Josephson Photodetectors via Temperature-to-Phase Conversion

We theoretically investigate the temperature-to-phase conversion (TPC) process occurring in dc superconducting quantum interferometers based on superconductor–normal-metal–superconductor ($S\text{−}N\text{−}S$) mesoscopic Josephson junctions. In particular, we predict the temperature-driven rearrangement of the phase gradients in the interferometer under the fixed constraints of fluxoid quantization and supercurrent conservation. This mechanism allows sizeable phase variations across the junctions for suitable structure parameters and temperatures. We show that the TPC can be a basis for sensitive single-photon sensors or bolometers. We propose a radiation detector realizable with conventional materials and state-of-the-art nanofabrication techniques. Integrated with a superconducting quantum-interference proximity transistor as a readout setup, an aluminum-based TPC calorimeter can provide a large signal-to-noise ratio $>100$ in the 10-GHz–10-THz frequency range and a resolving power larger than $10^2$ below 50 mK for terahertz photons. In the bolometric operation, electrical noise equivalent power of approximately ${10}^{-22}\mathrm{W}/\sqrt{\mathrm{Hz}}$ is predicted at 50 mK. This device can be attractive as a cryogenic single-photon sensor operating in the giga- and terahertz regime with applications in dark-matter searches.

Figure 1

(a) Scheme of a dc superconducting quantum interferometer realized with two $S\text{−}N\text{−}S$ Josephson junctions. Absorption of electromagnetic radiation of energy $h\nu$ increases the electronic temperature ($T_e$) in the $N$ region of the detector junction thereby suppressing the Josephson supercurrent circulating in the loop. The suppression leads to a variation of phase difference (${\phi }_{d,r}$) across both Josephson junctions and to a modification of the DOS in their $N$ regions. The DOS in the readout junction is probed via a SQUIPT used as the readout circuit. The structure, therefore, operates as a temperature-to-phase converter, i.e., $\delta T_e\to \delta {\phi }_{d,r}$. $L$ denotes the length of the junctions, and ${\mathbf{\Phi }}_{\mathrm{ext}}$ is the applied magnetic flux. (b) Enlargement of the detector in the vicinity of the weak-link region: Superconducting electrodes $S_1$ with energy gap ${\mathrm{\Delta }}_1\gg {\mathrm{\Delta }}_0$ (${\mathrm{\Delta }}_0$ is the zero-temperature energy gap in $S$) in good electric contact with the loop constitute Andreev mirrors for the energy absorbed by quasiparticles in the $N$ region. The Andreev reflection allows energy relaxation in the detector junction to occur predominantly via electron-phonon interaction with lattice phonons in the enclosed $N$ and $S$ regions (see text). $L_S$ is the length of the $S$ portions in direct contact with the $N$ nanowire.

Figure 2

(a) Josephson current $I^{d,r}$ vs $T_e=T_{\mathrm{bath}}$ calculated at ${\phi }_{d,r}=\pi /2$. $T_{\mathrm{bath}}$ and $T_c$ are the bath and critical temperature of the $S$ loop, respectively, whereas $I_c^{\pi /2}$ is the zero-temperature Josephson current at ${\phi }_{d,r}=\pi /2$. (b) Scheme of time ($t$) evolution of the electronic temperature $T_e$ in the detector junction after absorption of a photon at energy $h\nu$. $T_e^{\max }$ is the temperature reached in the weak link after the arrival of a single photonic event. ${\tau }_{\mathrm{diff}}$ represents the diffusion time across the $N$ wire, whereas ${\tau }_{e\text{−}\mathrm{ph}}$ is the electron-phonon relaxation time. In the present setup, ${\tau }_{\mathrm{diff}}\ll {\tau }_{e\text{−}\mathrm{ph}}$ due to the short length of the $N$ wire and the low temperature at which the detector is conceived to operate.

Figure 3

(a),(b) Contour plot of the phase ${\phi }_r$ across the readout junction vs $\alpha$ and $T_e$ calculated for ${\mathbf{\Phi }}_{\mathrm{ext}}=0.4{\mathbf{\Phi }}_0$ (a) and ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$ (b) at $T_{\mathrm{bath}}=0.01T_c$. (c),(d) Contour plot of the minigap ${ϵ}_g^r$ in the $N$ region of the readout junction vs $\alpha$ and $T_e$ calculated for ${\mathbf{\Phi }}_{\mathrm{ext}}=0.4{\mathbf{\Phi }}_0$ (c) and ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$ (d) at $T_{\mathrm{bath}}=0.01T_c$. Dashed lines in all panels positioned at $\alpha =3$ correspond to the value chosen for the interferometer in all forthcoming calculations unless differently stated. ${\mathrm{\Delta }}_0$ denotes the zero-temperature energy gap in corresponding to the critical temperature $T_c\simeq {\mathrm{\Delta }}_0/1.764k_B$.

