THz Mixing with High-T_C Hot Electron Bolometers: A Performance Modeling Assessment for Y-Ba-Cu-O Devices

Abstract

Hot electron bolometers (HEB) made from high-T_C superconducting YBa₂Cu₃O_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

First considered by Gershenzon et al. [1], the superconducting hot electron bolometer (HEB) principle was applied by these authors to low-T_C (T_C is the superconducting critical temperature) and high-T_C HEBs, Nb and YBa₂Cu₃O_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, MgB₂ 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].

The just mentioned experimental work was permanently sustained by modeling and simulation efforts to predict and interpret HEB performance. We now introduce those performance simulation aspects, because they indeed are our main concern in the present article.

Several steps in the HEB modeling approach have to be considered. In the “0D” or point bolometer model, only energy transfer between thermal reservoirs is evaluated. In the superconducting material (Figure 1a), the electron reservoir at temperature T_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.

Further modeling developments took the HEB device geometry into account to describe the electron temperature profile T_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].

The aim of this article is twofold. First, we shall provide a complete and detailed description of the YBCO HEB model and extend the results of [25,26,27] up to 4 THz. We shall also compare our modeling results in terms of the device performance as a THz heterodyne mixer with the few experimental data available for YBCO HEB mixers, which can be found in the literature.

Second, we shall consider the potential of such a device to provide a lightweight system for THz standoff detection for such applications as security and control. We shall therefore evaluate the standoff detection limits using passive imaging with YBCO HEB mixers. In this framework, we shall simulate well-defined and credible scenarios in terms of target distance, temperature resolution, atmospheric relative humidity, and transmission through clothing.

2. Models and Methods

2.1. Describing the Superconducting Transition at THz Frequencies

To have access to the resistance (and, more generally, the impedance) vs. electron temperature T_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_r⁴. 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 = σ₁ − jσ₂, assuming Ohm’s law is still valid in an impurity scattering description [29]:

$$t_{\mathrm{r}}<1:{\sigma }_1={\sigma }_{\mathrm{N}}{t}_{\mathrm{r}}^4+\frac{{\sigma }_2}{r},{\sigma }_2=\frac{1}{\omega {\mathsf{\mu }}_0{\lambda }_0^2}\frac{r^2}{1+r^2}\left(1-t_{\mathrm{r}}^4\right),$$

(1a)

$$t_{\mathrm{r}}\ge 1:{\sigma }_1={\sigma }_{\mathrm{N}},{\sigma }_2=0,$$

(1b)

where σ_N is the normal state conductivity, r = ωτ_S is the impurity scattering parameter (for scattering time τ_S), μ₀ is the permeability of a vacuum and λ₀ is the London’s penetration depth at t_r = 0.

As the two-fluid model depicts a homogeneous superconducting material, we have to describe the experimentally observed broadening of the superconducting transition; such a broadening is related to the film microstructure (e.g., grain boundary weak links). This is performed by introducing a Gaussian distribution of critical temperature values, centered at the mid-transition critical temperature T_C0, as discussed in Reference [27], with

$$g=\frac{1}{G_{\mathrm{N}}}\exp \left(-\frac{{\left(T_{\mathrm{e}}-T_{\mathrm{C}0}\right)}^2}{2{\left(\mathsf{\Delta }{T}_{\mathrm{G}}\right)}^2}\right),$$

(2)

where G_N is a normalization coefficient and ΔT_G is the critical temperature standard deviation, to be adjusted to fit the experimental transition.

After having expressed the two-fluid resistivity ${\rho }^{2\mathrm{F}}(\omega )={\rho }_1^{2\mathrm{F}}(\omega )+\mathrm{j}{\rho }_2^{2\mathrm{F}}(\omega )={\sigma }^{-1}$, we then fit the experimental DC resistivity transition [30] as a convolution product between ρ₁^2F (ω → 0) and the Gaussian function, so that

$${\rho }_1^{\exp }\left(T_{\mathrm{e}},T_{\mathrm{C}0},\mathsf{\Delta }{T}_{\mathrm{G}}\right)=\left({\rho }_1^{2\mathrm{F}}\left(\omega \to 0\right)\otimes g\right)\left(T_{\mathrm{e}},T_{\mathrm{C}0},\mathsf{\Delta }{T}_{\mathrm{G}}\right).$$

(3)

The so-obtained transition is then approximated by a Fermi-Dirac type function, as more appropriate to further numerical treatments (see below Section 2.2, hypothesis 8).

