hot}or T

_{cold}, and placed at the mixer input. Defining Y = (P

_{hot}/P

_{cold})

^{out}as the ratio between the associated mixer output powers (in the mixer bandwidth Δf), one obtains for the mixer input noise T

_{N}

^{in}= (T

_{hot}− YT

_{cold})/(Y − 1) in the Rayleigh-Jeans limit [39]. Moreover, the mixer gain is equal to the slope of the P

^{out}vs. k

_{B}T

_{BB}Δf straight line plot (k

_{B}is the Boltzmann constant).

#### 2.5. Standoff Detection Implementation with an HEB Heterodyne Detector

_{BB}emitted by a target of area A

_{T}and collected by a detector of area A

_{D}(the HEB focusing lens effective area). For a receiver of bandwidth 2Δν at center frequency ν

_{0}and for a temperature difference ΔT to be resolved at T

_{op}, the power difference is

_{ν}(T) is the blackbody spectral radiance (in $\mathrm{W}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{sr}}^{-1}\xb7{\mathrm{Hz}}^{-1}$), ε

_{T}the target emissivity, and d

_{T}the target to detector distance (possibly corrected to include imaging system primary and secondary mirrors).

_{T}/f

_{L}, approximately, where f

_{L}is the focal length of the focusing lens. We finally work out at wavelength λ:

_{int}= 1/(2πτ

_{int}), τ

_{int}being the (e.g., post-detection) integration time; t

_{opt}, t

_{atm}, and t

_{obs}are the transmission factors related to optics, atmosphere and obstacles, respectively (Figure 2). These factors are considered in the following.

- t
_{opt}includes the HEB planar antenna main lobe efficiency, the focusing lens and the detector cryostat window losses, and the LO injection losses (diplexer or beam splitter). - t
_{obs}represents the transmission through some obstacle existing in front of the target (cloth, cardboard, etc.).

## 3. Simulation Results

#### 3.1. YBCO Superconducting Transition in the THz Range

_{C}), the widening of the resistive transition (ΔT) and the appearance of a minimum resistivity (ρ

_{min}). To the first order, we write:

_{C}(f

_{THz}) = T

_{C}(0) × (1 − A

_{T}f

_{THz}),

_{THz}) = ΔT(0) × (1 + B

_{T}f

_{THz}),

_{min}(f

_{THz},t

_{r}= 0.8) ≅ 2ρ

_{min}(f

_{THz},t

_{r}= 0) = C

_{T}f

_{THz}.

_{T}, B

_{T}and C

_{T}parameters are introduced in the expression of the resistivity according to a Fermi-Dirac function (for convenience in our simulation). Their values are calculated for a given material and approximated for the 0 to 5 THz frequency range. Thus, in a first step, it is necessary to calculate C

_{T}from the expression of ρ

_{min}deduced from ρ

^{2F}(ω,t

_{r}→ 0), as presented in Equation (21c). The pair of parameters (A

_{T}, B

_{T}) is then calculated by determining the pair which minimizes the difference function between the gaussian fit and the current resistivity function |ρ

_{G}(T, ω) − ρ (T, ω)|, in both the 50 to 100 K and 0 to 5 THz ranges.

_{C}= 89 K [25,30], a transition width of 1.2 K and a conductivity σ

_{N}= 3.15 × 10

^{5}Ω

^{−1}·m

^{−1}. Such constrictions were made from YBCO films sputtered on MgO (001) single crystalline substrates. X-ray diffraction confirmed the mainly c-axis orientation of the films, with rocking curves exhibiting full width at half maximum values of ~ 0.18 degree for the (005) YBCO line. Weak parasitic lines were observed however, which indexation could also confirm the presence of a-axis growth [16]. The measured c-axis lattice parameter values ranged from 1.169 to 1.170 nm, which are close to the value of fully oxygenated Y

_{1}Ba

_{2}Cu

_{3}O

_{7}films [42]. Besides, in-plane twinning of the films is expected [43]. The film surface morphology was uniform with a typical roughness of about 4.5 nm rms independent of the YBCO thickness [44].

_{T}= 0.011 THz

^{−1}, B

_{T}= 0.186 THz

^{−1}and C

_{T}= 6.82 × 10

^{−8}Ω·m·THz

^{−1}. Those results are illustrated in Figure 3.

