Odd-frequency superconductivity

This article reviews odd-frequency (odd-$\omega$) pairing with a focus on superconducting systems. Since Berezinskii introduced the concept of odd-frequency order in 1974 it has been viewed as exotic and rarely occurring in nature. A view is presented in which the Berezinskii state is in fact a ubiquitous superconducting order that is both nonlocal and odd in time. This state appears under quite general circumstances in many physical settings including bulk materials, heterostructures, and dynamically driven superconducting states, and it is therefore important to understand the nature of odd-$\omega$ pairing. Presented are the properties of odd-$\omega$ pairing in bulk materials, including possible microscopic mechanisms, and definitions of the odd-$\omega$ superconducting order parameter and the unusual Meissner response of odd-frequency superconductors are discussed. Also presented is how odd-$\omega$ pairing is generated in hybrid structures of nearly any sort and its relation to Andreev bound states, spin-polarized Cooper pairs, and Majorana states is focused on. How odd-$\omega$ pairing can be applied to nonsuperconducting systems such as ultracold Fermi gases, Bose-Einstein condensates, and chiral spin nematics is overviewed. Because of the growing importance of dynamic orders in quantum systems also discussed is the emergent view that the odd-$\omega$ state is an example of phase coherent dynamic order. The recent progress made in understanding the emergence of odd-$\omega$ states in driven superconducting systems is summarized. A more general view of odd-$\omega$ superconductivity suggests an interesting approach to this state as a realization of the hidden order with inherently dynamic correlations that have no counterpart in conventional orders discussed earlier. The progress made in this rapidly evolving field is reviewed and an illustration of the ubiquity of the odd-$\omega$ states and the potential for future discoveries of these states in a variety of settings are given. The general rules or design principles, to induce odd-$\omega$ components in various settings, using the $S{P}^{*}O{T}^{*}$ rule, are summed up. Since the pioneering prediction of odd-$\omega$ superconductivity by Berezinskii, this state has become a part of every-day conversations on superconductivity. To acknowledge this, the odd-$\omega$ state is called a Berezinskii pairing as well in this article.

Figure 1

(a) Time reversal $\widehat{T}$ and (b) time permutation ${\widehat{T}}^{*}$ for two times. An odd-frequency superconductor has an order parameter that changes sign under time permutation ${\widehat{T}}^{*}$.

Figure 2

Splitting of pairing states for a pairing potential with $D_{4h}^{\mathrm{II}}$ symmetry. (a) $p$-wave spin singlet and (b) $s$-wave spin triplet. Adapted from [87].

Figure 3

Transition temperature $T_c$ for $p$- and $d$-wave spin-singlet pairing as a function of $\eta$. Adapted from [85].

Figure 4

Illustration of the composite Cooper pairs as a condensate that is occurring in the odd-$\omega$ state. The upper panel illustrates the nature of a composite fermion = fermion + boson (flux tubes as was shown to exist in the quantum Hall effect). Adapted from [67]. The lower panel illustrates composite Cooper pairs = Cooper pair + boson (spin or lattice) that condenses in the odd-$\omega$ state. Composite pairs are a natural extension of the concept of composite particles to Cooper pairs.

Figure 5

A nonexhaustive hierarchy of composite superconducting condensates is shown. We start with the conventional paired states as an even-$\omega$ state where the pairing correlator taken at equal time is proportional to the order parameter one can use in the Ginzburg-Landau description. One can extend the notion of superconducting states to the $2e+1$ composite boson condensate. This would correspond to the order parameter as a first derivative of the odd-$\omega$ amplitude. This line describes Berezinskii composite pairs as discussed in the text. One can continue with the process by taking higher order derivatives. The next step would be a paired state with a $2e$ pair and two bosons that would correspond to a second-in-time derivative and therefore to even-$\omega$ pairing. The third line corresponds again to the odd-$\omega$ state with three bosons attached to a pair, and so forth. The higher the order of the correlators, the more fragile the condensate will be. The situation is thus similar to the case of the fractional quantum Hall effect: the higher the fraction, the more fragile the fractional quantum Hall effect state is. We used the general labels even-$\omega$ and odd-$\omega$ and the specific labels based on $S{P}^{*}O{T}^{*}$ classification to underscore the fact that change in the time parity index leads to a new class of superconductors.