To obtain the THz dependence, we introduce ρ^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

For an HEB constriction of volume V_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:

$$\frac{\delta }{\delta x}\left[V_{\mathrm{C}}{\kappa }_{\mathrm{e}}\frac{\delta T_{\mathrm{e}}}{\delta x}\right]-V_{\mathrm{C}}{C}_{\mathrm{e}}\frac{\delta T_{\mathrm{e}}}{\delta t}=\frac{V_{\mathrm{C}}{C}_{\mathrm{e}}\left(T_{\mathrm{e}}^n-T_{\mathrm{p}}^n\right)}{n{\tau }_{\mathrm{ep}}{T}_{\mathrm{ec}}^{n-1}}-L\frac{\delta P_{\mathrm{J}-\mathrm{tot}}}{\delta x},$$

(4a)

$$\frac{\delta }{\delta x}\left[V_{\mathrm{C}}{\kappa }_{\mathrm{p}}\frac{\delta T_{\mathrm{p}}}{\delta x}\right]-V_{\mathrm{C}}{C}_{\mathrm{p}}\frac{\delta T_{\mathrm{p}}}{\delta t}=\frac{V_{\mathrm{C}}{C}_{\mathrm{p}}\left(T_{\mathrm{p}}^m-T_0^m\right)}{m{\tau }_{\mathrm{esc}}{T}_{\mathrm{pc}}^{m-1}}-\frac{V_{\mathrm{C}}{C}_{\mathrm{e}}\left(T_{\mathrm{e}}^n-T_{\mathrm{p}}^n\right)}{n{\tau }_{\mathrm{ep}}{T}_{\mathrm{ec}}^{n-1}},$$

(4b)

where indices e and p pertain to the electrons and phonons of the superconducting film, respectively (Figure 1a). κ_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 T₀), 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:

$$\delta P_{\mathrm{J}-\mathrm{tot}}\approx \frac{\rho (0,T_{\mathrm{e}}(x))}{S_{\mathrm{C}}}{I}_0^2\delta x+\frac{\rho ({\omega }_{\mathrm{LO}},T_{\mathrm{e}}(x))}{S_{\mathrm{C}}}\times \left[\frac{1}{2}{I}_{\mathrm{LO}}^2+I_{\mathrm{LO}}{I}_{\mathrm{S}}\cos \left(({\omega }_{\mathrm{LO}}-{\omega }_{\mathrm{S}})t\right)\right]\delta x.$$

(5)

In this equation, I₀ 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).

The hypotheses governing the resolution of Equation (4a,b) are given in the following list.

• 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. 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)	Ce (J·m−3·K−1)	Cp (J·m−3·K−1)	κe (W·m−1·K−1)	κp(W·m−1·K−1)
  A	100	10	2.5×104	6.5×105	1	10
  B	400	35	2.5×104	6.5×105	1	10
  Device	τep (ps)	τesc (ns)	σN1 (Ω–1·m–1)	JC2 (A·cm–2)	TC3 (K)	ΔT (K)	T0 (K)
  A	1.0	0.75	3.15×105	2.2×106	85	1.2	60
  B	1.7	2.6	3.15×105	2.2×106	89	1.2	70

  ¹ Normal state conductivity at ~ 100 K. ² Critical current density at 77 K. ³ Mid-transition critical temperature.

  ), we evaluate the electron diffusion length l_e ≈ π(τ_epκ_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 T₀, 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) = T₀.
• 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 T³ (Debye’s model).
• The YBCO to MgO substrate phonon escape time was deduced as τ_esc (ps) = 75 × θ (nm) [17]. In the case of PBCO/YBCO/PBCO trilayer constriction [30], where PBCO lies for PrBa₂Cu₃O_7–y, a correction should be applied to θ, as considered in Reference [13].
• For numerical computation convenience, the resistive superconductive transition has been approximated by a Fermi-Dirac function of the form ρ = ρ₀ [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^refS_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

In the two aforementioned Equation (4a,b), we set certain parameter values to limit to two the number of variables to find by the iterative solving method. The known parameters are V_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, I₀ and T₀, 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 d²T_e/dx² and d²T_p/dx² as well.