Figure 4

(a) Current ($I_p$) vs voltage ($V$) characteristics of the SQUIPT readout junction in the absence ($T_e=T_{\mathrm{bath}}$, $\nu =0$) and for finite incoming radiation ($T_e>T_{\mathrm{bath}}$, $\nu \ne 0$) calculated for ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$, $\alpha =1$, and $T_{\mathrm{bath}}=0.01T_c$. The voltage onset of large quasiparticle tunneling for zero ($h\nu =0$) and finite frequency ($h\nu \ne 0$) are indicated (black dashed arrows). The bias voltage of the SQUIPT readout junction is denoted by $V^{*}$, whereas $\delta I$ indicates the current variation due to the absorption of a photon of energy $h\nu$. (b) $I_p$ vs $V$ calculated for ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$, $T_{\mathrm{bath}}=0.01T_c$, $\nu =0$, and for several values of $\alpha$. The degree of asymmetry of the interferometer determines the onset of large quasiparticle tunneling in the absence of radiation.

Figure 5

(a) Current $I_p$ vs temperature $T_e$ characteristics of the SQUIPT readout junction calculated for a few values of $V$ at ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$. (b) Absolute value of the temperature-to-current transfer function $|{\tau }_I|$ vs $T_e$ calculated for the same parameters as in panel (a). (c) $I_p$ vs $T_e$ characteristics calculated for a few values of ${\mathbf{\Phi }}_{\mathrm{ext}}$ at $V=1.995{\mathrm{\Delta }}_0/e$. (d) Absolute value of the temperature-to-current transfer function $|{\tau }_I|$ vs $T_e$ calculated for the same parameters as in panel (c). In all these calculations, we set $\alpha =3$ and $T_{\mathrm{bath}}=0.01T_c$.

Figure 6

(a) Temperature sensitivity $s_T$ vs temperature $T_e$ calculated for a few values of $V$ at ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$. (b) $s_T$ vs $T_e$ calculated for a few values of ${\mathbf{\Phi }}_{\mathrm{ext}}$ at $V=1.995{\mathrm{\Delta }}_0/e$. In all calculations, we set $\alpha =3$ and $T_{\mathrm{bath}}=0.01T_c$.

Figure 7

(a) Behavior of the total electronic heat capacity $C_{\mathrm{tot}}^e$ in the detector junction vs temperature $T_e$ calculated at 10 mK for two values of the applied magnetic flux. Dashed line represents the total electronic heat capacity when the system is in the normal state. The inset shows an enlargement of $C_{\mathrm{tot}}^e$ in the low-temperature regime. (b) Electronic temperature in the detector junction $T_e$ vs $\nu$ calculated for several values of $T_{\mathrm{bath}}$ at ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$. (c) Relative variation of the electronic temperature $\delta T_e/T_e^0$ vs $\nu$ calculated for the same parameters as in panel (b). (d) Phase ${\phi }_r$ across the readout weak link vs absorbed energy $h\nu$ calculated for a few $T_{\mathrm{bath}}$ values at ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$ and $\alpha =3$.

Figure 8

(a) Detector time constant ${\tau }_{1/2}$ vs absorbed energy $h\nu$ of the incoming radiation calculated at different bath temperatures $T_{\mathrm{bath}}$. (b) Detector signal-to-noise ratio ($S/N$) vs voltage $V$ calculated at 10 mK for a few values of frequency of the incoming radiation. (c) Maximum SNR ($S/N_{\max }$) vs $\nu$ calculated for the same bath temperatures as in panel (a). (d) Detector resolving power $h\nu /\delta E$ vs $\nu$ calculated for the same $T_{\mathrm{bath}}$ as in panel (a). In the calculations of panels (a)–(c), we set ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$ and $\alpha =3$.

Figure 9

(a) Detector maximum signal-to-noise ratio $S/N_{\max }$ vs frequency $\nu$ calculated for several values of ${\mathbf{\Phi }}_{\mathrm{ext}}$. (b) $S/N_{\max }$ vs $\nu$ calculated for a few values of $\mathrm{\Gamma }$ at ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$. In all the calculations, we set $T_{\mathrm{bath}}=10\mathrm{mK}$.

Figure 10

(a) Weak-link total electron-phonon thermal conductance $G_{\mathrm{th}}^{d,\mathrm{tot}}$ of the detector vs $T_{\mathrm{bath}}$. $G_{\mathrm{th}\text{−}N}^d$ denotes the total $e$-ph thermal conductance in the normal state. The electronic heat conductivity of the SQUIPT $S$ tunnel probe and ring ($\mathrm{\Gamma }/{\mathrm{\Delta }}_0={10}^{-5}$, $G_{N,\mathrm{ring}}^r=10{\mathrm{\Omega }}^{-1}$) are also shown for comparison. (b) Electronic temperature $T_e$ in the detector junction vs optical power $P_{\mathrm{opt}}$ calculated for several values of $T_{\mathrm{bath}}$. (c) Relative variation of the electronic temperature $\delta T_e/T_e^0$ vs $P_{\mathrm{opt}}$ calculated for the same bath temperatures as in panel (c). (d) NEP vs $T_{\mathrm{bath}}$ for thermal-fluctuation noise (TFN) and shot noise (SQUIPT) at $V=1.995{\mathrm{\Delta }}_0$. In the calculations of panels (b)–(d), we set ${\mathbf{\Phi }}_{\mathrm{ext}}=0.499{\mathbf{\Phi }}_0$ and $\alpha =3$.