#### 3.2. DC Characteristics

#### 3.2.1. Temperature Profiles

_{e}(x) > T

_{p}(x), and (ii) the hot spot (i.e., normal state) region Δx

^{HS}such that T

_{e}(x) > T

_{C}for x ∈ Δx

^{HS}. We notice that T

_{e}(x) is closer to T

_{p}(x) for device B; this is due to the stronger phonon film to substrate escape efficiency, as testified by the ~3.5 times longer τ

_{esc}value for device B (Table 1).

_{C}material data for the NbN constriction considered in Reference [24]. We obtained T

_{e}(x) profiles very close to each other (less than 5% difference). This was verified for various P

_{LO}values in the quasi-static (QS) regime, i.e., at the low ω

_{LO}frequency limit of the constriction impedance (Equation (8)).

#### 3.2.2. Current-Voltage Plots

_{e}(x) profile allows to have access to the resistivity profile along the constriction, hence to the constriction resistance/impedance by integration (Equation (8)). As explained in Section 2.3, the I-V plots can then be deduced in the form of "clouds of points". This representation illustrates, in color/gray tone levels, the device DC response as a function of e.g., P

_{LO}, as shown in Figure 5 for devices A and B.

_{0}and exhibiting a short film to substrate escape time. Consequently, the largest voltage variations, as P

_{LO}varies, where a larger part of the constriction is still superconducting, are situated below ~ 30 mV. This effect is not observed for device B, due to the longer escape time in that latter case.

_{LO}= 220 nW is, however, noticeably different from that reported in Reference [24]. This is seemingly due to the difference in the computational method, but this difference does not affect the calculation of the dynamic performances (conversion gain and noise temperature) of the HEB.

#### 3.3. Mixer Performance

_{LO}; they were deduced from the heat equations solving results and presented in the form of “clouds of points” (see Section 2.3). To highlight the influence of P

_{LO}for a given DC bias power P

_{DC}, we have illustrated those results as P

_{DC}vs. P

_{LO}maps. We have considered four operating frequencies: The quasi-static (QS) regime, 500 GHz, 2.5 THz and 4 THz. The results are presented in the following.

#### 3.3.1. Noise Temperature

_{DC}vs. P

_{LO}area evolves in level and that the optimal P

_{LO}value also evolves (see Table 2 below). In particular, we notice that the secondary area of low noise temperature, starting at P

_{LO}≈ 35 μW in the QS mode, vanishes when the frequency increases.

_{LO}> 20 μW, whereas at 2.5 THz the low noise area is distributed differently, with improved noise level when 10 μW < P

_{LO}< 15 μW.

#### 3.3.2. Conversion Gain and Summary of Mixer Results

_{DC}increases, and is maximized at given P

_{LO}. Thus, as the frequency of the local oscillator increases, the conversion gain appears to be maximum around 2.5 THz and requires a local oscillator power of 12.5 μW (see Table 2 below).

_{imp}coupling losses, P

_{LO}should increase as 1/α

_{imp}. In fact, the present model introduces two physical properties, namely the influence of the non-uniform distribution of the dissipated LO power, and the impedance matching coefficient between the antenna and the nano-constriction, which alter the response of the simulated HEB.

#### 3.3.3. IF Bandwidth

_{IF}= (ω

_{LO}− ω

_{S})/2π, as originating from the last term in Equation (5). This is illustrated in Figure 10 for devices A and B. The main features are: (i) The low frequency regular bolometric response plateau, (ii) the regular bolometric cutoff associated with the escape time τ

_{esc}, (iii) a second plateau characterizing the HEB action, (iv) the HEB cutoff associated to the YBCO intrinsic relaxation time τ

_{ep}. Clearly, the bandwidth is limited by the τ

_{esc}value (Table 1), which is in favor of ultrathin and small volume constrictions [23,25]. It therefore appears that a bandwidth close to 1 GHz can be expected for device A.