All the above considerations give enough information that the second derivatives can be deduced at each point by our program when x varies from L/2 to L. Similarly, T_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) = T₀, which represents our criterion of success for the calculation.

Furthermore, we have made a significant improvement of this basic resolution process, by exchanging the roles of T_e^max and I₀, such that T_e^max is now a fixed parameter and I₀ 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 I₀ value (because I₀ 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).

Unlike the formal procedure just described, Equation (5) indicates that I_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 I₀ and having a certain resistance R (as well as all the other characteristics that we wish to deduce).

The goal of our approach is to obtain a “cloud of points” as homogeneous as possible, in order to extrapolate the operating configurations we are interested in. Typically, we recreated, by interpolation, a series of points that are the I-V responses to the different P_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.

The global solving loop process is summarized as follows. Starting with Equation (5), the local heat power dissipation leads to the global heterodyne power mixing expression:

$$P_{\mathrm{J}-\mathrm{tot}}\approx P_{\mathrm{DC}}+\alpha P_{\mathrm{LO}}+2\alpha \left[\sqrt{P_{\mathrm{LO}}{P}_{\mathrm{S}}}\cos \left(\left({\omega }_{\mathrm{LO}}-{\omega }_{\mathrm{S}}\right)t\right)\right],$$

(6)

where appears the intermediate frequency (IF) ${\omega }_{\mathrm{IF}}=\left|{\omega }_{\mathrm{LO}}-{\omega }_{\mathrm{S}}\right|$. As I_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:

$$I_{\mathrm{LO}}=\sqrt{\frac{2\alpha P_{\mathrm{LO}}}{R_{\mathrm{RF}}}},$$

(7)

the constriction resistance/reactance contributions being

$$R_{\mathrm{DC}}={\displaystyle {\int }_0^L\frac{\rho (0,T_{\mathrm{e}}(x))}{S_{\mathrm{C}}}\mathrm{d}x,}{R}_{\mathrm{RF}}={\displaystyle {\int }_0^L\frac{{\rho }_1({\omega }_{\mathrm{LO}},T_{\mathrm{e}}(x))}{S_{\mathrm{C}}}\mathrm{d}x,}\mathrm{and}{X}_{\mathrm{RF}}={\displaystyle {\int }_0^L\frac{{\rho }_2({\omega }_{\mathrm{LO}},T_{\mathrm{e}}(x))}{S_{\mathrm{C}}}\mathrm{d}x}.$$

(8)

It should be noticed that in Equation (7), α = α_imp × α’ (hypothesis 10 above) is a function of Z_RF, through the impedance matching coefficient

$${\alpha }_{\mathrm{imp}}=\left(1-{\left|\frac{Z_{\mathrm{RF}}-Z_{\mathrm{a}}^{*}}{Z_{\mathrm{RF}}+Z_{\mathrm{a}}}\right|}^2\right)\times \frac{\mathrm{Re}(Z_{\mathrm{RF}})}{\left|Z_{\mathrm{RF}}\right|},$$

(9)

where Z_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.

For a self-complementary antenna, Z_a = R_a + jX_a is purely real, with

$$Z_{\mathrm{a}}=R_{\mathrm{a}}=\frac{60\mathsf{\pi }}{\sqrt{\left(1+{\epsilon }_{\mathrm{r}}\right)/2}},$$

(10)

where ε_r is the substrate material dielectric constant [35,36].

2.4. HEB Mixer Performance

2.4.1. General Considerations and Conversion Gain

For the calculation of the mixer conversion gain and noise temperature, it is necessary to discriminate between: (i) The contribution originating from the DC bias power P_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:

$$K_{\mathrm{DC}}=\frac{\delta R}{\delta P_{\mathrm{DC}}}(\delta P_{\mathrm{LO}}=0),$$

(11a)

$$K_{\mathrm{RF}}=\frac{\delta R}{\delta P_{\mathrm{LO}}}(\delta P_{\mathrm{DC}}=0),$$

(11b)

where R is the constriction resistance (evaluated according to the adopted model), and assuming P_S << P_LO.

