Josephson radiation and shot noise of a semiconductor nanowire junction

TL;DR

This study investigates Josephson radiation and shot noise in an InSb nanowire junction, revealing detailed microwave environment effects and evidence of single-electron states through photon-assisted tunneling.

Contribution

It provides the first detailed measurement of Josephson radiation and shot noise in a semiconductor nanowire junction, including microwave environment characterization.

Findings

Zero frequency impedance Z(0)=492 Ω

Cutoff frequency f_0=33.1 GHz

Fano factor F≈1 indicating single-electron states

Abstract

We measured the Josephson radiation emitted by an InSb semiconductor nanowire junction utilizing photon assisted quasiparticle tunneling in an AC-coupled superconducting tunnel junction. We quantify the action of the local microwave environment by evaluating the frequency dependence of the inelastic Cooper-pair tunneling of the nanowire junction and find the zero frequency impedance $Z(0)=492\,\Omega$ with a cutoff frequency of $f_0=33.1\,$GHz. We extract a circuit coupling efficiency of $\eta\approx 0.1$ and a detector quantum efficiency approaching unity in the high frequency limit. In addition to the Josephson radiation, we identify a shot-noise contribution with a Fano factor $F\approx1$, consistently with the presence of single electron states in the nanowire channel.

Equations

$$I_s(t)=I_c\sin (2\pi ft),$$

$$I_{\text{PAT}}(V_{\text{det}})={\int }_0^{\infty }{S}_V(f){\left(\frac{e}{hf}\right)}^2{I}_{QP,0}\left(V_{\text{det}}+\frac{hf}{e}\right)\text{differential}f,$$

$$I_{\text{ICPT}}=\frac{I_c^2\text{Re}(Z(f))}{2V_{\text{NW}}},$$

$$\begin{split}\frac{\differential S_{I}(f)}{\differential V_{\textrm{NW}}}=\frac{F}{R_{\textrm{qp}}}\derivative{V_{\textrm{NW}}}\Bigl{(}&\frac{hf+eV_{\textrm{NW}}}{1-e^{-\beta(hf+eV_{\textrm{NW}})}}\\ &+\frac{hf-eV_{\textrm{NW}}}{1-e^{-\beta(hf-eV_{\textrm{NW}})}}\Bigr{)},\end{split}$$

Full text

The tunneling of Cooper pairs through a junction between two superconducting condensates gives rise to a dissipationless current Josephson (1962) with a maximum amplitude of the critical current, $I_{c}$ Ambegaokar and Baratoff (1963). Upon applying a finite voltage bias $V$, the junction becomes an oscillating current source

[TABLE]

with a frequency set by $hf=2eV$ where $h$ is the Planck constant and $e$ is the electron charge.

The Josephson radiation, defined by Eq. (1) has mostly been investigated for superconducting tunnel junctions Giaever (1965); Holst et al. (1994); Deblock et al. (2003), metallic Cooper-pair transistors Billangeon et al. (2007) and in circuit QED geometries Hofheinz et al. (2011); Basset et al. (2012a). Recently, it has also been proposed as a probe for topological superconductivity Pikulin and Nazarov (2012); San-Jose et al. (2012); Houzet et al. (2013), which requires gateable semiconductor Josephson junctions Doh et al. (2005).

In contrast to superconductor-insulator-superconductor (SIS) junctions, Josephson junctions with a semiconductor channel feature conductive modes of finite transmission probabilities Xiang et al. (2006); van Woerkom et al. (2017), leading to deviations from a sinusoidal current-phase relationship Della Rocca et al. (2007) and the universal ratio of the critical current and the normal-state conductance Ambegaokar and Baratoff (1963). Furthermore, soft-gap effects Takei et al. (2013) have been shown to result in excess quasiparticle current for subgap bias voltages, limiting prospective applications such as topological circuits Mourik et al. (2012) and gate-controlled transmon qubits de Lange et al. (2015).

