Abstract
Experiments have shown that several materials, including MgB_{2}, ironbased superconductors and monolayer NbSe_{2}, are multiband superconductors. Superconducting pairing in multiple bands can give rise to phenomena not available in a single band, including Leggett modes. A Leggett mode is the collective periodic oscillation of the relative phase between the phases of the superconducting condensates formed in the different bands. The experimental observation of Leggett modes is challenging because multiband superconductors are rare and because these modes describe charge fluctuations between bands and therefore are hard to probe directly. Also, the excitation energy of a Leggett mode is often larger than the superconducting gaps, and therefore they are strongly overdamped via relaxation processes into the quasiparticle continuum. Here, we show that Leggett modes and their frequency can be detected in a.c. driven superconducting quantum interference devices. We then use the results to analyse the measurements of such a quantum device, one based on a Dirac semimetal Cd_{3}As_{2}, in which superconductivity is induced by proximity to superconducting Al. These results show the theoretically predicted signatures of Leggett modes, and therefore we conclude that a Leggett mode is present in the twoband superconducting state of Cd_{3}As_{2}.
Similar content being viewed by others
Main
The superconducting state is well described by a complex order parameter Δ(r) = ∣Δ(r)∣e^{iϕ(r)}, characterized by an amplitude and a phase that, in general, depend on the position r. The time dependence of Δ(r) describes the collective, low energy excitations of a superconductor. In a standard singleband superconductor there are two types of collective excitations: Anderson–Higgs modes, corresponding to fluctuations of ∣Δ∣ and pseudoGoldstone modes, corresponding to fluctuations of the phase ϕ. The Fermi surface of a multiband metal is formed by several generally disconnected Fermi pockets. In this case, at low temperatures, the metal can become a multiband superconductor characterized by different order parameters Δ_{i} for different Fermi pockets^{1,2,3,4,5}, as shown schematically in Fig. 1a. It was pointed out^{6} that a multiband superconductor will have additional collective modes corresponding to fluctuations of the phase difference between the order parameters of different Fermi pockets. So far, evidence of Leggett modes has been obtained only via direct spectroscopy techniques in MgB_{2} (refs. ^{7,8,9,10,11}) and, more recently, in an Febased superconductor^{12}. Using an approach of limited applicability, it had been theorized that in Josephson junctions (JJs) in which one lead is formed by a singleband superconductor and the other by a twoband superconductor, signatures of a Leggett mode could be present^{13}.
Here we show, using a different method, how the presence of Leggett modes can be observed in JJs and in a.c.driven superconducting quantum interference devices (SQUIDs) in which all the leads are formed from the same multiband superconducting material. This opens a new approach to the detection and characterization of Leggett modes. In addition, we show that measurements on SQUIDs based on the superconducting Dirac semimetal (DSM) Cd_{3}As_{2} display the theoretically predicted unique signatures associated with the presence of a Leggett mode. This adds Cd_{3}As_{2} to the list of the few materials exhibiting the presence of Leggett modes and points to the unusual multiband character of the superconducting state in this material and possibly more generally in DSMs.
For a twoband superconductor, the dynamics of the Leggett mode can be described by the effective Lagrangian
where ℏ is the reduced Planck’s constant, C_{12} is the interband capacitance, e is the electron’s charge, I_{12} is the effective interband critical Josephson current^{6} and ϕ_{0} is the equilibrium value of ϕ. From equation (1) we obtain that when ϕ − ϕ_{0} ≪ 1, ϕ will oscillate with frequency \({\omega }_{\rm{L}}=\sqrt{(2e/\hslash ){I}_{12}/{C}_{12}}\) around ϕ_{0}.