Figure 6

Schematic of a driven superconducting system with a 2D superconducting region lying between two insulating slabs each capped by a conducting electrode configured in such a way as to generate an electric field. The ac voltage acts as a time-dependent drive. Such a device could be realized by sandwiching a thin-film superconductor, like Pb and other superconductors, between two insulating wafers. Adapted from [204].

Figure 7

In the left (right) column, we plot the even-$\omega$ (odd-$\omega$) terms of the real part of the Wigner transform of the anomalous part of the Green’s function, $⟨{\widehat{F}}^{\mathrm{R}}(\omega ,T=\pi /2{\mathrm{\Omega }}_0)⟩$, in black (solid), where we have taken the average value of ${\widehat{F}}^{\mathrm{R}}(\mathbf{k};\omega ,T=\pi /2{\mathrm{\Omega }}_0)$ at $|\mathbf{k}|=k_{\mathrm{F}}^{(a)}$ and $|\mathbf{k}|=k_{\mathrm{F}}^{(b)}$. In each case we also plotted the parity-preserving terms (green dashed) and parity-reversing terms (red dash-dotted). (a) The diagonal component for band $a$, (b) the diagonal component for band $b$, and (c) the interband component. (d) The components of the drive, plotted in the time domain over a full period, the green vertical line denotes the time $T_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}\equiv T=\pi /2{\mathrm{\Omega }}_0$ at which all plots in this figure are evaluated. The parameters used to describe the driven multiband superconductor in this case are the effective masses $m_a=0.5{\AA }^{-2}/\mathrm{eV}$ and $m_b=1{\AA }^{-2}/\mathrm{eV}$; chemical potentials ${\mu }_a={\mu }_b=2\mathrm{eV}$; $s$-wave gaps ${\mathrm{\Delta }}_{aa}=2\mathrm{meV}$, ${\mathrm{\Delta }}_{bb}=7\mathrm{meV}$, ${\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_{ba}=0$, consistent with ${\mathrm{MgB}}_2$ ([47]); interband scattering $\mathrm{\Gamma }=10\mathrm{meV}$; dissipation described by $\eta =1\mathrm{meV}$; and a drive $U(t)=U_0\cos ({\mathrm{\Omega }}_{0}t)$ with $U_0=10\mathrm{meV}$ and ${\mathrm{\Omega }}_0=1\mathrm{meV}$ (242 GHz). Adapted from [205].

Figure 8

(a), (b) The 2D DOS computed using effective masses $m_a=0.5{\AA }^{-2}/\mathrm{eV}$ and $m_b=1{\AA }^{-2}/\mathrm{eV}$; chemical potentials ${\mu }_a={\mu }_b=2\mathrm{eV}$; $s$-wave gaps ${\mathrm{\Delta }}_{aa}=2\mathrm{meV}$, ${\mathrm{\Delta }}_{bb}=7\mathrm{meV}$, ${\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_{ba}=0$, consistent with ${\mathrm{MgB}}_2$ ([47]); interband scattering, $\mathrm{\Gamma }=10\mathrm{meV}$; dissipation described by $\eta =0.1\mathrm{meV}$; and a drive with $U_0=10\mathrm{meV}$, and ${\mathrm{\Omega }}_0=1\mathrm{meV}$ (242 GHz). In both panels we show the case for no drive in black (solid), and the cases with the drive at times $T_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}\equiv T=0$ and $T=\pi /2{\mathrm{\Omega }}_0$ in green (dashed) and red (dash-dotted), respectively. (a) We focus on the states near the Fermi surface, in (b) we focus on the range of energies near the crossing of the two bands at which we find the driven DOS at $T=0$ possesses two peaks shifted from the avoided crossing at $E_0$ by $\pm {\mathrm{\Omega }}_0/2$. (c), (d) The 3D DOS plotted for the same parameters as in (a) and (b). Note that the main difference is that in 3D the driven DOS at $T=0$ is slightly suppressed relative to the undriven DOS (see inset). (e) We plot the spectrum of the two band superconductor given by ${ϵ}_{\pm }(\mathbf{k})$. The horizontal gray line denotes the avoided crossing (see inset) at $E_0$, due to the finite interband scattering $\mathrm{\Gamma }$. (f) We show the drive plotted in the time domain over a full period, the green vertical line at $T=0$ denotes the beginning of the period where the drive has maximum amplitude, while the red line denotes $T=\pi /2{\mathrm{\Omega }}_0$ where the drive amplitude is zero. The horizontal line (dashed) shows $U_0=0$. Adapted from [205].