Here we investigate the high-frequency radiation signatures of a voltage-biased semiconductor Josephson junction Doh et al. (2005) by directly measuring the frequency-resolved spectral density. As a frequency-sensitive detector, we utilize a SIS junction, where the photon-assisted tunneling current Deblock et al. (2003) is determined by the spectral density of the coupled microwave radiation Tien and Gordon (1963). In addition to the detection of the monochromatic Josephson radiation, we demonstrate the presence of a broadband contribution, attributed to the shot noise of the nanowire junction Blanter and Büttiker (2000), similarly to earlier experiments on carbon nanotube quantum dots Onac et al. (2006); Basset et al. (2012b).

Our setup follows the geometry of earlier experiments utilizing SIS junctions Deblock et al. (2003). In contrast, our microwave radiation source is an InSb nanowire (NW) Plissard et al. (2012) Josephson junction (Fig. 1d) with a channel length of $100\,$nm. The junction leads (in brown in Fig. 1d) are created by removing the surface oxides by Ar ion milling and then in-situ sputtering of NbTiN superconducting alloy. Owing to the highly transparent contacts, this procedure enables induced superconductivity in the semiconductor channel Mourik et al. (2012); de Lange et al. (2015). A predefined gate structure (purple regions in Fig. 1d) provides electrostatic control of the semiconductor channel and is covered by sputtering a $20\,$nm thick SiNx dielectric layer.

The $I(V)$ characteristics of the two junctions are measured in a standard four point probe geometry via highly resistive Pt feedlines effectively decoupling the on-chip elements (Fig. 1) thermally anchored at $20\,$mK from the measurement setup. In order to gain access to a wider $V_{\textrm{NW}}$ range, we use $R_{1}=1\,$k$\Omega$ in the nanowire biasing lines and $R_{2}=6\,$k$\Omega$ in the voltage measurement leads (see Fig. 1b).

The detector SIS split junction is shown in Fig. 1f and is fabricated using standard shadow evaporation techniques Dolan (1977). The typical normal state resistance was measured to be $20\,\textrm{k}\Omega$ for a nominal junction area of $100\times 100\,$nm2. The bottom and top Al layer thicknesses are $9$ and $11\,$nm, respectively. The split junction geometry enables the flux control of the total Josephson coupling of the detector. To measure the quasiparticle tunneling response, we set $\Phi=\Phi_{0}/2$, with $\Phi_{0}=h/2e$ the flux quantum, to minimize the Josephson coupling. We note that the minimal detector critical current is negligible compared to that of the nanowire junction. Finally, we utilize two parallel plate capacitors of $C_{c}\approx 400\,$fF with sputtered SiNx dielectric which couple the nanowire junction to the detector in the frequencies of interest (Fig. 1e), yet enable independent voltage biasing and current measurements in the DC domain.

The mesoscopic noise source under consideration is characterized by its current noise density, $S_{I}(f)$ Blanter and Büttiker (2000), which results in the voltage noise density $S_{V}(f)=S_{I}(f)|Z(f)|^{2}$, where $Z(f)$ is the complex frequency-dependent impedance of the coupling circuit. In Fig. 1b, we depict a parallel $RC$ network resulting in $Z(f)=R(1-jf/f_{0})/(1+f^{2}/f_{0}^{2})$ with $2\pi f_{0}=(RC)^{-1}$ in the limit of negligible detector admittance, $r_{\textrm{det}}^{-1}=dI_{\textrm{det}}/dV_{\textrm{det}}\ll R^{-1}$.

We deduce the voltage noise density $S_{V}(f)$ starting from the equation for the photon-assisted current in the SIS detector Tucker and Feldman (1985); Deblock et al. (2003):

[TABLE]

which describes the DC current contribution at an applied voltage $V_{\textrm{det}}<2\Delta$. Crucially, this equation holds if the quasiparticle current in the absence of radiation has a well-defined onset, $I_{QP,0}(V_{\textrm{det}}<2\Delta)=0$ Deblock et al. (2003) and in the limit of weak coupling, where multiphoton processes do not contribute to the quasiparticle current Tien and Gordon (1963). In addition, a detector with a sharp quasiparticle current onset can reach the quantum limit Tucker and Feldman (1985) where each absorbed photon results in the tunneling of one quasiparticle.