A SQUID, see Fig. 1b, is formed by two JJs connected in parallel and encircling a finite size area. Let \({\theta }_{i}\equiv {\phi }_{i}^{\rm{R}}{\phi }_{i}^{\rm{L}}\) be the difference between the superconducting order parameter in the right and left lead for band i. Then the current across the JJ’s leads, for a JJ with lowmedium transparency^{14}, is given by \(I={I}_{1}\sin ({\alpha }_{1}{\theta }_{1})+{I}_{2}\sin ({\alpha }_{2}{\theta }_{2})\) where I_{i} is the critical supercurrent for band i, and α_{i} is equal to 1 for a standard JJ and 1/2 for a topological JJ^{15,16,17}. For biased hightransparency topologically trivial JJs, Landau–Zener transitions can induce a current–voltage response equivalent to a topological junction^{18,19,20}.
To understand the effect of a Leggett mode on the dynamics and voltage–current (V–I) characteristic of a JJ, we first present a simplified analysis of a voltagebiased JJ. A rigorous analysis of the realistic case of a currentbiased JJ is presented later (see also Supplementary Section 1). The dynamics of the relative phase ϕ can induce oscillations in the phase difference ψ ≡ (θ_{1} − θ_{2})/2. Let \({\phi }_{\rm{R}}\equiv {\phi }_{1}^{\rm{R}}{\phi }_{2}^{\rm{R}}\) and \({\phi }_{\rm{L}}\equiv {\phi }_{1}^{\rm{L}}{\phi }_{2}^{\rm{L}}\), Fig. 1c, so that \(\psi =({\phi }_{\rm{R}}{\phi }_{\rm{L}})/2={\psi }_{0}+\tilde{\psi}{(t)}\), where ψ_{0} is the equilibrium value of ψ and \(\tilde{\psi}{(t)}\) the time dependent part. We can write θ_{1} = θ_{A} + ψ, θ_{2} = θ_{A} − ψ, with θ_{A} = (θ_{1} + θ_{2})/2. In the presence of a voltage V across the JJ’s leads, we have dθ_{A}/dt + dψ/dt = 2eV/ℏ. We consider the case when \(V(t)={V}_{\rm{d.c.}}+{V}_{\rm{a.c.}}\cos \omega t\). When the Leggett mode is driven, directly or indirectly, by a periodic drive, we can assume \(\tilde{\psi }\approx {\hat{A}}_{\omega }\sin (\omega t)\), with \({\hat{A}}_{\omega }\approx {A}_{0}{{{\varGamma }}}_{\rm{L}}\omega /({({\omega }^{2}{\omega }_{\rm{L}}^{2})}^{2}+{{{\varGamma }}}_{\rm{L}}^{2}{\omega }^{2})\) the amplitude of the mode and Γ_{L} its broadening. In the limit when ω ≈ ω_{L} so that \({\hat{A}}_{{\omega }_{\mathrm{L}}}\gg {V}_{\mathrm{a.c.}}/\omega\) we obtain:
where J_{n}(x) is the nth Bessel function of the first kind. When α_{1} = α_{2} = 1, depending on the value of ψ_{0}, we can have suppression of the odd or even Shaprio spikes^{19}. For ψ_{0} = 0, we have suppression of the odd steps. In this case, for ω ≈ ω_{L} the Shapiro steps’ structure is qualitatively the same as the one obtained at low frequencies and powers in the presence of a topological superconducting channel (α_{1} = α_{2} = 1/2), or Landau–Zener processes in highly transparent junctions^{18}. For small ω_{L} and nonnegligible Γ_{L}, it might be difficult to pinpoint reliably the cause of the missing odd Shapiro steps. However, for the case when ψ_{0} = π/2, equation (2) leads to a suppression of the even Shapiro spikes, a phenomenon that cannot be attributed to the topological nature of the JJ or to Landau–Zener processes. In the remainder, we assume α_{1} = α_{2} = 1 and discuss a concrete situation when we can expect ψ_{0} ≠ 0.