#### 3.4. Standoff Detection Performances Requirements

_{int}= 1 Hz. Five frequencies centered in atmospheric transmission windows were considered: 670 GHz, 1.024 THz, 1.498 THz, 2.522 THz, and 3.436 THz. Besides, the detection scenario was chosen according to the following parameters:

- Target emissivity ε
_{T}= 1. - Optical losses were evaluated according to the antenna main lobe efficiency (−2 dB), the focusing lens and the detector cryostat window losses (−1 dB), and the LO injection losses (beam splitter: −3 dB), amounting to t
_{opt}≅ 24%. - t
_{atm}was determined for various relative humidity values (RH, in the 10% to 70% range) at operating temperature T_{op}= 295 K, at sea level and with clear atmospheric conditions (e.g., no dust).

_{DSB}≈ 85 K is required for a system operating at 2.5 THz at RH level of 40 %). In order to reach a viable standoff passive detection operation with T

_{DSB}values technologically reachable, it is therefore necessary to reduce either the target distance or the temperature resolution.

_{DSB}≈ 110 K for ΔT = 1 K, a restrictive parameter which could be relaxed to T

_{DSB}approaching 1100 K for ΔT = 10 K.

## 4. Discussion

#### 4.1. Inhomogeneous P_{LO} Hypothesis Effect on I-V Plots at Low DC Voltage

_{imp}between the terahertz antenna and the nano-constriction, of impedance R

_{RF}+ jX

_{RF}. This impedance is very small at low P

_{DC}and P

_{LO}power levels (say < 1 μW). In that case, α

_{imp}≅ 0 and therefore the contribution of αP

_{LO}in the heat equations seems to be negligible: If it were true, the I-V plots would be very similar at various P

_{LO}levels.

_{LO}values. The reason is related to the method of calculation, as considered now.

- In the conventional hot spot method (homogeneous P
_{LO}dissipation), the LO power expression, including impedance matching, is written as:$$\alpha {P}_{\mathrm{LO}}={\alpha}^{\prime}\frac{4{R}_{\mathrm{a}}R}{{({R}_{\mathrm{a}}+R)}^{2}}{P}_{\mathrm{LO}},$$_{a}is the antenna resistance, P_{LO}is the power actually applied prior to losses, and is deduced from αP_{LO}which is the power used for the hot spot calculation. When R → 0, αP_{LO}→ 0 at all P_{LO}values (Figure 12a). - In the hot spot method with RF current (inhomogeneous P
_{LO}dissipation), the terahertz power is determined from the intensity of the terahertz current, as outlined in Section 2.3 (Equations (6)–(10)). After some manipulations, still with the constriction low impedance value hypothesis, the following I_{LO}vs. P_{LO}relationship can be worked out [27]:$${I}_{\mathrm{LO}}\approx \sqrt{{\alpha}^{\prime}\frac{8{P}_{\mathrm{LO}}}{{R}_{\mathrm{a}}\sqrt{1+{r}^{2}}}}.$$_{DC}and P_{LO}levels, we observe there is a non-zero I_{LO}∝ P_{LO}^{1/2}along the constriction in the superconducting state. If P_{LO}is strong enough, the critical temperature T_{C}(I_{LO}) < T_{0}and R_{RF}is no longer negligible. Consequently, the P_{LO}dissipation causes the increase of the electron temperature and therefore a non-zero resistance even at low current (Figure 12b).

_{DC}and α

_{imp}P

_{LO}are quite distinct in the RF current hot spot model, which makes it possible to obtain results consistent with the typical HEB I-V measurements (Figure 12b).

#### 4.2. Taking the Resistivity Limit in the Superconducting Transition into Account

^{2F}(ω,t

_{r}→ 0)/ρ

_{N}> 1% criterion; starting from Equation (1a,b), we can work out:

_{THz}> ρ

_{N}× 9.5 × 10

^{4}, which is the case of our study, where the typical normal state resistivity is ρ

_{N}= 3.2 × 10

^{−6}Ω·m and our calculation frequency f

_{THz}≥ 0.3 THz.

_{2}

^{2F}(kinetic inductance effect). A calculation result of the constriction impedance R

_{RF}(T

_{e}) + jX

_{RF}(T

_{e}) is shown in Figure 14. We notice that the reactive part reaches its maximum near the middle of the resistive transition, where the detectors are considered the most sensitive, because dR

_{RF}/dT

_{e}is optimal.