K_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

$$\Delta R=\Delta P_{\mathrm{LO}}{K}_{\mathrm{RF}}+\Delta P_{\mathrm{DC}}{K}_{\mathrm{DC}},\mathrm{whence}{K}_{\mathrm{RF}}=\frac{\Delta R}{\Delta P_{\mathrm{LO}}}-K_{\mathrm{DC}}\frac{\Delta P_{\mathrm{DC}}}{\Delta P_{\mathrm{LO}}}.$$

(12)

The expression of the conversion gain G = (IF power delivered to the load R_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]):

$$G=\frac{R_{\mathrm{L}}{\tilde{I}}^2/2}{P_{\mathrm{S}}}=\frac{2I_0^2{R}_{\mathrm{L}}{K}_{\mathrm{RF}}^2{P}_{\mathrm{LO}}}{{\left(R_0+R_{\mathrm{L}}\right)}^2{\left[1-\frac{K_{\mathrm{DC}}{I}_0^2\left(R_{\mathrm{L}}-R_0\right)}{\left(R_{\mathrm{L}}+R_0\right)}\right]}^2}$$

(13)

where R₀ = R − ΔR and I₀ is the constriction current when P_S = 0.

It is to be noticed that the gain is also involved in the noise expression. Let us consider the noise measured at the mixer input (index “in”) and the noise measured at the mixer output (index “out”); the corresponding relationship for the noise temperatures is Tⁱⁿ = T^out/G.

2.4.2. Noise Temperature Contributions

The two main noise contributions of internal origin in a superconducting HEB are the Johnson (Jn) noise and the thermal fluctuation (TF) noise [3,37].

Johnson noise is considered as an additional voltage source v_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:

$$T_{\mathrm{Jn},\mathrm{DSB}}^{\mathrm{in}}=\frac{R_0{T}_0}{I_0^2{K}_{\mathrm{RF}}{P}_{\mathrm{LO}}}.$$

(14)

The noise spectral density originating from thermal fluctuations of the electrons involves the electron-electron interaction time τ_ee and is expressed as follows [24]:

$$\langle T_{\mathrm{e}}\rangle =\sqrt{\frac{4{\mathrm{k}}_{\mathrm{B}}{T}_{\mathrm{e}}^2{\tau }_{\mathrm{ee}}}{C_{\mathrm{e}}{V}_{\mathrm{C}}}}.$$

(15)

A current ${\tilde{I}}_{\mathrm{TF}}$ results from these fluctuations hence the noise power $P_{\mathrm{TF}}^{\mathrm{out}}={\mathrm{k}}_{\mathrm{B}}{T}_{\mathrm{TF}}^{\mathrm{out}}\Delta f$ dissipated in the load resistance R_L. In the same manner as previously, one obtains the DSB expression of the mixer input thermal fluctuation noise:

$$T_{\mathrm{TF},\mathrm{DSB}}^{\mathrm{in}}=\frac{1}{K_{\mathrm{RF}}^2{P}_{\mathrm{LO}}}{\left(\frac{\mathrm{d}R}{\mathrm{d}{T}_{\mathrm{e}}}\right)}^2\frac{T_{\mathrm{e}}^2}{C_{\mathrm{e}}{V}_{\mathrm{C}}}{\tau }_{\mathrm{ee}}.$$

(16)

The global expression of the mixer noise temperature is therefore

$$T_{\mathrm{N},\mathrm{DSB}}^{\mathrm{in}}=T_{\mathrm{Jn},\mathrm{DSB}}^{\mathrm{in}}+T_{\mathrm{TF},\mathrm{DSB}}^{\mathrm{in}}+\frac{T_{\mathrm{IF}}}{2G},$$

(17)

where we have introduced the IF amplifier output noise temperature T_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 T₀ of the HEB [38].

In practice, the mixer performance can be accessed by widespread techniques, such as the Y-factor method [39], which consists in using a blackbody (BB) source alternately adjusted at temperatures T_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_Nⁱⁿ = (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_BT_BBΔf straight line plot (k_B is the Boltzmann constant).

2.5. Standoff Detection Implementation with an HEB Heterodyne Detector

Passive imaging techniques rely both on collecting naturally emitting radiation from a scene and detecting the contrast between warmer and colder targets. The human being exhibits an emissivity of about 95% at frequencies above 600 GHz, which makes such a target appear warm relatively to a metal object of low emissivity.