To describe the dynamics of an a.c.currentbiased 2band JJ, we use a resistively and capacitively shunted junction (RCSJ) model^{21,22}. When placing a lead on the surface of a DSM, the states of the lead couple strongly to the DSM’s surface states and weakly to the DSM’s bulk states. In this situation, the supercurrent in the bulk band (band 2) is mediated by interband processes (Fig. 1d), and a nonzero ψ_{0} is expected. In particular, we expect ϕ_{L} = π/2 and ϕ_{R} = −π/2 so that ψ_{0} = π/2. In this scenario, the values of ϕ_{L} and ϕ_{R} are not accidental but are the result of selftuning in JJs based on DSMs in which superconductivity is induced via the proximity effect by a superconductor placed on the surface of the DSM. In this case, the current’s lowest energy path to the bulk is via a Josephson supercurrent, \({I}_{12}^{\;({\rm{s}})}\), between the surface band and the bulk band. Considering that in general for a JJ, we have the current–phase relation \({I}^{({\rm{s}})}\propto \sin (\phi )\), we see that to maximize the supercurrent between surface and bulk the system will selftune in a state in which on the left lead ϕ_{L} = π/2 and on the right lead ϕ_{R} = −π/2, given that on the right lead \({I}_{12}^{\;({\rm{s}})}\) has to flow in the opposite direction, from bulk to surface, Fig. 1d (see also Supplementary Fig. 1 and Supplementary Section 1). The capacitance between the two leads is very small compared to the normal resistances R_{i} across the leads, so it can be neglected. Conversely, for the interband charge flow within the same lead, we can neglect the resistive channel, considering the nonnegligible interband capacitance C_{12}. The resulting effective RCSJ model is shown in Fig. 1d.
In the presence of the current bias \({I}_{\rm{B}}={I}_{\rm{d.c.}}+{I}_{\rm{a.c.}}\cos (\omega t)\), the dynamics of the RCSJ model shown in Fig. 1c are described by the equations
where ω_{J} ≡ 2eRI_{1}/ℏ, τ ≡ ω_{J}t, R = R_{1}R_{2}/(R_{1} + R_{2}), ξ ≡ (R_{1} − R_{2})/(R_{1} + R_{2}), \(\hat{\omega }\equiv \omega /{\omega }_{\rm{J}}\), i_{B} ≡ I_{B}/I_{1}, i_{2} ≡ I_{2}/I_{1} and \({\hat{A}}_{0}\equiv {\omega }_{\rm{L}}^{2}{R}_{1}/({\omega }_{\rm{J}}^{2}{i}_{12}({R}_{1}+{R}_{2}))\). In the remainder, we set ω_{L}/ω_{J} = 0.005, Γ_{L}/ω_{L} = 7.5 × 10^{−5}, \({\hat{A}}_{0}=0.0045\), ξ = − 0.6 and i_{2} = 1.5.
The dynamics of the SQUID can be obtained starting from equations (3) and (4) for each of the two JJs. In the remainder, we will denote by \({X}_{i}^{(j)}\) the quantity X for band i in arm j of the SQUID, see Fig. 1c. We assume the SQUID to be symmetric; the parameters entering the JJs’ RCSJ model and the selfinductance L are assumed to be the same for the left and right arm of the SQUID. In experiments, some asymmetry between left and right JJs is expected. We have checked the effect of asymmetries in the SQUID and found that: (1) small asymmetries simply cause the structure of the Shapiro steps to be slightly asymmetric with respect to the biasing current, (2) large asymmetries can give rise to a complicated subharmonic step structure arising from higher harmonic terms and (3) asymmetries alone cannot be responsible for suppression of nonzero even Shapiro steps before entering the Bessel regime. For the kth band, the phase difference \(\eta \equiv ({\theta}_{k}^{(2)}{\theta}_{k}^{(1)})/2\uppi ={\hat{\varPhi}}+\beta ({i}^{(1)}{i}^{(2)})\) where \({\hat{\varPhi}}={\varPhi}_{\rm{ext}}/{\varPhi}_{0}\) is the normalized external flux threading the SQUID, Φ_{0} = h/2e, β = I_{1}L/Φ_{0} and i^{(j)} = I^{(j)}/I_{1} with I^{(j)} the total current flowing through arm j. Using equations (3) and (4) and considering current conservation and the flux quantization for η, in the limit β ≪ 1, in terms of the phases \({\theta }_{\rm{s}}\equiv {\sum }_{ij}{\theta }_{i}^{(j)}/4\), \(\psi ={\psi }^{(1)}={\psi }^{(2)}={\psi }_{0}+\tilde{\psi} (t)\), we find (see Supplementary Section 2) that the dynamics of the SQUID are described by the equations
in conjunction with equation (4).
