---
title: Second-Order Josephson Effect
url: https://www.emergentmind.com/topics/second-order-josephson-effect
type: topic
---

# Second-Order Josephson Effect

Searching arXiv for recent and foundational papers on the second-order Josephson effect.
The second-order Josephson effect denotes a current–phase relation (CPR) in which the leading superconducting supercurrent varies as $\sin 2\phi$ rather than $\sin \phi$, or in which a substantial $\sin 2\phi$ harmonic coexists with the conventional first harmonic. In Fourier form, the CPR may be written as $I(\phi)=I_1\sin\phi+I_2\sin 2\phi+\cdots$, with the second-order effect corresponding either to a dominant or pure $I_2$ term, or to physical phenomena specifically associated with the second harmonic [1401.5253], [2105.05786]. Across contemporary literature, this effect appears in several distinct settings: spin-filter Josephson junctions with purely second-harmonic transport, diffusive SIS tunnel junctions where $I_2$ emerges as a self-consistent correction, high-transparency semiconductor and topological junctions where large $I_2/I_1$ is measured, voltage-biased junctions where Higgs-mode dynamics enhance the $2\omega_J$ AC component, multicomponent superconductors with second-order intercomponent couplings, and excitonic insulators where the effective Josephson coupling is intrinsically $\cos 2\theta$ rather than $\cos\theta$ [1401.5253], [2211.07119], [2403.19445], [2402.13074], [2605.28221], [2102.10455].

## 1. Definition and current–phase structure

In the conventional Josephson effect, the supercurrent through a weak link is described by
$$
I(\phi)=I_1\sin\phi,
$$
where $\phi$ is the gauge-invariant phase difference and $I_1$ is the first-harmonic critical current [1401.5253]. More generally,
$$
I(\phi)=I_1\sin\phi+I_2\sin 2\phi+I_3\sin 3\phi+\cdots,
$$
with higher harmonics becoming relevant in the presence of transmission resonances, multiple Andreev reflections, or magnetic and spin-active interfaces [1401.5253].

The term “second-order Josephson effect” is used in two closely related senses in the literature. In one sense, it refers to a CPR whose dominant nonlinear contribution is the second harmonic $\sin 2\phi$, as in spin-filter, semiconductor, and topological Josephson junctions [1401.5253], [2211.07119], [2403.19445]. In another sense, it refers to an underlying coupling energy proportional to $\cos 2\phi$ or $\cos 2\theta$, implying a $\pi$-periodic phase dependence and two degenerate minima at phase differences differing by $\pi$, as in excitonic insulators and in second-order intercomponent couplings of multicomponent superconductors [2102.10455], [2605.28221].

A central consequence is doubled phase periodicity. A junction with $I(\phi)=I_2\sin 2\phi$ has period $\pi$ rather than $2\pi$ [1401.5253]. In AC transport, the second harmonic produces a supercurrent oscillating at $2\omega_J$ when the condensate phase evolves as $\phi(t)=\omega_J t$ under voltage bias [2402.13074]. In interference phenomena, a dominant $\sin 2\phi$ term leads to half-periodic SQUID oscillations and halved Fraunhofer-like magnetic periods [1401.5253], [2211.07119].

## 2. Microscopic origins in superconducting junctions

One microscopic route to a second-order CPR is suppression of the singlet first harmonic by spin filtering. In NbN/GdN/NbN mesa devices, GdN acts as a spin-dependent tunnelling barrier with polarization $P>80\%$ and up to approximately $89\%$ at $4.2\,\mathrm{K}$ [1401.5253]. As $P\to 1$, singlet Cooper-pair tunnelling requires both spin channels and the first-harmonic contribution $I_1\to 0$ [1401.5253]. If spin mixing at the superconducting interfaces converts singlets into equal-spin triplets, coherent transport of two triplet pairs yields directly
$$
I(\phi)=I_2\sin 2\phi,
$$
with no first-harmonic term [1401.5253]. In the Green’s-function formulation quoted there,
$$
I \propto \int dE\,\mathrm{Tr}\{\sigma_z[g_L T g_R T^\dagger]^2\},
$$
where the square denotes coherent transfer of two triplet pairs, and the amplitude $I_2$ scales as $\sim \Delta^2\sin^2\theta_m$ [1401.5253].