We have checked the feasibility of standoff target detection operating in the passive mode with a YBCO HEB THz mixer. We have considered the simplified schematic in Figure 2, which illustrates losses of various origins. The noise equivalent power (NEP) requirement for the detector was deduced from the blackbody power P_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 ν₀ and for a temperature difference ΔT to be resolved at T_op, the power difference is

$$\Delta P_{\mathrm{BB}}(\Delta \nu ,\Delta T)={\displaystyle {\int }_{{\nu }_0-\Delta \nu }^{{\nu }_0+\Delta \nu }\left[B_{\nu }(T_{\mathrm{op}}+\Delta T)-B_{\nu }(T_{\mathrm{op}})\right]\mathrm{d}\nu }{\epsilon }_{\mathrm{T}}\frac{A_{\mathrm{D}}{A}_{\mathrm{T}}}{d_{\mathrm{T}}^2},$$

(18)

where B_ν(T) is the blackbody spectral radiance (in $\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{sr}}^{-1}·{\mathrm{Hz}}^{-1}$), ε_T the target emissivity, and d_T the target to detector distance (possibly corrected to include imaging system primary and secondary mirrors).

The NEP was then obtained by assuming the resolution to be diffraction limited for the detector (i.e., at the focusing lens focal plane). Consequently, the resolved distance at the target can be related to the Airy disk diameter at the detector focal plane by a factor d_T/f_L, approximately, where f_L is the focal length of the focusing lens. We finally work out at wavelength λ:

$$NEP=\frac{{\displaystyle {\int }_{{\nu }_0-\Delta \nu }^{{\nu }_0+\Delta \nu }\left[B_{\nu }(T_{\mathrm{op}}+\Delta T)-B_{\nu }(T_{\mathrm{op}})\right]\mathrm{d}\nu }}{\sqrt{\delta {\nu }_{\mathrm{int}}}}{\epsilon }_{\mathrm{T}}{\left(\frac{\mathsf{\pi }1.22\lambda }{2}\right)}^2{t}_{\mathrm{opt}}{t}_{\mathrm{atm}}{t}_{\mathrm{obs}},$$

(19)

where δν_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_atm was extracted from the atmospheric attenuation factor α_atm (in dB/m) in various conditions, using the Atmospheric Effects Model (AEM) described in Reference [40]. Such extracted values were also in line with values deduced from data in Reference [41].
• t_obs represents the transmission through some obstacle existing in front of the target (cloth, cardboard, etc.).

The double sideband noise temperature required for the HEB was deduced from the NEP in the Rayleigh-Jeans limit as [39]:

$$T_{\mathrm{DSB}}=\frac{NEP}{2{\mathrm{k}}_{\mathrm{B}}\sqrt{2\mathsf{\Delta }\nu }}.$$

(20)

3. Simulation Results

3.1. YBCO Superconducting Transition in the THz Range

As shown in Section 2.3, we have been able to link the experimental DC transition with a two-fluid model representation of the resistive transition when ω → 0. From this result, we perform at terahertz frequencies a similar convolution as in Equation (3), which will therefore modify the modeled resistive transition accordingly. We approach this modification as the combination of three phenomena: The decrease of the average mid-transition critical temperature (T_C), the widening of the resistive transition (ΔT) and the appearance of a minimum resistivity (ρ_min). To the first order, we write:

T_C(f_THz) = T_C(0) × (1 − A_T f_THz),

(21a)

ΔT(f_THz) = ΔT(0) × (1 + B_T f_THz),

(21b)

ρ_min(f_THz,t_r = 0.8) ≅ 2ρ_min(f_THz,t_r = 0) = C_T f_THz.

(21c)

In practice, the A_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.

From an experimental point of view, our current technological process allows to achieve YBCO constrictions of length and width L = w = 400 nm and thickness θ = 35 nm, with a mid-transition critical temperature T_C = 89 K [25,30], a transition width of 1.2 K and a conductivity σ_N = 3.15 × 10⁵ 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 Y₁Ba₂Cu₃O₇ 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].

For a YBCO constriction exhibiting such dimensions, our simulation results lead to: A_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

For our simulations, we have considered two HEB devices (data gathered in Table 1). The constriction dimensions of device A are close to the limits achievable with e-beam lithography; these dimensions are those considered in the "0D" HEB model [22]. The constriction dimensions of device B are those achievable with our currently available technological process, combining optical and e-beam lithography [25]. Device B is also the constriction example taken in Section 3.1.