Using equations (4), (5) and (6), we obtain \({V}_{\rm{d.c.}}=\bar{V}=\mathop{\lim }\nolimits_{{t}_{f}\to \infty }(1/{t}_{f})\) \(\int\nolimits_{0}^{{t}_{f}}[(\hslash /2e){\rm{d}}{\theta }_{\rm{s}}/{\rm{d}}t]{\rm{d}}t\) where t_{f} is the total integration time. Let’s first consider \({\hat{\varPhi}}\,{{\mathrm{mod}}}\,\,2=0\) and set β = 0.05π. For ∣ω − ω_{L}∣ ≫ 1, the dependence of V_{d.c.} with respect to I_{d.c.} exhibits the standard Shapiro steps: all steps are present if either α_{1} or α_{2} is equal to 1, but only even steps are present if α_{1} = α_{2} = 1/2. For ω = ω_{L}, ψ_{0} = 0 and α_{1} = α_{2}, we have that the odd steps are strongly suppressed, see Fig. 2a, so that the structure of the Shapiro steps resembles the structure expected for a topological JJ for which a channel with α = 1/2 dominates. However, for ψ_{0} = π/2, and α_{1} = α_{2} = 1, we have the unusual situation that only the even Shapiro steps are suppressed, as shown in Fig. 2b. This behaviour is present as long as ω = 2πf is within the inverse lifetime, Γ_{L}, of the Leggett mode frequency f_{L} = ω_{L}/2π. When ℏω_{L} = 2πℏf_{L} < Δ_{sc} we can expect Γ_{L} to be quite small. We can calculate the width W of the Shapiro steps by binning the y axis of Fig. 2b for a fixed power. Figure 2c shows the width of the steps, W, as a function of V_{d.c.} and a.c. frequency f, assuming Γ_{L} = 0.05f_{L}. We see that for ∣f − f_{L}∣ ≪ Γ_{L}, the even steps are suppressed while the odd steps are strong; we also note that for f far from the resonance, we recover a voltage–current profile in which all the steps are present (apart from small corrections due to higher harmonics).
We can investigate the effect of the Leggett mode on the Shapiro steps when the SQUID is threaded by a nonzero magnetic flux Φ_{ext}. For the case when \({\hat{\varPhi}}\,{{\mathrm{mod}}}\,\,2\ne 0\), we first note that for \({\hat{\varPhi}}\,{{\mathrm{mod}}}\,\,2=1\), the second term vanishes. In this case, we find that the SQUID’s V–I curve exhibits the same Shapiro steps as for the case \({\hat{\varPhi}}=0\). When Φ_{ext} is a halfinteger of Φ_{0}, the first term on the righthand side of equation (6) vanishes and the term proportional to β affects the dynamics of the SQUID. In this case, when ψ ≈ 0, the factor of 2 in the argument of the sine causes the appearance of halfinteger Shapiro steps, as in standard SQUIDs^{23} when α_{1} = α_{2} = 1 and the appearance of the odd Shapiro steps when α_{1} = α_{2} = 1/2.
When f ≈ f_{L}, so that \(\tilde{\psi }\) is not negligible and Φ_{ext} is not a multiple of Φ_{0}, the SQUID’s V–I features are difficult to predict from a simple analysis of the equations. Numerically, for the case when f = f_{L}, ψ_{0} = π/2 and Φ_{ext} = Φ_{0}/2, we find that the SQUID has a fairly unique V–I curve, as shown in Fig. 3. Contrary to the case of a single JJ, the odd step at V = (hf/2e) is absent, and a new fractional step at V = 3/2(hf/2e) appears together with a step at V = 4(hf/2e), while the step at V = 3(hf/2e) survives. Figure 3b shows the range of values of Φ_{ext} around Φ_{0}/2 for which this step structure is present, and Fig. 3c shows how the step structure and the width of the steps depend on the a.c. frequency f, for f ≈ f_{L}, when Φ_{ext} = Φ_{0}/2.
