Microwave control of the superconducting proximity effect and minigap in magnetic and normal metals

We demonstrate theoretically that microwave radiation applied to superconducting proximity structures controls the minigap and other spectral features in the density of states of normal and magnetic metals, respectively. Considering both a bilayer and Josephson junction geometry, we show that microwaves with frequency ω qualitatively alters the spectral properties of the system: inducing a series of resonances, controlling the minigap size Emg, and even replacing the minigap with a strong peak of quasiparticle accumulation at zero energy when ω = Emg. The interaction between light and Cooper pairs may thus open a route to active control of quantum coherent phenomena in superconducting proximity structures.

In this work, we show that shining light on superconducting hybrid structures offers a way to control the proximity effect in both normal metals and magnetic materials. We discover that an oscillating electric field (t) applied transversely to the junction induces a series of resonances in the density of states, and that it can be used to control the size of the minigap E mg in both bilayer superconductor/normal-metal (SN) and Josephson (SNS) junctions. The light interaction even inverts the minigap, generating a peak of quasiparticle accumulation at E = 0 when the frequency of the light is tuned to ω = E mg . These findings give interesting prospects for transistor-like functionality via light-superconductor interactions since the density of states controls the availability of charge-and spin-carriers. Providing both analytical and numerical results, including the case of a magnetic exchange-field being present in the metal or in the superconductor, we show how the interaction between light and Cooper pairs controls the low-energy density of states, offering a new way to manipulate superconducting correlations. This may open a new pathway to active control of quantum coherent phenomena in superconducting proximity structures.

Theory
We use the time-dependent quasiclassical Keldysh-Usadel theory [26][27][28][29] to describe the superconductivity of these systems in the diffusive limit. We begin with the SN bilayer, in which case superconducting correlations leak into the normal metal via the proximity effect. The electric field (t) =ωA 0 sin (ωt) =−∂A/∂t is accounted for by the gauge field A = A 0 cos(ω t). The Usadel equation in N then reads: Here, D is the diffusion coefficient, =ĝ g x E ( , ) is the quasiclassical time-averaged Green function, E is the quasiparticle energy, α = DA /4 0 2 is a measure of the strength of the interaction with light, ω is the driving fre- . The derivation of this equation is shown in the Methods section and is valid when α ω  . We assume that the field is screened in the S region, which is taken to have a size and thickness far exceeding the superconducting coherence length ξ and penetration depth λ , allowing us to use the bulk superconducting Green function ĝ BCS there. Practically, our proposed setup could be realized by depositing a thick superconductor to partially cover a thin normal metal layer, such that the microwave field penetrates the normal layer where it is not covered by a superconductor whereas it is shielded in the superconductor (see the inset of e.g. Fig. 1). Such a lateral geometry should be well described by an effective 1D model, as done in ref. 30. The thickness of the N layer should be much smaller than the skin depth and penetration depth λ , which is experimentally feasible (typical values for the skin depth of a normal metal such as Cu is of order μm at microwave frequencies, whereas λ Nb ~ 50 nm and λ Al ~ 20 nm). From Eq. (1), we derive the following Ricatti-parametrized 31,32 Usadel equation: The Green function ĝ can then be calculated from the 2 × 2 matrix γ in spin space, the normalization matrix ≡ − γγ −  (1 ) 1  , and their tilde-conjugates defined by ). An equivalent equation for γ  can be found by tilde-conjugation of Eq. (2). In Eq. (2), we have also incorporated the possibility of a magnetic exchange field h = |h| which allows us to later consider the case of a ferromagnetic metal. The other quantities in the equation are the inelastic scattering rate δ , and the short-hand notations . From these equations, physical quantities of interest may be computed, such as the proximity-modified density of states The Usadel equation is supplemented by the Kupriyanov-Lukichev boundary conditions 33 , which are valid at low-transparency tunneling interfaces.
We now have at hand a coupled set of non-linear partial differential equations which are non-local in energy space. A numerical solution can be obtained via iteration. After discretizing the energy space, the equations are initially solved for α = 0. The procedure is then repeated with α ≠ 0 until self-consistency is achieved, using the solutions γ and γ  from the previous iteration to approximate G and F. In this way, we are able to compute the quasiclassical Green function in the presence of microwave radiation, α ≠ g ( 0), and access the density of states N/N 0 in the proximate metal.