Figure 9

(a) Diamagnetic Meissner response for a ring with conventional superconducting correlations, and (b) paramagnetic Meissner response that can occur for a ring with odd-$\omega$ superconducting correlations. In the event of a paramagnetic supercurrent response, the odd-$\omega$ superconductor experiences an attractive force to the underlying magnet, causing superconducting antilevitation. Adapted from [137].

Figure 10

Results for the DOS and Cooper pair symmetry near the vortex core of a conventional $s$-wave BCS superconductor. (a) Normalized local DOS around the vortex at $E=0$. The center of the vortex is situated at $x=y=0$. Spatial dependences of (b) even-$\omega$ singlet (ESE) and (c) odd-$\omega$ singlet (OSO) correlations at $E=0$. $f_j$ corresponds to the different angular momentum components of the anomalous Green’s function $f={\sum }_n{f}_n{e}^{in\theta }$, and all have a spin-singlet symmetry. Adapted from [222].

Figure 11

Schematic of the conventional Josephson junction is shown. Interlead tunneling induces diagonal pairing amplitudes $F_{LL},F_{RR}\sim T_{\mathrm{tun}}^2$, $T_{\mathrm{tun}}$ being the tunneling matrix element. We also indicate the presence of odd-$\omega$ interlead pairing amplitude $F_{LR}\sim T_{\mathrm{tun}}({\mathrm{\Delta }}_L-{\mathrm{\Delta }}_R)\omega$ that is odd-$\omega$, odd under $L\to R$ permutation, while preserving the product $S{P}^{*}O{T}^{*}=-1$. In addition to the conventional Cooper pairs present in each of the leads tunneling induces the interlead superconducting correlations. Traditional textbook analysis predicts the corrections to the intralead pairing and explains the Josephson effect as an induction of the $T_{\mathrm{tun}}^2\mathrm{Re}\{{\mathrm{\Delta }}_R^{*}{\mathrm{\Delta }}_L\}$ term in free energy. The interlead pairing amplitude is much larger at small tunneling amplitudes $T_{\mathrm{tun}}$. From [18].

Figure 12

(a) The $H\text{−}{T}_{\mathrm{temp}}$ phase diagram for the even-$\omega$ superconducting order parameter in ${\mathrm{MgB}}_2$. Dashed (solid) lines indicate a second (first) order phase transition. (b) $H\text{−}{T}_{\mathrm{temp}}$ phase diagram for the odd-$\omega$ superconducting order parameter. The insets in both (a) and (b) show the Matsubara frequency dependences of the order parameters for different magnetic field values. The color bar max and min values are 7 and 0 meV for the even-$\omega$ amplitude and 0.3 and $0.0\mathrm{meV}$ for the odd-$\omega$ amplitude. (c) The band-resolved field dependence of the even-$\omega$ and odd-$\omega$ order parameters at low temperature. The lines correspond to the maximum values in Matsubara space of the momentum averaged superconducting order parameters on each band, which is equivalent to the peaks in the insets of (a) and (b). The two upper lines, as measured from $H=0$, show ${\mathrm{\Delta }}_e$, whereas the two lines starting from zero amplitude show $10\times {\mathrm{\Delta }}_0$. Adapted from [12].

Figure 13

Qualitative dependence of the most stable superconducting pairing symmetries on the degree of one dimensionality and spin or charge fluctuations. Adapted from [188].