The discussion above shows that when the equilibrium phase difference, \({\psi }_{0}\,{{\mathrm{mod}}}\,\,2\uppi\), between the two superconducting order parameters is 0, the microwave response of a SQUID in which an undamped Leggett mode is present, for ω ≈ ω_{L}, is similar to one obtained when the single JJs forming the SQUID have a current–phase relation that is 4π periodic, due either to the presence of a topological superconducting channel or to Landau–Zener processes. The analysis also shows that when ψ_{0} ≈ π/2, the SQUID’s microwave response, both in the absence and presence of an external magnetic flux Φ_{ext}, exhibits unique qualitative features that cannot be attributed to topological superconducting pairing or Landau–Zener processes.
In a DSM such as Cd_{3}As_{2}, the bulk threedimensional conduction and valence electronic bands touch at isolated points, and a projection of the spectral density onto a surface Brillouin zone reveals Fermi arcs connecting the bulk Dirac points^{24,25}. DSMs with proximityinduced superconductivity are predicted to be able to realize exotic nonAbelian anyons that can be used to develop topologically protected qubits^{26} and can be used in microwave singlephoton detection for sensing applications^{27,28,29}. Another aspect of DSMs that has received less attention in the literature concerns the multiband properties of superconducting DSMs^{30,31,32}. By placing a superconducting material on the surface of Cd_{3}As_{2}, superconducting pairing can be induced in the Cd_{3}As_{2} (ref. ^{30,31,32}). The pairing has been shown to be characterized by two order parameters Δ_{1} and Δ_{2}. Leggett modes result from oscillations of the difference between the phases of the superconducting gaps of different bands, and therefore their presence is always allowed, regardless of the mechanism—intrinsic as in MgB_{2}, or via proximity effect as in our devices—responsible for the superconducting pairing. In addition, recent experiments on single JJs formed by superconducting leads based on Al/Cd_{3}As_{2} have shown compelling signatures of an equilibrium phase difference θ_{1} − θ_{2} between the two phases across the junction, arising from the two superconducting order parameters, being equal to π, implying ψ_{0} = π/2 (ref. ^{32}). Motivated by these results and the theoretical analysis above, we have investigated the microwave response of a SQUID based on Al/Cd_{3}As_{2}. Details about the fabrication and measurement of the device can be found in the Methods section and Supplementary Section 5.
At frequency f = 2 GHz and Φ_{ext} = 0, the SQUID’s measured dV/dI exhibits peaks and valleys consistent with the standard Shapiro steps’ structure (Fig. 4a). However, for f = 7 GHz and f = 9 GHz, for all the microwave powers values considered, the first and third steps are clearly visible, but the second step is strongly suppressed (Fig. 4b). Considering that our device shows no hysteretic features in the current–voltage characteristic and no evidence of a biasdependent normal resistance, mechanisms for missing Shapiro steps due to hysteresis^{33} or biasdependent resistance^{34} are not relevant. When the zeroth step’s width approaches zero for I_{a.c.} ≈ I_{c}, the system begins to enter the ‘Bessel regime’, where oscillations in step widths with increasing power I_{a.c.} > I_{c} regularly occur and can lead to missing steps. Our measurements are not in the Bessel regime, given that for f ≈ 9 GHz, (1) the zeroth step is clearly nonzero at all powers and (2) the second step is missing at low powers, as shown in Fig. 4a,b (see also Supplementary Fig. 8c) and does not reappear as the power increases. We find it is very difficult to explain the suppression of even steps at low powers without invoking the presence of a Leggett mode.
We can estimate the value of ω_{L} in our device, as discussed in Supplementary Section 3. We find that ω_{L} ≈ 10 GHz is quite smaller than the value of ω_{L} ≈ 2.3 THz in MbB_{2} (ref. ^{8}), due to the high density of states of the bands of Cd_{3}As_{2}. We notice, however, that the precise value of ω_{L} depends on bands parameters whose accurate estimate is hard to obtain from experiments.