In the presence of a monochromatic radiation, where $S_{V}(f)\sim\delta(f-\mathcal{F})$, Eq. (2) describes the shift of the initial $I_{QP,0}(V_{\textrm{det}})$ quasiparticle current by $\delta V_{\textrm{det}}=h\mathcal{F}/e$. This is the case of the Josephson radiation Deblock et al. (2003) with $S_{I}(f)=\frac{I_{c}^{2}}{4}\delta(f-\mathcal{F})$, where $h\mathcal{F}=2eV_{\textrm{NW}}$ with $V_{\textrm{NW}}$ the applied voltage bias on the emitter junction with a critical current $I_{c}$. On the other hand, the nonsymmetrized quasiparticle shot noise is characterized by $S_{I}=eIF$ in the zero frequency and zero temperature limit with $I$ being the applied current. The Fano factor, $F$ is characteristic to the mesoscopic details of the junction Blanter and Büttiker (2000).

Note that Eq. (2) can be handled as a convolution of $S_{V}(f)/(hf)^{2}$ and $I_{QP,0}(V_{\textrm{det}})$. However, the inverse problem leading to $S_{V}(f)$ is unstable due to the noise in the experimental data. To this end, we use Tikhonov regularization Trong et al. (2014) to extract the noise density measured by the detector (see raw for details). It is to be noted that the measured $I_{\textrm{det,0}}$ (see inset of the Fig. 2b) exhibits backbending due to the self-heating effects in the leads of the superconducting tunnel junction, therefore we used a monotonous $I_{QP,0}(V_{\textrm{det}})$ centered around the same quasiparticle onset. However, the uncertainity of $I_{QP,0}(V_{\textrm{det}})$ prevents the determination of the exact lineshape of $S_{V}(f)$ which could indicate the linewidth of the Josephson radiation Dahm et al. (1969).

We demonstrate the detection of the Josephson radiation in Fig. 2. In panel (a), we plot the PAT current contribution as a function of the DC bias voltages $V_{\textrm{det}}$ and $V_{\textrm{NW}}$. In Fig. 2b, we show line traces $I_{\textrm{PAT}}(V_{\textrm{det}})$ exhibiting well-defined onset values corresponding to a monochromatic Josephson radiation tuned by $V_{\textrm{NW}}$. Thus, we can extract the radiation frequency based on Eq. (2) (orange dots in Fig. 2a). By evaluating the relation between $V_{\textrm{NW}}$ and the radiation frequency (black line in Fig. 2a), we find a ratio of $475\pm 4.2\frac{\textrm{MHz}}{\mu\textrm{V}}$ which is in reasonable agreement with $\frac{2e}{h}\sim 484\frac{\textrm{MHz}}{\mu\textrm{V}}$ expected for the case of Cooper-pair tunneling Parker et al. (1967). The intersect for $f=0$ is set by the quasiparticle current onset to be $2\Delta/e=480\,\mu$V (see inset of Fig. 2b).

The impedance $Z(f)$ of the environment results in a finite power dissipation $I_{c}^{2}\textrm{Re}(Z(f))/2$ which gives rise to a DC current due to inelastic Cooper-pair tunneling (ICPT) processes in the NW Josephson junction (see Fig. 1a) Holst et al. (1994). This effect has been first addressed to calculate the shape of the supercurrent branch in overdamped SIS junctions and purely resistive environments Ivanchenko and Zil’berman (1969). Later, the theory was adapted for high channel transmissions Chauvin et al. (2007). It has also been shown that for an arbitrary $Z(f)\ll h/4e^{2}\approx 6.5\,$k$\Omega$, the ICPT contribution can be evaluated as Holst et al. (1994)

[TABLE]

with a critical current $I_{c}$ and an applied voltage $V_{\textrm{NW}}$. Here, the junction effectively probes the real component of the impedance $Z(f)$ at a frequency $f=2eV_{\textrm{NW}}/h$.