Results and Discussion
The light-interaction with the proximity-induced condensate has a strong effect on the spectral properties of the quasiparticles. We show this in what follows, considering an SN bilayer in Fig. 1, an SNS junction in Fig. 2, and an SF bilayer in Fig. 3. In each case, we have provided results for different system parameters in order to demonstrate the robustness of the microwave radiation influence.
Starting with the SN bilayer, it is seen that by tuning the microwave frequency ω , the density of states takes on qualitatively different characteristics. At ω /Δ 0 = 0.4, there is a strong quasiparticle accumulation at E = 0, diametrically opposite to the hallmark minigap that usually is present in SN bilayers. Increasing ω gradually to ω /Δ 0 = 1.0 causes the density of states to revert to a minigap structure, albeit with a much reduced magnitude. We Scientific RepoRts | 6:38739 | DOI: 10.1038/srep38739 will later in this manuscript describe the precise condition leading to the appearance of the quasiparticle accumulation peak and its physical origin, providing also analytical results which supports the underlying explanation. In the plots, we have set α /Δ 0 = 0.1, which gives a maximum ratio of α /ω = 0.25, so that α is always considerably smaller than ω . The criterion α ω  is, however, more strictly satisfied at the higher frequency range considered in the figures.
The minigap itself is monotonically tuned with ω , as shown in Fig. 2 for the SNS case. At zero phase difference φ , the minigap is gradually reduced as ω increases, demonstrating that the driving frequency can be used to tailor the minigap size. At a finite phase difference, the light-interaction again inverts the minigap for certain frequencies, and generates a peak of quasiparticle accumulation at E = 0, similarly to the bilayer case [see Fig. 1(e)]. This can be seen in Fig. 2(c) for φ /π = 0.5. Finally, we show results for when an exchange field is present, i.e. a magnetic metal h ≠ 0, in Fig. 3, in which case the microwave field also alters the modulation of the density of states. To facilitate comparison with experiments, we note that for a typical diffusion constant of e.g. D = 7 × 10 −3 m 2 /s in Cu 34 , the requirement ω  De A /4 2 0 2 2  (having reinstated e and ħ) corresponds to ω .  0 3GHz for a modest electric field magnitude of 0.1 V/m, which is feasible. Moreover, for a superconducting gap Δ 0 = 0.5 meV, the parameter choice ħω /Δ 0 = 0.4 corresponds to a frequency ω  300GHz.
Besides the control and inversion of the minigap, another particularly noteworthy feature that all the abovementioned structures have in common is that the low-energy density of states features a series of spectral features resembling weak resonances, which vanish as soon as the microwave field is turned off (α = 0). To gain insight into the physical origin of these features seen in the density of states, we provide an analytical solution which is permissible in the ferromagnetic case, but which also seems to account for the nature of the light interaction with the superconducting condensate in the normal case (h = 0). In the weak proximity effect regime, the linearized equation governing the behavior of the spinless f s and spin-polarized f t Cooper pairs reads } 0 , as is usually the case for ferromagnets, one can solve the above equation via Fourier-transformation.
, a straight-forward calculation leads to Inserting this into our expression for ± t ( )  and performing an inverse Fourier-transformation, we end up with the final expression for f ± (E): 1 and k ± = k ± (t). We note that p ± (t) is a periodic function in t, while f BCS → 0 when E → ± ∞ . In the absence of microwave radiation (α = 0), k ± becomes independent of t, and the above simplifies to the usual result . To solve the integral Eq. (8) in the general case, we make use of the periodicity of p ± (t). The period is T = 2π /ω , so we can write the Fourier series n n n, BCS Numerically, we find that it is usually sufficient with ~15 Fourier-coefficients p n,± to obtain a perfect representation of p ± (t). Using the same procedure as above, one can also find an expression for the anomalous Green function in a Josephson geometry consisting of a superconductor/ferromagnet/superconductor trilayer. The only difference is the expression for p ± (t), which takes the form where φ is the phase difference between the superconductors. From the analytical expression, it is clear that resonances should be expected whenever = ∆ ± ω E n formally diverges, although this divergence is in practice diminished due to inelastic scattering. The weight of these resonances, i.e. the magnitude of their spectral peak, is in turn governed by the Fourier series coefficients p n which depends on the other system parameters. We note that, very recently, similar features were reported for a narrow and thin dirty superconducting strip subject to microwave radiation in ref. 13. In the present proximity-system, there is an additional minigap E mg in the system, and one might expect to have similar resonances at E = E mg ± nω . The density of states plots in Fig. 2 [see for instance (a) for ω /Δ 0 = 0.4] are consistent with this statement, demonstrating how additional spectral features, which are not present in the absence of light, occur at such excitation energies. It actually turns out that these resonances are the physical origin behind the transition from the minigap to the quasiparticle accumulation peak at E = 0. To be exact, the transition from fully gapped DOS to a strong zero-energy peak occurs precisely when ω = E mg . We show an example of this behavior at the bottom of Fig. 1. It is intriguing that the light-interaction actually induces a second, inner minigap which upon closing generates this feature, whereas the outer minigap E mg remains [see e.g. Fig. 2(a) showing a particularly clear example of the inner and outer minigaps].
The fact that the microwave radiation induces a series of weak resonances shifted with ± nω from the conventional spectral peaks (E = Δ and E = E mg in the normal metal case) has interesting consequences when a finite magnetic field splits the density of states in the superconductor 35 , since the exchange field in the superconductor h S itself produces a similar shift in the spectral peaks from Δ 0 to Δ 0 ± h S . We show the corresponding proximity-induced density of states in Fig. 4, where the combined influence of the exchange field and the light interaction produce a very rich subgap structure in the density of states. Since the superconductor in this particular case, unlike the previous systems considered in this work, has to be sufficiently thin to permit the homogeneous penetration of a magnetic field, the microwave field is not completely shielded by the superconductor and we thus here assumed that (t) is applied only to the non-superconducting part.
The most remarkable feature is nevertheless the influence of the microwave field on the minigap in the SN case, controlling its magnitude and even transforming it into a quasiparticle accumulation peak at E = 0. These results may represent the first step toward a different way to control the superconducting proximity effect, and thus the available spin-and charge-carriers, in normal and magnetic metals, by using microwave radiation. One advantage of this is the fact that the control is in situ and that the length of the system (setting the Thouless energy scale), which normally changes the minigap, does not have to be altered, which would inevitably require fabrication of multiple samples. The zero-energy peak induced by the light-interaction resembles the type of spectral feature that is characteristically seeen in the density of states of conventional SF structures due to odd-frequency superconductivity [36][37][38] , but in this case it occurs without any such pairing at all. It could also be of interest to examine the consequences of the predictions made herein with regard to conductivity experiments 39 and non-equilibrium Josephson contacts 40 .