Figure 4c shows the voltage across the SQUID as a function of the perpendicular magnetic field B in the d.c. limit, I_{a.c.} = 0. SQUID oscillations of periodicity 1.8 mT are observed, which correspond to an effective SQUID ring area of 1.14 μm^{2}. Enveloping the SQUID oscillations is the Fraunhofer diffraction pattern of the JJs. Anomalous oscillations can also be observed for B such that the flux threading a single JJ, Φ_{J} = B × A_{JJ}, A_{JJ} being the area of the JJ, is a multiple of Φ_{0}/2. The presence of these oscillations is consistent with a πperiodic supercurrent in each of the JJs forming the SQUID because ψ_{0} = π/2.
In Fig. 4d we present as a colour plot the measured dV/dI as a function of \(\bar{V}\) and B in the presence of an a.c. component of the current with f = 9 GHz and relative power −22 dBm. Besides the periodicity of the Shapiro steps with respect to B, with a period consistent with the periodicity observed in the d.c. limit, Fig. 4c, we observe interesting features for B ≈ 1 mT corresponding to Φ_{ext} = Φ_{0}/2. To more clearly identify these features, we show in Fig. 4e the dV/dI traces for B = 0 mT and B = 1 mT. We see that for B = 1 mT, that is, Φ_{ext} = Φ_{0}/2, both the first and second Shapiro steps are suppressed and a 3/2 subharmonic step emerges, features that are remarkably consistent with the theoretical results shown in Fig. 3. To better understand the evolution of the Shapiro steps’ structure with Φ_{ext} when f = 9 GHz, in Fig. 4f we plot the measured width of the steps at V = hf/2e and V = (3/2)(hf/2e) as a function of B. We see that when B ≈ 1 mT, Φ_{ext} = Φ_{0}/2, the width of the first step is suppressed, whereas the width of the 3/2 step is enhanced around B ≈ 1 mT. The evolution of the 1 and 3/2 steps with Φ_{ext} is in good qualitative agreement with the theoretical results, shown in Fig. 4g.
Our theoretical and experimental results show how the response to microwave radiation of JJs and SQUIDs formed by multiband superconductors can be used to identify the presence of Leggett modes in such superconductors. By showing that qualitative signatures in the response due to Leggett modes appear only when the microwave frequency is close to the frequency of the Leggett mode, and when, at equilibrium, the phase difference between superconducting order parameters is not zero, the results also allow experimentally obtaining an estimate of the Leggett mode’s frequency and its broadening and of the relative phases between superconducting gaps, all quantities that are otherwise challenging to measure experimentally. When the density of states is large, the energy of the Leggett mode can be well below the superconducting gap making it underdamped and therefore more easily observable and more relevant for the low energy behaviour of the superconductor. This should make our results, and more generally the physics of Leggett modes, relevant for the superconducting states of flat band systems, such as the recently realized twisted bilayers, that in the metallic phase have multiple bands crossing the Fermi energy. Finally, our results suggest that the superconducting state induced in the DSM Cd_{3}As_{2} by the proximity of a standard swave superconductor might be characterized by a nonzero difference between the phases of the order parameters^{35}, making such state very interesting from a fundamental point of view and for possible technological applications.
Methods
Fabrication
Mechanical exfoliation is used to obtain flat and shiny Cd_{3}As_{2} thin flakes of thickness ~200 nm from an initial bulk ingot material^{29}, synthesized via a chemical vapour deposition method^{36}. The SQUID structure is fabricated by first depositing the Cd_{3}As_{2} thin flake on a Si/SiO_{2} substrate with a 1μmthick SiO_{2} layer. Next, ebeam lithography is used to define 300nmthick Al electrodes. Additional details about the device can be found elsewhere^{32}.
Measurements
To measure the sample resistance, an approximately 11 Hz phasesensitive lockin amplifier technique is used with an excitation current of 10 nA. To measure the differential resistance, a large direct current up to ±2 μA is added to the a.c. current. The entire device is immersed in a cryogenic liquid at a temperature of approximately 0.25 K, well below the device’s superconducting transition temperature. To measure the microwave response of the device, an Agilent 83592B sweep generator is used to generate microwaves, which are conducted through a semirigid coax cable.