In the following, we use a circuit model where the two independently measured current values $I_{\textrm{PAT}}(V_{\textrm{det}})$ and $I_{\textrm{ICPT}}(V_{\textrm{NW}})$ depend on the same microwave enviroment, characterized by $Z(f)$. This model applies provided that the linear resistance of the nanowire and the impedance of the detector, $r_{\textrm{det}}$, are much higher than the effective shunt resistance of the circuit, depicted by $R$ in Fig. 1b. In addition, the lumped element description of Fig. 1b is valid if the circuit is much smaller than the characteristic wavelength $c/f\sim 1\,$mm. Our structure, $~{}50\,\mu$m in size (see Fig. 1e), fulfills this condition. Note that this is in contrast to a prior work Basset et al. (2012a) where the sample and detector were embedded in a transmission line resonator and thus the effective impedance values were measured to be different.

It is important to notice that the PAT current decreases with increasing frequency (Fig. 2b). By correcting for the $\sim f^{-2}$ dependence in Eq. (2), we find that the fluctuation amplitude $\delta V=I_{c}|Z(f)|\sim\sqrt{S_{V}}$ exhibits a characteristic cutoff frequency (Fig. 3a), even though the current oscillation amplitude of the Josephson junction is constant, see Eq. (1). Thus, we can attribute this cutoff to the coupling circuit impedance, $Z(f)$. We find a good agreement between the experimental data and the impedance of a single-pole $RC$ network (solid blue line in Fig. 3a) yielding to a cutoff frequency $f_{0}=(2\pi RC)^{-1}=33.1\,$GHz.

Next, we turn to the measured $I(V)$ trace of the nanowire Josephson junction. The inset of Fig. 3b shows the raw curve, which exhibits a supercurrent peak around zero $V_{\textrm{NW}}$ and a linear branch. The latter fits to a linear slope of $R_{\textrm{NW}}=14.03\,$k$\Omega$ (solid green line). We then extract the $I_{\textrm{ICPT}}(V_{\textrm{NW}})$ component by subtracting this slope from the raw measured data (black dots in Fig. 3b), which is an additive component to the supercurrent peak unless the device has channels of transmission very close to unity Chauvin et al. (2007). In order to find the critical current and the noise temperature of the junction, we use the finite temperature solution of Ivanchenko and Zil’bermann Ivanchenko and Zil’berman (1969) with substituting $|Z(f)|$ as the impedance of the environment raw . With this addition, we find an excellent agreement with the experimental data (blue solid line in Fig. 3b), with $I_{c}=9.38\,$nA critical current. Notably, with the now determined value of $I_{c}$, we can extract $R=492\,\Omega$ and $C=9.8\,$fF fully characterizing the microwave environment of the junctions. In addition, we find $I_{c}R_{\textrm{NW}}=132\,\mu$V, which indicates the induced superconducting gap in the nanowire channel. This value is close to the induced gap values measured earlier in similar devices Mourik et al. (2012); Gül et al. (2017). We also extract an effective noise temperature $T=132\,$mK, which is higher than the substrate temperature of $20\,$mK, similarly to earlier experiments Chauvin et al. (2007).

Thus far, we evaluated $I_{\textrm{ICPT}}(V_{\textrm{NW}})$ at $V_{\textrm{det}}\approx 50\,\mu$V$\ll 2\Delta/e=480\,\mu$V, where $I_{\textrm{PAT}}\approx 0$, thus the detector load is negligible. However, depending on $V_{\textrm{NW}}$, we find a negative $\Delta I_{\textrm{ICPT}}(V_{\textrm{det}})$, i.e. a reduction of the emitter current, when the detector threshold is on resonance with the emitted frequency (Fig. 3c). We can understand this effect by the reduction of $Z(f)$ in Eq. (3) in the presence of a finite $r_{\textrm{det}}$ in parallel with $R$. In first order, we find $\Delta I_{\textrm{ICPT}}/I_{\textrm{ICPT}}=-\textrm{Re}({Z(f)})/r_{\textrm{det}}\approx-R/r_{\textrm{det}}$. By using the measured DC current values, we evaluate the efficiency of the coupling circuit to be the ratio of the absorbed and emitted power $\eta=P_{\textrm{det}}/P_{\textrm{emi}}=2I_{\textrm{PAT}}/I_{\textrm{ICPT}}$ (Fig. 3d). We find typical values spanning $0.1-0.2$, an order of magnitude improvement over earlier reported values Deblock et al. (2003); Billangeon et al. (2006), however $\eta<1$ owing to the resistive losses of the device. Furthermore, the decrease of $\eta$ with increasing $f$ is consistent with the low-pass nature of the coupling circuit. We also calculate the detector quantum efficiency $Q=P_{\textrm{det}}/\Delta P_{\textrm{emi}}=2I_{\textrm{PAT}}/\Delta I_{\textrm{ICPT}}$ (Fig. 3e) and find values scattering around unity. This value directly measures the ratio of electron and photon rate passing the detector junction, thus confirming that it is in the quantum limit Tucker and Feldman (1985).