Figure 14

Spatial dependence of the pair potential normalized against its bulk (solid line) and the even-$\omega$ spin-singlet pair amplitudes $E_s(x)$ ($s$-wave channel, dash-dotted line) and $E_{p_x}(x)$ ($p$-wave channel, dash-dotted line) for an $S/N$ ballistic bilayer. The $x$ axis extends into the superconducting layer. The odd-$\omega$ pair amplitudes in the corresponding angular momentum channels are denoted $O_s(x)$ and $O_{p_x}(x)$ and are shown as dashed lines. The parameter $Z$ quantifies the junction transparency, with $Z=0$ corresponding to a perfect interface and $Z\gg 1$ corresponding to the tunneling limit. (a) The superconductor is of the conventional $s$-wave BCS type whereas in (b) the superconductor is of the $p_x$ type. The ballistic superconducting coherence length is $\xi =v_F/\mathrm{\Delta }$. Adapted from [200].

Figure 15

The arrows illustrate the trajectories of scattered quasiparticles at the interface between a diffusive normal metal and an unconventional superconductor with a (a), (b) $d$-wave symmetry and (c), (d) a $p$-wave symmetry. The angle $\alpha$ denotes the angle between the normal to the interface and the crystal axis in the $d$-wave case and the lobe direction in the $p$-wave case. The angle $\varphi$ denotes the injection angle of quasiparticles as measured from the $x$ axis. Adapted from [219].

Figure 16

(a) Critical supercurrent as a function of temperature for different separation distances between the superconducting electrodes. (b) Schematic setup of the studied devices, consisting of a lateral Josephson junction with two superconducting electrodes deposited on the half-metal ${\mathrm{CrO}}_2$. (c) Scanning electron micrograph of a typical final device. (d) Illustration of the alignment of the current direction with respect to the magnetization axes: $I$ is the current, $H$ is the applied magnetic field, and $M$ is the magnetization. Adapted from [123].

Figure 17

Starting out with a conventional $s$-wave even-$\omega$ superconductor described by a wave function ${\psi }_0$, a proximity coupling to a homogeneous diffusive ferromagnet creates short-ranged odd-$\omega$ Cooper pairs with a wave function ${\psi }_{\text{short}}$. These rapidly decay in an oscillatory manner inside the magnetic region. In the presence of a magnetic inhomogeneity at the interface, long-ranged odd-$\omega$ Cooper pairs ${\psi }_{\mathrm{long}}$ which are spin polarized (triplet) emerge which penetrate a much longer distance compared to ${\psi }_{\text{short}}$. Adapted from [139].

Figure 18

(a) The sample structure on which the STM measurements were performed: an Au/Nb/Ho/Nb multilayer. (b) The magnetization of Ho at zero field (remanent magnetization $M_r$, red line) and with the set field $H$ switched on (blue line). The vertical (black) lines separate different magnetic phases of Ho: a bulk helix (region 1), coexisting helix and $F$ component (region 2), and $F$ state (region 3). (c), (d) Typical subgap features obtained in the normalized conductance. Adapted from [63].

Figure 19

(a) Setup used for observation of the paramagnetic Meissner effect due to odd-$\omega$ triplets: an Au/Ho/Nb trilayer exposed to an external field $\mathit{B}$. Low-energy muons injected in Au provided information about the local magnetization profile. (b) Experimental measurement and theoretical fit to the local magnetization signal $B_{\mathrm{loc}}$ as well as the theoretically computed spatial distribution of the shielding current density $J_x$ throughout the system. From [64].

Figure 20

(a) Schematic measurement setup used by [216]: two perpendicularly linearly polarized lights emerging from the fiber become circularly polarized and focus on the sample using a lens. The electric field $E$ penetrates a short distance $\ll d_S$ into the superconductor. (b) Kerr effect measurement of an Al/(Co-Pd) bilayer system with a 50 nm Al sample. Adapted from [216].

Figure 21

Suggested expermiental setup for electrically controlled odd-$\omega$ pairing in a double-dot three-terminal device. (a) The quantum dots have level positions ${ϵ}_{L,R}$ and are contacted by a superconducting lead $S$ and two normal leads $L$ and $R$. (b) Illustration of a local Andreev reflection process where the Cooper pair electrons tunnel into a normal lead through one dot. (c) Nonlocal Andreev reflection (AR) where the two electrons making up the Cooper pair tunnel into different leads. The blue lines refer to the pair amplitudes $F_{LL}$ and $F_{RR}$ in the case of local AR whereas the red lines illustrate the nonlocal amplitude $F_{LR}$ which is odd $\omega$ on the resonance point ${ϵ}_L={ϵ}_R$. Adapted from [41].