Simulations
The numerical integration of the dynamical equations has been performed using the adaptive Runge–Kutta methods of order four and five.
Data availability
The data that support the findings of this study are publicly available at https://doi.org/10.6084/m9.figshare.24871635 (ref. ^{37}). Source data are provided with this paper.
Code availability
All the codes used to obtain the numerical results presented are available upon reasonable request.
References
Tsuda, S. et al. Evidence for a multiple superconducting gap in MgB_{2} from highresolution photoemission spectroscopy. Phys. Rev. Lett. 87, 177006 (2001).
Souma, S. et al. The origin of multiple superconducting gaps in MgB_{2}. Nature 423, 65–67 (2003).
Stewart, G. R. Superconductivity in iron compounds. Rev. Mod. Phys. 83, 1589–1652 (2011).
Ugeda, M. M. et al. Characterization of collective ground states in singlelayer NbSe_{2}. Nat. Phys. 12, 92–97 (2015).
Xi, X. et al. Ising pairing in superconducting NbSe_{2} atomic layers. Nat. Phys. 12, 139–143 (2016).
Leggett, A. J. Numberphase fluctuations in twoband superconductors. Prog. Theor. Phys. 36, 901–930 (1966).
Brinkman, A. et al. Charge transport in normal metal–magnesiumdiboride junctions. J. Phys. Chem. Solids 67, 407–411 (2006).
Blumberg, G. et al. Observation of Leggett’s collective mode in a multiband MgB_{2} superconductor. Phys. Rev. Lett. 99, 227002 (2007).
Klein, M. V. Theory of Raman scattering from Leggett’s collective mode in a multiband superconductor: application to MgB_{2}. Phys. Rev. B 82, 014507 (2010).
Mou, D. et al. Strong interaction between electrons and collective excitations in the multiband superconductor MgB_{2}. Phys. Rev. B 91, 140502 (2015).
Giorgianni, F. et al. Leggett mode controlled by light pulses. Nat. Phys. 15, 341–346 (2019).
Zhao, S. Z. et al. Observation of soft Leggett mode in superconducting CaKFe_{4}As_{4}. Phys. Rev. B 102, 144519 (2020).
Ota, Y., Machida, M., Koyama, T. & Matsumoto, H. Theory of heterotic superconductorinsulatorsuperconductor Josephson junctions between single and multiplegap superconductors. Phys. Rev. Lett. 102, 237003 (2009).
Beenakker, C. W. J. in LowDimensional Electronic Systems Springer Series in SolidState Sciences Vol. 111 (eds Bauer, G. et al.) 78–82 (Springer, 1992); https://doi.org/10.1007/9783642848575_7
Fu, L. & Kane, C. L. Josephson current and noise at a superconductor/quantumspinHallinsulator/superconductor junction. Phys. Rev. B 79, 161408 (2009).
Wiedenmann, J. et al. 4πperiodic Josephson supercurrent in HgTebased topological Josephson junctions. Nat. Commun. 7, 10303 (2016).
Dartiailh, M. C. et al. Phase signature of topological transition in Josephson junctions. Phys. Rev. Lett. 126, 036802 (2021).
Dartiailh, M. C. et al. Missing Shapiro steps in topologically trivial Josephson junction on InAs quantum well. Nat. Commun. 12, 78 (2021).
Shapiro, S. Josephson currents in superconducting tunneling: the effect of microwaves and other observations. Phys. Rev. Lett. 11, 80 (1963).
Domínguez, F. et al. Josephson junction dynamics in the presence of 2π and 4πperiodic supercurrents. Phys. Rev. B 95, 195430 (2017).
Barone, A. & Paternò, G. Physics and Applications of the Josephson Effect (Wiley, 1982); https://onlinelibrary.wiley.com/doi/book/10.1002/352760278X
Romeo, F. & De Luca, R. Shapiro steps in symmetric πSQUIDs. Physica C 421, 35–40 (2005).