3.2.1. Temperature Profiles

The temperature profiles (solutions of Equation (4a,b)), shown in Figure 4 for devices A and B exhibit features, typically representing the HEB phenomena, namely: (i) The hot electron condition T_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).

As a check of the validity of our simulation approach, we run our procedure with low-T_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

The T_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.

With regard to device A, we notice that the I-V dependence can be hysteretic: For a given bias current value, there are three distinct HEB voltage operating points. Indeed, in the 5 mV to ~ 15 mV range, the differential resistance is negative, this behavior being typical of HEBs operating at low T₀ 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.

For further verification, we also compared, for an NbN device, our I-V results with those reported in Reference [24]. As shown in Figure 6, the calculated values are close to the HEB measurements at bias currents above 20 μA. The result of our calculation for P_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

The noise temperature and the conversion gain were determined at given local oscillator power P_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

We observe in Figure 7 that the minimum noise P_DC vs. P_LO area evolves in level and that the optimal P_LO value also evolves (see

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	IDC (μA)	R (Ω) 1	PDC (µW)	PLO (μW)	G (dB)	TDSB (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.

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.

This consideration can be furthered by separating the sources of noise for the two cases of QS regime and LO frequency at 2.5 THz (Figure 8). The contribution of each noise source informs us about the evolution of the area where the noise is minimum. We notice that in the QS regime, the fluctuation noise is minimal when P_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

We then resume the comparison for the conversion gain at the four working frequencies, as shown below in Figure 9. The gain decreases as P_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).

As compared to previous results [25], the present model highlights a sharp decrease in the local oscillator power needed to operate the HEB at the optimum point. Intuitively, one might be inclined to think that after taking into account α_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.

Indeed, the dissipation of the radiofrequency current is concentrated in the resistive zone of the nano-constriction, where dominates the dissipation by Joule effect (hot spot). Thus, if for example a quarter of the nano-constriction is resistive, then the power will be mainly dissipated in this zone, thus reducing a prerequisite of 35 μW (calculated in the model of the "regular" hot spot adapted to YBCO [25]) down to 7.5 μW.

3.3.3. IF Bandwidth

Small signal analysis (as formulated in Reference [25]) was performed to determine the IF response in terms of conversion gain as a function of the intermediate frequency f_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

We have run the simulation for the standoff system, as presented in Section 2.5 (Equations (18)–(20)), with detector double sideband width 2Δν = 2 GHz and integration bandwidth δν_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).
• t_obs was calculated for light cotton clothing from data in Reference [46] for frequencies below 1 THz, and in Reference [47] for frequencies above 1 THz. Values of t_obs for the five selected frequencies are collected in

  Table 3. Transmission coefficient through an obstacle of light cotton cloth [46,47].

  Frequency (THz)	0.67	1.02	1.49	2.52	3.44
  tobs (%)	75	47	21	2	0.1

  .

Various values of temperature resolution ΔT were taken into account, from 0.5 K to 10 K, with respect to the results obtained as a function of the pertinent parameters (frequency, target distance).

Figure 11a exhibits the required HEB mixer noise temperature for a passive mode detection system as a function of frequency and atmospheric relative humidity level for a 10 m target distance and a resolution ΔT = 1 K. The influence of relative humidity strongly occurs at frequencies higher than 2.5 THz (for instance, a low value of T_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.

Figure 11b exhibits the required noise temperature as a function of target distance and resolution temperature at 2.5 THz and highly stringent RH level of 70%. Thus, for instance, at 5 m target distance, the detection system would require T_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

In this discussion, we focus on a limited number of points specific to our present modeling approach, where it differs from either our previous modeling hypotheses or previously published models. We wish also to compare between our simulations and some experimental results, unfortunately very few in the case of YBCO HEBs.

4.1. Inhomogeneous P_LO Hypothesis Effect on I-V Plots at Low DC Voltage

Our model introduces the matching factor α_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.