_{2}

^{2F}(ω,T

_{e}). It is therefore necessary to know the YBCO temperature distribution along the constriction and solve the complete convolution calculation to estimate ρ

_{2}

^{2F}and to deduce the imaginary part of the constriction impedance. We therefore opted for this procedure as part of the hot spot model as presented above.

#### 4.3. Mixer Noise: Effect of Taking the Impedance Matching Factor into Account

_{LO}≈ 10 μW, as observed in Section 3.3.1, was not present in our previous model without impedance matching [25]; we therefore studied the influence of taking into account the impedance matching factor between the antenna and the nano-constriction on the noise temperature.

_{imp}(shown previously in Figure 7) and compare it to the distribution of noise temperatures with constant α

_{imp}(= 1), as shown below (Figure 15). We notice a significant change in the position of the better operation area. In the quasi-static mode, the best operation area at P

_{LO}≈ 10 μW is no longer present and, in general, the noise temperatures are higher when the impedance matching is not taken into account (see Table 4).

_{LO}along the nano-constriction. We clearly observe (Table 4): (i) A decrease in the prerequisite of P

_{LO}, and (ii) an improvement of the noise temperature when the operating frequency increases. This is due to the influence of the distribution of P

_{LO}along the nano-constriction which is in question: This distribution is more localized in the quasi-static regime than at 4 THz, to take the extreme cases.

#### 4.4. Comparing Simulated Mixer Noise with Experimental Results

_{C}superconducting ultrathin films: Nb HEBs mainly below 1 THz, and (100) NbN HEBs above 1.4 THz. For example, values of T

_{DSB}= 600 K at 2.5 THz, with a large noise bandwidth in excess of 7 GHz [51], and T

_{DSB}= 815 K at 4.7 THz [52] were reported for NbN HEBs. However, T

_{C}values for these HEB devices remain below 11 K. An interesting alternative can be offered with HEBs made from c-axis oriented MgB

_{2}ultrathin films [20,53], due to increased operation temperature (T

_{C}= 39 K) and expected instantaneous bandwidth up to 10 GHz.

_{DSB}= 3600 K and G = − 11 dB at P

_{LO}= 1 μW. It may be compared to our performance prediction for a device B (Table 1) under not optimized conditions at 750 GHz (T

_{DSB}= 3000 K and G = − 10.8 dB at P

_{LO}= 9 μW [27]). Besides, the very large T

_{DSB}values reported in Reference [13] were attributed to inadequate experimental conditions by the authors, who predicted an improvement of T

_{DSB}< 10

^{4}K up to 6 THz (system noise temperature), a figure compatible with our predicted T

_{DSB}= 5 × 10

^{3}K (mixer noise temperature, not optimized [27]) at 2.5 THz. Our present optimized predictions (solid curve in Figure 16) have therefore to be taken as a limit that could be reached with improved nano-structuring of c-axis oriented YBCO ultrathin films [54,55].

_{RF}for an optimal fit between their noise simulations and measurements on NbN HEBs. After having applied this same correction in our model, we also checked a very good (and even better) fit with the noise temperature measurements reported in Reference [24]. In fact, our results benefited from computational software improvements: Fewer errors accumulated when solving the differential equations, interpolation between points to obtain K

_{DC}, and smoothing algorithms. The physical explanation for this overestimation of the resistance variation induced by the RF power remains, however, uncertain for the authors in Reference [24], and ourselves. These considerations nevertheless encouraged us to adapt the NbN HEB hot spot model to YBCO HEBs. Clearly the adjustment of the K

_{RF}correction coefficient should require a comparative and systematic study with YBCO HEB mixer experimental tests, which is a potential point of progress in the present approach.

_{LO}≈ 150 μW at 750 GHz (see Reference [57] for instance). Up to 2.5 THz, another room temperature alternative relies on tunable backward wave oscillators (BWO). However, the operation of BWOs requires heavy weight magnets, making the BWOs bulky; they deliver, e.g., P

_{LO}≈ 25 μW at 2 THz (see Reference [58] for instance). At and above 2.5 THz, the LO power can be obtained from optoelectronic sources, such as compact mm-size THz quantum cascade lasers (QCL) operating at 70 K, with P

_{LO}in the 50 μW to 1 mW range [59,60]. The temperature operation of 70 K, matching the HEB operation, makes it possible to integrate both components within similar cryogenic systems.