A different microscopic regime is the diffusive SIS tunnel junction. In a planar SIS geometry with diffusive superconducting banks, a fully self-consistent perturbation theory in the small parameter $\alpha=(\xi g_N)/\sigma\ll 1$ yields
$$
J(\delta\phi)=J_0\sin\delta\phi\,[1-4\alpha(1-\cos\delta\phi)V(T)]
=I_1\sin\delta\phi+I_2\sin 2\delta\phi,
$$
with
$$
I_1=J_0[1-4\alpha V(T)],\qquad I_2=2J_0\alpha V(T)
$$
[2105.05786]. In that formulation, the second harmonic is a small positive correction whose magnitude is explicitly temperature dependent. The same treatment shows that at second order the spectral phase $\chi(x,\omega)$ departs from the condensate phase $\phi(x)$, i.e. $\chi_2\neq \phi_2$ [2105.05786].

High transparency provides another established route. In a short single-mode ballistic SNS junction with transparency $T$, the Andreev bound-state spectrum
$$
E_\pm(\phi)=\pm \Delta\sqrt{1-T\sin^2(\phi/2)}
$$
yields a nonsinusoidal CPR whose harmonic expansion contains a sizable $I_2$ term [2211.07119]. In planar Al/InAs-quantum-well junctions, a two-component model with $I(\phi)=I_1\sin\phi+I_2\sin 2\phi$ fits multiple measurements, and analysis including loop inductance suggests that the sign of the second harmonic is negative [2211.07119]. In 1T-PtTe$_2$, the second harmonic is attributed to topological spin-momentum-locked states that promote coherent Andreev processes, with the relative phase between the $2e$ and $4e$ components tunable by magnetic field [2403.19445].

The sign of $I_2$ is not merely a fitting detail. In altermagnetic Josephson junctions, the truncated CPR
$$
J_s(\phi)\approx I_1\sin\phi+I_2\sin 2\phi
$$
can be forward- or backward-skewed depending on the sign of $I_2$ [2407.19413]. For $t_J=0.03\,t$, the fit gives $I_2/I_1\approx 0.37$ with a minus sign, while for $t_J=0.44\,t$, $I_2/I_1\approx 0.64$ with a plus sign [2407.19413]. This directly links second-harmonic physics to skewness, $\varphi$-junction behavior, and field-tuned $0$–$\pi$ transitions.

## 3. Experimental signatures and diagnostic criteria

The most direct experimental signatures of a second-order Josephson effect are half-periodicity in magnetic interference, half-integer Shapiro steps, and explicit Fourier extraction of the $\sin 2\phi$ component. In spin-filter NbN/GdN/NbN junctions, devices with GdN thickness at or above $2.0\,\mathrm{nm}$ show a measured interference period
$$
\Delta H_{\mathrm{obs}}\simeq \frac{1}{2}\Delta H_{\mathrm{conventional}},
$$
and the ratio of second-lobe width $H_2^+$ to first-lobe width $H_1$ converges to unity once the barrier’s internal flux is accounted for, matching the prediction for $I(\phi)=I_2\sin 2\phi$ [1401.5253]. No evidence of $\sin\phi$ contributions is reported in those devices [1401.5253].

In planar superconductor–semiconductor junctions, the second harmonic is identified through three complementary probes [2211.07119]. First, dc-SQUIDs exhibit half-periodic oscillations tunable by gate voltages and magnetic flux. Second, single-junction diffraction patterns show kinks near half-flux quantum. Third, microwave irradiation produces half-integer Shapiro steps. In the cited Al/InAs devices, the best fit in a symmetric SQUID configuration gives $I_2/I_1=0.4$ for both junctions, while similar signatures are also observed in Sn/InAs devices [2211.07119].

In the Ta$_2$Pd$_3$Te$_5$ asymmetric edge interferometer, half-integer Shapiro steps appear at voltages
$$
V_{m/n}=(m/n)\,\frac{hf}{2e}
$$
with $n=2$, directly signaling a non-zero $I_2$ term [2306.08478]. Those steps persist over a broad range of microwave drive powers [2306.08478]. The same work reports antisymmetric second-harmonic transport in lock-in measurements, where
$$
V=R_\omega I_{ac}\sin\omega t+R_{2\omega}I_{ac}^2\cos 2\omega t+\cdots,
$$
and $R_{2\omega}(B_z)$ is antisymmetric about $B_z=0$, closely tracking the diode-current asymmetry $\Delta I_c(B_z)$ [2306.08478].