In Figure 12, however, we observe two very different distributions between the I-V plots, according to P_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:

  $$\alpha P_{\mathrm{LO}}={\alpha }^{\prime }\frac{4R_{\mathrm{a}}R}{{(R_{\mathrm{a}}+R)}^2}{P}_{\mathrm{LO}},$$

  (22)

  where R is the constriction resistance, R_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]:

  $$I_{\mathrm{LO}}\approx \sqrt{{\alpha }^{\prime }\frac{8P_{\mathrm{LO}}}{R_{\mathrm{a}}\sqrt{1+r^2}}}.$$

  (23)

  Even at low P_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) < T₀ 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).

Figure 12a illustrates how impedance matching is not correctly formulated with conventional HEB models, in which powers of both DC and RF origins are confounded. The resulting DC plots (the I-V curves seem to be merging) are observed neither in the NbN nor YBCO-based HEB measurements. P_DC and α_impP_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).

Another verification has been possible through DC I-V plots simulated at 650 GHz and compared to measurements performed on a YBCO device (L = w = 300 nm, θ = 30 nm) made available to us [48,49], and further considered in Reference [50]. Although not entirely quantitative, the constriction pumping effect by the LO power is clearly visible in Figure 13. We also observe a better similarity between those results and the I-V curves in Figure 12b, rather than those in Figure 12a, so in favor of our complete model with RF current.

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

The limits of the THz superconducting transition model appear when the normal state resistivity is very high (because YBCO tends to deteriorate with aging [16]); in this case, the effects of the working frequency are of little significance. It is possible to calculate an order of magnitude of this resistivity limit by considering the ρ^2F(ω,t_r → 0)/ρ_N > 1% criterion; starting from Equation (1a,b), we can work out:

$$\frac{{\rho }^{2\mathrm{F}}(\omega ,t_{\mathrm{r}}\to 0)}{{\rho }_{\mathrm{N}}}\approx \frac{\omega {\mathsf{\mu }}_0{\lambda }_0}{3}.$$

(24a)

According to the limit criterion and Equation (21c) we obtain:

$$\frac{{\rho }^{2\mathrm{F}}(f_{\mathrm{THz}},t_{\mathrm{r}}\to 0)}{{\rho }_{\mathrm{N}}}\approx \frac{f_{\mathrm{THz}}\times 1.05\times {10}^{-7}}{{\rho }_{\mathrm{N}}}>1\%.$$

(24b)

The resistive transition model should therefore be used at frequencies such that f_THz > ρ_N × 9.5 × 10⁴, 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.

Besides, it is possible to carry out the calculation further, because our two-fluid dispersive model also makes it possible to express the imaginary part of the resistivity ρ₂^2F (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.

However, there is no obvious function to represent the variations of ρ₂^2F (ω,T_e). It is therefore necessary to know the YBCO temperature distribution along the constriction and solve the complete convolution calculation to estimate ρ₂^2F 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

The best operation area at P_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.

As already mentioned, our approach shows that there may be cases where the loss of coupling efficiency, due to impedance mismatch, may increase the sensitivity of the device. This influence is more evident by considering the distribution of calculated noise temperatures with non-constant α_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. 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	IDC (μA)	R (Ω) 1	PDC (µW)	PLO (μW)	G (dB)	TDSB (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.

).

In this discussion, where the influence of impedance matching is removed, the noise distribution as a function of frequency is only subject to change according to the distribution of P_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.

As part of a model to optimize the manufacturing of future HEBs, this result demonstrates the importance of taking impedance matching into account. The parameters to consider for manufacturing an HEB are the dimensions of the nano-constriction and the quality of the film, but also the resistive characteristic of the antenna, whose appropriate adjustment of the impedance should allow a gain in performance.

4.4. Comparing Simulated Mixer Noise with Experimental Results

A significant number of HEB heterodyne mixer performance experimental results are gathered in Figure 16.

The state-of-the-art HEB mixers are made from epitaxially grown low-T_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 MgB₂ ultrathin films [20,53], due to increased operation temperature (T_C = 39 K) and expected instantaneous bandwidth up to 10 GHz.

We notice the small number of published data concerning YBCO devices. As already mentioned in our introduction, this situation can be attributed to YBCO nanostructuration challenges, therefore a performance degradation associated with aging effects, even on highly c-axis oriented YBCO ultrathin films. Among these, the result at 585 GHz reported in Reference [15] concerns a rather large constriction (L = 1 µm, w = 2 µm, θ = 100 nm) exhibiting T_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 < 10⁴ K up to 6 THz (system noise temperature), a figure compatible with our predicted T_DSB = 5 × 10³ 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].