Finally, we note that the measured reduction $\Delta I_{\textrm{ICPT}}/I_{\textrm{ICPT}}\ll 1$ directly confirms our initial assumption of negligible detector load on the circuit. This proves that the analysis based on a circuit model with the same $Z(f)$ for the nanowire junction and the SIS detector is consistent.

We now turn to the shot-noise contribution to $I_{\textrm{PAT}}$. We observe a monotonous increase in $I_{\textrm{PAT}}$ with increasing $V_{\textrm{NW}}$ at any $V_{\textrm{det}}$ consistently with the broadband $S_{I}$ (Fig. 4a). Note that, in contrast with the data shown in Fig. 2b, here the contribution of the Josephson radiation is negligible. To quantify the shot-noise contribution, we consider the derivative of the nonsymmetrized expression with respect to $V_{\textrm{NW}}$ Aguado and Kouwenhoven (2000):

[TABLE]

where $\beta=1/k_{\textrm{B}}T$ is the inverse temperature 111Note the we omitted the voltage-independent terms in Aguado and Kouwenhoven (2000). We can then calculate $\differential{I_{\textrm{PAT}}}/\differential{V_{\textrm{NW}}}$ by subsituting $\differential S_{I}(f)/\differential V_{\textrm{NW}}$ in place of $S_{I}(f)$ in Eq. (2). Using the effective temperature $T=132\,$mK extracted earlier we find a confidence interval of $F=0.8\ldots 1.3$ (Fig. 4b). Considering that the channel length of $100\,$nm is similar to the mean free path found earlier in the same nanowires Gül et al. (2015), this result is consistent with ballistic transport which is dominated by single electron channels of low transmission where $F=1$ de Jong and Beenakker (1997); Blanter and Büttiker (2000). In contrast, $F=1/3$ characteristic of diffusive normal transport Beenakker and Büttiker (1992) does not fit our data.

Furthermore, the measured $I_{\textrm{NW}}(V_{\textrm{NW}})$ and $I_{\textrm{PAT}}(V_{\textrm{NW}})$ do not agree with a transport dominated by multiple Andreev reflections, where a subgap structure is anticipated both in the current Scheer et al. (1997) and in the shot noise Cron et al. (2001) depending on the channel transmissions. Our experiment thus provides insight into the nature of the charge transport at finite voltage bias in the nanowire Josephson junction and concludes that the finite subgap current can be attributed to single electron states inside the induced superconducting gap.

In conclusion, we built and characterized an on-chip microwave coupling circuit to measure the microwave radiation spectrum of an InSb nanowire junction with NbTiN bulk superconducting leads. Our results clearly demonstrate the possibility of measuring the frequency of the Josephson radiation in a wide frequency range, opening new avenues in investigating the $4\pi$-periodic Josephson effect Lutchyn et al. (2010) in the context of topological superconductivity Oreg et al. (2010). Based on the Fano factor, the shot-noise contribution to the measured signal demonstrates the presence of subgap quasiparticle states and excludes multiple Andreev reflection as the source of subgap current of the nanowire Josephson junction.

The authors acknowledge D. Bouman, A. Bruno, O. Benningshof, M. C. Cassidy, M. Quintero-Pèrez and R. Schouten for technical assistance, and R. Deblock for fruitful discussions. This work has been supported by the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO) by a Veni grant, Microsoft Corporation Station Q, and a Synergy Grant of the European Research Council.