In 1T-PtTe$_2$, the CPR is extracted from Fraunhofer critical-current oscillations under $B_z$ and Fourier-transform methods. The data exhibit both a fundamental $\Phi_0$ period from $I_1$ and a strong $\Phi_0/2$ component from $I_2$ [2403.19445]. In one junction, $\Delta I_{c,\max}\approx 34\,\mu\mathrm{A}$ at $B_y=24\,\mathrm{mT}$ implies $I_2\approx 17\,\mu\mathrm{A}$ and $I_2/I_1\approx 0.37$ [2403.19445].

These signatures are not interchangeable in evidentiary strength. One source explicitly describes the magnetic interference pattern $I_c(H)$ as the “most unambiguous probe of the CPR” [1401.5253], while another notes that half-integer Shapiro steps can also arise from vortex phase locking or non-equilibrium quasiparticles and should therefore be treated as corroborating rather than primary evidence [2211.07119]. This suggests that robust identification of a second-order Josephson effect is strongest when multiple probes concur.

## 4. Dynamical and AC manifestations

Under DC voltage bias, the phase evolves as
$$
\phi(t)=\frac{2eV}{\hbar}t\equiv \omega_J t,
$$
and a second-harmonic contribution produces current oscillations at $2\omega_J$ [2402.13074]. In transparent junctions between single-band $s$-wave superconductors, allowing for order-parameter amplitude dynamics modifies the Josephson current to
$$
I(t)\approx J[\Delta_L(t)\Delta_R(t)]\sin\phi(t),
$$
with $\Delta_j(t)=\Delta_{0,j}+\delta\Delta_j(t)$ [2402.13074]. If $\delta\Delta_j(t)=A_j\cos\omega_J t$, then
$$
I(t)=I_1\sin\omega_J t+I_2\sin 2\omega_J t+\cdots,
$$
where
$$
I_1=J\Delta_{0,L}\Delta_{0,R},\qquad
I_2=\frac{1}{2}J[\Delta_{0,L}A_R+\Delta_{0,R}A_L]
$$
[2402.13074].

The enhancement mechanism is resonant excitation of the superconducting Higgs or amplitude mode. In the effective Lagrangian,
$$
\mathcal{L}_j=(\partial_t\delta\Delta_j)^2-\omega_{H,j}^2\delta\Delta_j^2+\cdots,\qquad
\mathcal{L}_J=-2J\Delta_L\Delta_R\cos(\omega_J t),
$$
with $\omega_{H,j}=2\Delta_{0,j}$ [2402.13074]. The Josephson term acts as a periodic drive, and the response is filtered by the retarded Higgs susceptibility $\chi_{\Delta\Delta,j}(\omega)$, which peaks at $\omega=\omega_{H,j}$ [2402.13074]. Near resonance, the $2\omega_J$ component can exceed the $\omega_J$ component when the equilibrium gaps are sufficiently asymmetric and the transparency is high [2402.13074].

The same paper gives a numerical illustration in a two-dimensional model with $N_L=36$, $N_R=6$, $N_w=12$, $\mu=0$, bandwidth $\zeta=5$, transparency $\mathcal{T}=0.4\zeta$ and damping $\Gamma=0.02\zeta$, where the $2\omega_J$ current eventually exceeds the $\omega_J$ component when $\Delta_{0,L}/\Delta_{0,R}\lesssim 0.1$ [2402.13074]. This is a dynamical second-harmonic dominance rather than a static pure $\sin 2\phi$ CPR.

AC manifestations also arise in non-Higgs contexts. In diffusive SIS junctions, the small $\sin 2\delta\phi$ term shifts the phase of maximum critical current away from $\delta\phi=\pi/2$ and can produce half-harmonic Shapiro steps [2105.05786]. In excitonic insulators, by contrast, the phase equation under strong drive gives $\dot\theta=-\phi_a$, and because the current is $J\propto \sin 2\theta$, the AC Josephson frequency remains $f=2eV/h$ despite the $\pi$-periodic phase dependence [2102.10455].