#### 4.5. Standoff Detection Limits with YBCO HEB Mixers

_{T}is expected, the lower T

_{DSB}floor of the receiver system is required.

_{T}vs. ΔT relationship illustrating a constant T

_{DSB}value. T

_{DSB}values were chosen at 1000 K and at 2000 K, corresponding to an optimal operating point (in terms of local oscillator power in particular) for YBCO device B, as shown in Table 2. For instance, with a 2.5 THz receiver exhibiting T

_{DSB}= 2000 K, we could expect d

_{T}≈ 3.5 m at ΔT = 10 K.

_{LO}= 9 µW), the mixer noise temperature increases (see values of T

_{DBS-MIN}in Table 5, extracted from Reference [27]). In this context of non-optimal operating point, Table 5 presents various potential scenarios for accommodating the HEB mixer performance at three frequencies. Calculations were also performed by relaxing the obstacle attenuation constraint. As an example, for a 2.5 THz receiver exhibiting T

_{DSB}= 4150 K, we should expect d

_{T}≈ 1.6 m at ΔT = 10 K and RH = 70%. If the obstacle transmission parameter is relaxed (t

_{obs}= 1), the achievable target distance increases: For the same receiver, we could expect d

_{T}≈ 11.5 m at ΔT = 10 K and RH = 70%.

## 5. Conclusions and Future Plans

_{DC}vs. P

_{LO}coordinates made it possible to clearly illustrate the optimal operating conditions as a function of the THz frequency for a YBCO HEB. For instance, it was shown that T

_{DSB}≈ 1000 K could ultimately be achieved at P

_{LO}≈ 13 μW for a constriction of dimensions compatible with our technological process. The IF instantaneous bandwidth was shown to lie below 1 GHz for a 10 nm thick YBCO constriction, and to be roughly inversely proportional to the thickness. Some comparisons with the few available published results could be made. It has been noted that some flux flow effect could be involved in the detection mechanism and increase the bandwidth, which is a point to be further considered. These models will also provide a useful guide to refine our ongoing HEB fabrication process.

_{LO}level) was also discussed. Typically, detection at 3 m through cotton cloth could be readily achieved in moderate humidity conditions with 10 K target temperature resolution.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Superconducting hot electron bolometers (HEB): (

**a**) Illustration of the point bolometer thermal model with two-reservoirs (electrons and phonons of the superconductor); (

**b**) Sketch of a YBCO HEB constriction (of length L, width w, and thickness θ) connected to the arms of a THz planar antenna.

**Figure 2.**Simplified schematic for HEB standoff THz detection arrangement. The target resolved area is deduced from the Airy pattern at the HEB location (Equation (19)).

**Figure 3.**Frequency-dependent YBCO superconducting transition illustrated by the constriction resistance vs. frequency. The DC plot (f = 0) is a fit from experiment [30]. The gaussian curves describe the T

_{C}distribution as discussed in the text. Fermi-Dirac fits have been used for the device performance simulation as more appropriate to the convergence of numerical solutions. Redrawn after [27].

**Figure 4.**For devices A and B, electron temperature T

_{e}(x) (solid curves) and phonon temperature T

_{p}(x) (dashed curves) profiles: (

**a**) For device A (P

_{LO}= 5 μW); (

**b**) for device B (P

_{LO}= 35 μW). T

_{0}and T

_{C}are the reference (cold finger) and mid-transition critical temperatures, respectively. Arrows on T

_{C}lines delimit the hot spot regions Δx

^{HS}(see text). Curve labels indicate the DC bias current I

_{0}values.

**Figure 5.**For devices A and B, DC current vs. DC voltage maps according to P

_{LO}values, at 400 GHz LO frequency: (

**a**) For device A (T

_{0}= 60 Κ); (

**b**) for device B (T

_{0}= 70 Κ). Redrawn after [45].

**Figure 7.**For an HEB constriction of dimensions L = w = 400 nm and θ = 35 nm (Figure 1), maps exhibiting double sideband noise temperature T

_{DSB}levels in DC bias power vs. LO power coordinates. Impedance matching coefficient α

_{imp}with the antenna was included.