Vanneste, C. et al. Shapiro steps on currentvoltage curves of d.c. SQUIDs. J. Appl. Phys. 64, 242–245 (1988).
Wehling, T., BlackSchaffer, A. & Balatsky, A. Dirac materials. Adv. Phys. 63, 1–76 (2014).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in threedimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131 (2001).
Chi, F. et al. Photonassisted transport through a quantum dot sidecoupled to Majorana bound states. Front. Phys. 8, 254 (2020).
Chatterjee, E., Pan, W. & Soh, D. Microwave photon number resolving detector using the topological surface state of superconducting cadmium arsenide. Phys. Rev. Res. 3, 023046 (2021).
Pan, W., Soh, D., Yu, W., Davids, P. & Nenoff, T. M. Microwave response in a topological superconducting quantum interference device. Sci. Rep. 11, 8615 (2021).
Wang, A.Q. et al. 4πperiodic supercurrent from surface states in Cd_{3}As_{2} nanowirebased Josephson junctions. Phys. Rev. Lett. 121, 237701 (2018).
Huang, C. et al. Proximityinduced surface superconductivity in Dirac semimetal Cd_{3}As_{2}. Nat. Commun. 10, 2217 (2019).
Yu, W. et al. π and 4π Josephson effects mediated by a Dirac semimetal. Phys. Rev. Lett. 120, 177704 (2018).
Shelly, C. D., See, P., Rungger, I. & Williams, J. M. Existence of Shapiro steps in the dissipative regime in superconducting weak links. Phys. Rev. Appl. 13, 024070 (2020).
Mudi, S. R. & Frolov, S. M. Model for missing Shapiro steps due to biasdependent resistance. Preprint at https://arxiv.org/abs/2106.00495 (2021).
Ng, T. K. & Nagaosa, N. Broken timereversal symmetry in Josephson junction involving twoband superconductors. Europhys. Lett. 87, 17003 (2009).
Ali, M. N. et al. The crystal and electronic structures of Cd_{3}As_{2}, the threedimensional electronic analogue of graphene. Inorg. Chem. 53, 4062–4067 (2014).
Cuozzo, J. J. et al. Leggett modes in a Dirac semimetal. Figshare https://doi.org/10.6084/m9.figshare.24871635 (2024).
Acknowledgements
This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DESC0022245 (J.J.C, W.P. and E.R.). J.J.C. also acknowledges support from the Graduate Research Fellowship awarded by the Virginia Space Grant Consortium (VSGC). The authors acknowledge the Cd_{3}As_{2} material synthesis work of D. X. Rademacher and helpful discussions with J. Schirmer. The work at Sandia is supported by a Laboratory Directed Research & Development (LDRD) project. Device fabrication was performed at the Center for Integrated Nanotechnologies, a US Department of Energy (DOE), Office of Basic Energy Sciences (BES), user facility. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc., for the US DOE’s National Nuclear Security Administration under contract DENA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the US DOE or the United States Government.
Author information
Authors and Affiliations
Contributions
J.J.C. and E.R. developed the theoretical model. J.J.C. carried out the numerical simulations. W.Y., P.D., T.M.N., D.B.S. and W.P. conceived the experiment and contributed to material growth, device fabrication, electronic transport measurements and experimental data analysis. W.P. coordinated the experiment. All authors contributed to interpreting the data. The manuscript was written by J.J.C., W.P. and E.R., with suggestions from all other authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks Alexander Golubov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Figs. 1–9 and Sections 1–7.
Supplementary information
Source Data Fig. 2
Simulation data.
Source Data Fig. 3
Simulation data.
Source Data Fig. 4
Experimental data sets and one set of simulation data.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cuozzo, J.J., Yu, W., Davids, P. et al. Leggett modes in a Dirac semimetal. Nat. Phys. (2024). https://doi.org/10.1038/s41567024024124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41567024024124
This article is cited by

SpinSelective PointContact Spectroscopy of Leggett Modes in Superconducting ProximityCoupled MgB2/La0.67Sr0.33MnO3 Nanocomposites
Journal of Superconductivity and Novel Magnetism (2024)