## 5. Alternative formulations beyond conventional superconducting weak links

The second-order Josephson concept extends beyond ordinary Cooper-pair tunnelling through a weak link. In electron–hole bilayers with excitonic order formed by orbitals of opposite parity, integrating out fermions to second order in the interlayer tunnelling yields an effective phase Lagrangian
$$
L=\frac{1}{2}\nu\left[-(\partial_t\theta+\phi_a)^2+v_g^2(\nabla\theta-A_a)^2-\frac{\Delta_p^2}{D}\cos\bigl(2(\theta-dA_z)\bigr)\right]
$$
[2102.10455]. The free energy therefore contains $-\cos 2\theta$, and the interlayer current obeys
$$
J_z(\theta)=\frac{\nu}{D}\Delta_p^2\sin 2\theta\equiv J_c\sin 2\theta
$$
[2102.10455]. The minima at $\theta=0,\pi$ are degenerate, and a voltage pulse with $\int \phi_a dt\simeq \pi$ switches between them [2102.10455]. In Ta$_2$NiSe$_5$, reflection-symmetry analysis is argued to place the system in precisely this class [2102.10455].

In multicomponent superconductivity, second-order Josephson couplings occur between condensates rather than across a spatial junction. In the three-component Ginzburg–Landau model compatible with the 3Q pair-density-wave state, the free-energy density includes terms
$$
-\eta_{jk}n_jn_k\cos[2(\theta_k-\theta_j)]
$$
with no conventional $\cos(\theta_k-\theta_j)$ coupling [2605.28221]. These couplings are invariant under both time reversal and $\pi$-phase flip, but in frustrated regimes the ground state can spontaneously break both symmetries [2605.28221]. The theory identifies five distinct ground states: an 8-fold degenerate frustrated state and four 4-fold degenerate non-frustrated phase-locked states, and numerical analysis finds a Higgs–Leggett mode unique to the frustrated region [2605.28221]. This is a second-order Josephson effect in the intercomponent phase sector.

A related but distinct framework is the bilayer XY model with second-order Josephson coupling
$$
H=-J\sum_{\ell=1,2}\sum_{\langle ij\rangle}\cos(\theta_{\ell,i}-\theta_{\ell,j})
-J_2\sum_i\cos[2(\theta_{1,i}-\theta_{2,i})]
$$
[2507.19401]. The term $\cos[2(\Delta\theta)]$ pins the interlayer phase difference to $0$ or $\pi$, leaving a common $U(1)$ and a discrete $\mathbb{Z}_2$ symmetry [2507.19401]. In the dual formulation, the second-order Josephson term maps to a two-dimensional noncompact $U(1)$ gauge field, and the resulting theory predicts that the only transition out of the low-temperature ordered phase is an Ising transition driven by condensation of $\mathbb{Z}_2$ domain-wall loops [2507.19401]. The paper argues that point-defect-based Coulomb-gas methods miss the relevant excitations in this regime [2507.19401].

## 6. Device consequences, nonreciprocity, and circuit applications

A large or dominant second harmonic strongly alters device functionality. In the Ta$_2$Pd$_3$Te$_5$ edge interferometer, the generalized interferometric CPR includes both first and second harmonics on each branch,
$$
I(\phi,\Phi)=\sum_{n=1}^2\left[I_n\sin\!\left(\phi+2\pi(n-1)\Phi/\Phi_0\right)
+\alpha_n I_n\sin\!\left(2\phi+4\pi(n-1)\Phi/\Phi_0\right)\right],
$$
and the second-harmonic term is an important element in generating a Josephson diode effect [2306.08478]. The reported diode efficiency reaches approximately $45\%$ in one device and up to approximately $73\%$ in another at $B_z\approx 8.4\,\mathrm{mT}$, with switching powers of approximately $6.4\,\mathrm{pW}$ and approximately $0.56\,\mathrm{pW}$, respectively [2306.08478].

In 1T-PtTe$_2$, the second harmonic is directly correlated with a large intrinsic Josephson diode effect, and the relative phase between the $2e$ and $4e$ harmonics is tunable with in-plane magnetic field [2403.19445]. The free-energy expansion quoted there,
$$
F[\phi]=-\alpha_1\cos\phi-\alpha_2\cos 2\phi-\gamma B_y\sin\phi+\cdots,
$$
implies a field-tunable phase shift of the second-harmonic component [2403.19445]. The diode efficiency reaches approximately $32\%$ at $B_y\approx 24\,\mathrm{mT}$ [2403.19445].