**Figure 8.**For an HEB constriction of dimensions L = w = 400 nm and θ = 35 nm (Figure 1), maps exhibiting Johnson noise and thermal fluctuation noise contributions to T

_{DSB}levels in DC bias power vs. LO power coordinates. Impedance matching coefficient α

_{imp}with the antenna was included.

**Figure 9.**For an HEB constriction of dimensions L = w = 400 nm and θ = 35 nm (Figure 1), maps exhibiting G levels in DC bias power vs. LO power coordinates. Impedance matching coefficient α

_{imp}with the antenna was included.

**Figure 11.**Standoff detection double sideband noise temperature requirements: (

**a**) As a function of operating frequency for various atmospheric humidity contents (at fixed ΔT and d

_{T}); (

**b**) as a function of target distance for various target temperature resolutions (at fixed ν

_{0}and RH).

**Figure 12.**Comparing low-voltage behavior of DC I-V plots taking impedance matching factor with antenna into account: (

**a**) Regular hot-spot model with uniform P

_{LO}power dissipation along the constriction; (

**b**) our hot-spot model approach with I

_{LO}as an input parameter, i.e., non-uniform P

_{LO}power dissipation.

**Figure 13.**For a YBCO HEB, I-V map, with P

_{LO}levels, from our simulation with 650 GHz RF current (non-uniform LO power dissipation). Measurement points, without and with LO power applied at the same frequency, are also indicated (courtesy J. Raasch [48]).

**Figure 14.**Example of the real and imaginary parts of a total THz impedance of a constriction; the temperature was assumed uniform in this case.

**Figure 15.**For an HEB constriction of dimensions L = w = 400 nm and θ = 35 nm (Figure 1), maps exhibiting double sideband noise temperature T

_{DSB}levels in DC bias power vs. LO power coordinates. Impedance matching coefficient with the antenna was not included (α

_{imp}= constant = 1).

**Figure 16.**For HEB heterodyne receivers or mixers, double sideband noise temperature as a function of operating frequency. DC and PC are for diffusion cooled and phonon cooled devices, respectively. Hot spot (HS) model results are those of Table 2 (optimized) and Reference [27] (fixed LO power). QL: Quantum limit hν/(2k

_{B}). Redrawn and updated after [56] and [45].

**Figure 17.**Standoff detection DSB requirements: (

**a**) Required noise temperature as a function of ν

_{0}for various distances at specified ΔT and fixed humidity; simulated T

_{DSB}values for device B are also shown at both optimal P

_{LO}conditions - solid curve (Table 2) and at fixed P

_{LO}= 9 µW - dashed curve [27]; (

**b**) Required distance vs. temperature difference relationship to achieve T

_{DSB}= 1000 K or 2000 K, at fixed RH. Symbols: Computed values, dotted curves: Best fits (according to functions indicated).

**Table 1.**For constrictions A and B, device characteristics and YBCO physical parameters used in the simulations. Unless otherwise stated, the parameter values are those of [22].

Device | L, w (nm) | θ (nm) | C_{e} (J·m^{−3}·K^{−1}) | C_{p} (J·m^{−3}·K^{−1}) | κ_{e} (W·m^{−1}·K^{−1}) | κ_{p}(W·m^{−1}·K^{−1}) | |

A | 100 | 10 | 2.5×10^{4} | 6.5×10^{5} | 1 | 10 | |

B | 400 | 35 | 2.5×10^{4} | 6.5×10^{5} | 1 | 10 | |

Device | τ_{ep} (ps) | τ_{esc} (ns) | σ_{N}^{1} (Ω^{–1}·m^{–1}) | J_{C}^{2} (A·cm^{–2}) | T_{C}^{3} (K) | ΔT (K) | T_{0} (K) |

A | 1.0 | 0.75 | 3.15×10^{5} | 2.2×10^{6} | 85 | 1.2 | 60 |

B | 1.7 | 2.6 | 3.15×10^{5} | 2.2×10^{6} | 89 | 1.2 | 70 |

^{1}Normal state conductivity at ~ 100 K.

^{2}Critical current density at 77 K.