Figure 22

(a) Scanning electron microscopy image of the $\mathrm{Al}/{\mathrm{Al}\mathrm{O}}_x/\mathrm{Fe}$ sample used by [112] along with the measurement scheme. (b) Differential conductance spectrum for the structure at zero magnetic field ($B=0$), together with a theoretical fit (red line). (a), (b) Adapted from [112]. (c) Differential conductance ($dI/dV$) measurements normalized to the normal-state value of a 100 nm NbN/3 nm GdN/30 nm TiN tunnel junction demonstrating the evolution of a zero-energy peak with decreasing temperature.) Adapted from [165].

Figure 23

(a) Andreev bound state (ABS) formed at the interface between a normal metal and an $s$-wave superconductor separated by a magnetic barrier, e.g., a ferromagnetic insulator. The spin-dependent phase shifts arising due to the magnetic barrier give rise to an interface state which appears at zero energy for strong enough phase shifts. (b) ABS formed at the interface between a normal metal and a $d$-wave superconductor separated by a nonmagnetic barrier, e.g., an insulator. The electronlike and holelike excitations experience different signs of the pair potential $\mathrm{\Delta }(\mathbf{k})$ upon scattering, leading to the formation of a zero-energy state. In both (a) and (b), the bound state at the interface is accompanied by a strong increase in the magnitude of the odd-$\omega$ correlations, quantified via the anomalous Green’s function $f$. The dashed line indicates how quasiparticles are Andreev reflected back toward the interface by the pair potential $\mathrm{\Delta }(\mathbf{k})$ after penetrating a distance $\sim {\xi }_S$ into the superconductor.

Figure 24

Two Majorana modes localized at the end of a wire with a finite hybridization ${\mathrm{\Gamma }}_{\mathrm{hyb}}$. From [111].

Figure 25

Schematic setup of multiple superconducting wires hosting Majorana fermions on their edges. The fermions are represented by operators ${\mu }_a^i$ (denoted ${\gamma }_a^i$ in the main text) where $i$ is the wire index and $a=1$, 2 label the two edges of each wire. The dashed line illustrates possible couplings between Majorana fermions on neighboring wires. Adapted from [111].

Figure 26

Phase diagram of the normal and pairing states of collection of Majorana states is shown. For the large coupling between the ends of the wire, the system will have strong pairing fluctuations in the odd-$\omega$ channel. The phase diagram is drawn for the mean-field solution of the pairing state. As explained, Majorana states have a strong propensity to form the odd-$\omega$ state. The solid green line shows the critical temperature $T_c\propto g^2$. From [111].

Figure 27

The normal ($g_{\uparrow \uparrow }$) and anomalous $(f_{\uparrow \uparrow })$ Green’s functions plotted at the superconducting interface (lattice position $j=10$ in this model) for $E=0$ as a function of the exchange field strength $V_{\mathrm{ex}}$ normalized to the hopping amplitude $t$. At the topological phase transition $V_{\mathrm{ex}}=V_c$, where the Majorana fermion emerges, the odd-$\omega$ amplitude has a sharp increase. Adapted from [16].

Figure 28

Schematic usage of the Majorana STM. The tip contains a Majorana bound state $\gamma$ which probes odd-$\omega$ superconductivity in the quantum dot (QD) via a tunnel coupling. Superconductivity exists in the QD via proximity to a host $s$-wave superconducting material, and the pairing can be tuned between even-$\omega$ and odd-$\omega$ via an external magnetic field. Adapted from [120].

Figure 29

Critical temperature $T_c$ vs the scaled fermion-phonon coupling $\gamma$. Solid lines: critical temperature for $s$-wave odd-$\omega$ pairing. Dashed lines: critical temperature for $p$-wave pairing. We have defined $c_S$ as the phonon speed of sound and $\xi =\sqrt{{\xi }_0^2+\gamma /12q_F^2}$, where ${\xi }_0$ is the boson coherence length. Adapted from [119].