In altermagnetic junctions, electric and Zeeman fields tune not only the magnitude of $I_2$ but also its sign and the skewness of the CPR [2407.19413]. The ratio $a_2(t_J)\equiv I_2/I_1$ crosses unity near $t_J\simeq 0.13\,t$, where the junction becomes a $\varphi$-junction, and can be tuned by gate potential $V_z$ or in-plane field $B_y$ [2407.19413]. The same work reports field-induced $0$–$\pi$ transitions and a regime where the critical current increases with Zeeman field, which it describes as surprising because supercurrents are typically suppressed by magnetic fields [2407.19413].

The second harmonic also affects superconducting circuits more broadly. In a Josephson traveling-wave parametric amplifier with generalized CPR
$$
I(\phi)=I_1\sin\phi+I_2\sin 2\phi,
$$
the Josephson potential becomes
$$
U(\phi)=-I_1\cos\phi-\frac{I_2}{2}\cos 2\phi
$$
[2502.00804]. Numerical simulations for an array of $N=990$ cells with $C_J=200\,\mathrm{fF}$, $R_J=20\,\mathrm{k}\Omega$, $C_g=24\,\mathrm{fF}$, $L_g=120\,\mathrm{pH}$, pump frequency $7\,\mathrm{GHz}$ and signal frequency $6\,\mathrm{GHz}$ show that the weighting of the second harmonic changes the gain profile, phase-space structure, and stability; the maximum gain reaches approximately $13\,\mathrm{dB}$ without dispersion engineering, peaking near $g=J_{c2}/J_{c1}\approx -0.6$ [2502.00804]. A plausible implication is that second-harmonic engineering is relevant not only to equilibrium CPR physics but also to nonlinear microwave design.

## 7. Conceptual distinctions, limitations, and open issues

The second-order Josephson effect should not be conflated with a single universal microscopic mechanism. In spin-filter junctions, the effect is linked to suppression of singlet tunnelling and coherent transfer of two triplet pairs [1401.5253]. In diffusive SIS junctions, it is a perturbative self-consistent correction to conventional tunnelling theory [2105.05786]. In high-transparency semiconductor and topological junctions, it is associated with Andreev-bound-state structure and coherent higher-order transport [2211.07119], [2403.19445]. In Higgs-driven AC transport, it arises from nonequilibrium order-parameter dynamics [2402.13074]. In excitonic insulators and multicomponent superconductors, it is built into the effective phase energy as a fundamental $\cos 2\theta$ coupling [2102.10455], [2605.28221].

A recurring point of contention concerns interpretation of the second harmonic as coherent $4e$ transport. One source explicitly defines the $\sin 2\phi$ term as a distinct four-electron process in 1T-PtTe$_2$ [2403.19445], while another frames large $I_2$ in planar semiconductor junctions more cautiously, stating that the microscopic origins remain to be understood and that alternative explanations can account for some but not all evidence [2211.07119]. This suggests that the language of “quartets” or “$4e$ transfer” is well motivated in some settings but not yet universally established across all large-$I_2$ experiments.

Another important distinction concerns robustness. Standard theory predicts higher harmonics to be extremely sensitive to barrier thickness and temperature, especially near $0$–$\pi$ transitions. The spin-filter experiments report instead a purely second-harmonic CPR that is insensitive to barrier thickness beyond approximately $2.0\,\mathrm{nm}$ and persists from $4.2\,\mathrm{K}$ to at least $15\,\mathrm{K}$, which is taken to imply that standard spin-inactive tunnelling theory is not applicable to spin-dependent barriers [1401.5253]. By contrast, in self-consistent SIS theory the second harmonic is small and parametrically controlled by $\alpha$ or $\gamma=G_N/G_D(T)$ [2105.05786].

The broader significance of the second-order Josephson effect lies in its combination of doubled periodicity, symmetry-selective coupling, and nonlinear tunability. The literature connects it to intrinsically protected $\phi=0$–$\pi$ qubits, built-in phase bias elements, rapid single-flux-quantum logic, protected phase slips, superconducting diodes, low-power memory, higher-order coherent transport, and spectroscopic access to collective modes [1401.5253], [2306.08478], [2403.19445], [2102.10455]. A plausible implication is that the second harmonic has moved from being a small correction in Josephson phenomenology to a design principle spanning superconducting weak links, multicomponent condensates, and correlated electron-hole systems.

Source: https://www.emergentmind.com/topics/second-order-josephson-effect