^{3}Mid-transition critical temperature.

**Table 2.**For an HEB of dimensions L = w = 400 nm and θ = 35 nm at different operating frequencies, with matching coefficient α

_{imp}included, compared performances at an optimal operating point with respect to the noise temperature.

Frequency | I_{DC} (μA) | R (Ω) ^{1} | P_{DC} (µW) | P_{LO} (μW) | G (dB) | T_{DSB} (K) |
---|---|---|---|---|---|---|

QS | 454 | 21.3 | 4.4 | 7.5 | −9.0 | 1781 |

500 GHz | 464 | 21.4 | 4.6 | 7.5 | −7.1 | 1208 |

2.5 THz | 394 | 14.6 | 2.3 | 12.5 | −6.1 | 1013 |

4 THz | 379 | 13.0 | 1.9 | 13.5 | −6.5 | 1093 |

^{1}RF resistance.

Frequency (THz) | 0.67 | 1.02 | 1.49 | 2.52 | 3.44 |

t_{obs} (%) | 75 | 47 | 21 | 2 | 0.1 |

**Table 4.**For an HEB of dimensions L = w = 400 nm and θ = 35 nm at different operating frequencies, with unit impedance matching coefficient (α

_{imp}= 1), compared performances at an optimal operating point with respect to the noise temperature.

Frequency | I_{DC} (μA) | R (Ω) ^{1} | P_{DC} (µW) | P_{LO} (μW) | G (dB) | T_{DSB} (K) |
---|---|---|---|---|---|---|

QS | 241 | 27.9 | 1.6 | 35 | −13.8 | 2607 |

500 GHz | 271 | 26.4 | 1.9 | 30 | −13.6 | 2704 |

2.5 THz | 392 | 19.2 | 3.0 | 12 | −10.8 | 2332 |

4 THz | 389 | 18.2 | 2.7 | 12.5 | −10.1 | 2021 |

^{1}RF resistance.

**Table 5.**Standoff detection scenarios (with and without obstacle) designed to accommodate YBCO HEB mixer performance (device B) in terms of minimum T

_{DSB}at P

_{LO}= 9 µW [27].

Frequency (THz) | T_{DSB-MIN} (K) [27] | Obstacle | ΔT (K) | RH (%) | D_{T} (m) | NEP (fW/Hz^{1/2}) | T_{DSB} (K) |
---|---|---|---|---|---|---|---|

0.67 | 2480 ^{1} | Yes | 0.5 | 70 | 78 | 3.1 | 2495 |

0.67 | 2480 ^{1} | No | 0.5 | 70 | 90.8 | 3.1 | 2485 |

1.02 | 2730 | Yes | 1.0 | 70 | 19.0 | 3.4 | 2760 |

1.02 | 2730 | No | 1.0 | 70 | 26.5 | 3.4 | 2775 |

2.52 | 4150 | Yes | 10 | 30 | 3.8 | 5.1 | 4160 |

2.52 | 4150 | Yes | 10 | 70 | 1.6 | 5.1 | 4150 |

2.52 | 4150 | No | 10 | 70 | 11.5 | 5.2 | 4245 |

^{1}Simulated at 750 GHz.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ladret, R.; Dégardin, A.; Jagtap, V.; Kreisler, A.
THz Mixing with High-*T*_{C} Hot Electron Bolometers: A Performance Modeling Assessment for Y-Ba-Cu-O Devices. *Photonics* **2019**, *6*, 7.
https://doi.org/10.3390/photonics6010007

**AMA Style**

Ladret R, Dégardin A, Jagtap V, Kreisler A.
THz Mixing with High-*T*_{C} Hot Electron Bolometers: A Performance Modeling Assessment for Y-Ba-Cu-O Devices. *Photonics*. 2019; 6(1):7.
https://doi.org/10.3390/photonics6010007

**Chicago/Turabian Style**

Ladret, Romain, Annick Dégardin, Vishal Jagtap, and Alain Kreisler.
2019. "THz Mixing with High-*T*_{C} Hot Electron Bolometers: A Performance Modeling Assessment for Y-Ba-Cu-O Devices" *Photonics* 6, no. 1: 7.
https://doi.org/10.3390/photonics6010007