Concluding remarks
Building on these results, an interesting future direction to explore would be the influence of light on supercurrents and the critical temperature in magnetic proximity systems, to see if the microwave radiation may be used to manipulate these quantities as well, which we intend to explore in a future work. The interaction between light and Cooper pairs could in this way open a different route to active control of quantum coherent phenomena in superconducting proximity structures.

Derivation of the Usadel equation incorporating microwave radiation. The time-dependent
Usadel equation may be written as 3 where we defined the gauge-covariant derivative ] , and the associated product E E E T T T Above, e is the electron charge, E is the quasiparticle energy, and A is the time-dependent vector potential which describes, in our case, an ac electric field E = − ∂ A/∂ t. We note that a useful property of the -product is that: We set |e| = 1 in what follows for brevity of notation and also apply the electric field perpendicularly to the junction direction, so that ∇ ⋅ = ⋅ ∇ = .
A A 0 (17) In this case, the left hand side of Eq. (12) becomes Since Moreover, the Green function satisfies the normalization condition = . ˆĝ g 1 (20) This brings us to Scientific RepoRts | 6:38739 | DOI: 10.1038/srep38739 in the Usadel equation, which in its complete form reads: The next step is the obtain the Fourier-transformed version of the above equation in energy-space. To accomplish this, we make use of similar approximations as in ref. 13. In the presence of a driving field A(T), we take into account A up to second order by deriving an equation for the harmonic Green function at zero frequency (see Appendix of ref. 13) which is essentially the time-averaged Green function. Higher order harmonic time-dependent terms in ĝ are induced by A and thus correspond to fourth order in A and higher. This approximation is valid when We now average Eq. (25) over a period 2π /ω , which means that all terms that go like e ±2iωT are removed since ĝ is the time-averaged Green function. After laborious calculations, using for instance that ous exchange fields, we find that f σσ = 0 whereas f s and f t can be non-zero. Inserting Eq. (29) into Eq. (2) in the main manuscript produces the linearized equation x 2 with f ± = f t ± f s . This governs the behavior of the spinless f s and spin-polarized f t Cooper pairs induced in the normal metal.