As a corollary remark, we notice that the authors in Reference [24] applied a factor 0.3 to K_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.

We finally comment on the availability of LO sources. Below 1 THz, the solid state based electronic sources (frequency multiplier chains) offer compactness, tunability and room temperature operation with, e.g., delivered P_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

We are discussing here the results on standoff detection requirements (Section 3.4) in the light of achievable noise performance (Section 3.3 and Section 4.3). Figure 17a summarizes results presented in Section 3.4 by showing the dependence on the target distance of the standoff detection system in terms of required mixer noise temperature. Constraints of a high relative humidity level and transmission through a cotton tissue have been considered to match a realistic scenario, thus leading to reduce temperature resolution (i.e., increasing ΔT values) at higher frequencies. The higher d_T is expected, the lower T_DSB floor of the receiver system is required.

This latter point is emphasized in Figure 17b, which exhibits the d_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.

However, if the HEB mixer operates at a non-optimal point, e.g., due to a low available LO power (e.g., P_LO = 9 µW), the mixer noise temperature increases (see values of T_DBS-MIN in

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)	TDSB-MIN (K) [27]	Obstacle	ΔT (K)	RH (%)	DT (m)	NEP (fW/Hz1/2)	TDSB (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.

, 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%.

As a final comment, we compare the chosen instantaneous bandwidth Δν = 1 GHz to the expected HEB IF cutoff related to the phonon escape time (Figure 10, Section 3.3.3). This choice is close to the simulated bandwidth for device A. For device B, however, the expected bandwidth approaches ~200 MHz, which falls short of the standoff mixer bandwidth parameter. However, it should be emphasized that measurements of YBCO-based devices have exhibited several GHz bandwidth values (e.g., 7 GHz [15]). This discrepancy could be ascribed to a vortex flow mechanism, also invoked to interpret YBCO detectors response, which would superimpose to the regular HEB action, leading to an improved bandwidth value [12].

5. Conclusions and Future Plans

Our first contribution in this paper was to provide a physical description of the superconducting transition enabling to introduce the YBCO THz conductivity in the HEB equations solving loop. This description allowed us to have access to the full impedance of the nano-constriction as a function of temperature, highlighting in particular: (i) The shift and broadening of the superconducting transition as the THz frequency increases, (ii) a residual resistance in the low temperature limit. The knowledge of this impedance also allowed to introduce the THz power matching factor with the antenna, thus leading to a relevant discussion on the influence of the LO power level to optimize the HEB performance.

Our second contribution was to approach in a realistic way the LO power dissipation in the constriction, by making the assumption of a uniform current along this constriction, as delivered by the antenna. The resulting non-uniform LO power dissipation along the constriction, in the frame of a hot spot model, has a clear influence on the shape of the DC I-V plots that gave us the opportunity to check our simulations with published measurement results.

Our third contribution was to simulate the gain, noise temperature and IF bandwidth of an HEB THz mixer. The presentation of the results in the form of level maps in P_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.

Our fourth contribution consisted in evaluating the performances required to implement a YBCO HEB in a standoff detection system in passive mode. The knowledge of the optimized performances of the HEB made it possible to discuss the conditions of detection (at the diffraction limit) according to the parameters of the standoff system (distance to the target, absorption and atmospheric humidity, temperature resolution, obstacles). The impact of non-optimized conditions (with respect to P_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.

This set of modeling studies will support ongoing new imager design and evaluation. Future work will include the development of a 2.5 THz coherent detection system prototype, based on a YBCO HEB mixer as a heterodyne detector and a QCL source as a local oscillator, each device being implemented in its own cryostat or, even better, in its own miniaturized cryo-generator [61]. The main challenges concern: (i) The production of high quality c-axis oriented YBCO ultrathin films and antenna-coupled nano-constrictions made from these films, which are particularly sensitive to technological steps; (ii) the implementation of a THz LO source of sufficient power, while limiting its thermal dissipation; (iii) the development of a low noise/broadband cooled IF amplifier chain; (iv) the implementation of optical components in the cryostats; (v) the consideration of an adequate strategy for the imaging system with YBCO HEB mixers, e.g., scanning vs. staring array options.