Pressure-Induced 0-\pi Transitions and Supercurrent Crossover in Antiferromagnetic Weak Links

# Pressure-Induced 0−π Transitions and Supercurrent Crossover in Antiferromagnetic Weak Links

## Abstract

We compute self-consistently the Josephson current in a superconductor-antiferromagnet-superconductor junction using a lattice model, focusing on transitions occurring when the width of the antiferromagnetic region changes from an even to an odd number of lattice sites. Previous studies predicted transitions when alternating between an even and an odd number of sites for sufficiently strong antiferromagnetic order. We study numerically the magnitude of the threshold value for this to occur, and also explain the physics behind its existence in terms of the phase-shifts picked up by the quasiparticles constituting the supercurrent in the antiferromagnet. Moreover, we show that this threshold value allows for pressure-induced transitions by destroying the antiferromagnetic nesting properties of the Fermi surface, a phenomenon which has no counterpart in ferromagnetic Josephson junctions, offering a new way to tune the quantum ground state of a Josephson junction without the need of multiple samples.

###### pacs:
73.20.-r 73.40.Gk 74.25.F- 85.25.Cp

Introduction. The study of the interplay between superconductivity and magnetism has been of considerable interest in condensed matter physics over the last decades. Phenomena such as the -transition (1) in ferromagnetic Josephson junctions has received much attention both from a fundamental quantum physics point of view in addition to being suggested as a potential basis for qubits. (2). While most of the focus in the above context has been on ferromagnetic (F) order, antiferromagnetic (AF) Josephson junctions are also of fundamental interest, due to the close relationship between the superconducting (S) phase and the antiferromagnetic phase in for instance high-temperature cuprate and iron-pnictide superconductors. Superconductivity and antiferromagnetism spin-density wave states may even coexist in the superconducting pnictides. (3) Similar to SFS junctions, antiferromagnetic Josephson junctions (SAFS) have been predicted to display -transitions (4). However, for SAFS these transitions display a high sensitivity to the exact number of atomic layers (even vs. odd number) in the antiferromagnet. Ref. (5) reported that an antiferromagnetic Josephson junction is in a -state for an odd number of layers, while it is in the -state for an even number of layers provided that the antiferromagnetic order is much stronger than the superconducting order. An even-odd effect has also been observed in Josephson junctions with magnetic impurities in the middle layer. (6)

In this Rapid Communication, we report on a novel aspect of antiferromagnetic Josephson junctions which allows for control over transitions within a single sample in a way which has no counterpart in SFS structures. We first compute numerically the threshold value for the antiferromagnetic order parameter at which the even-odd effect occurs. Below this threshold, even and odd junctions behave qualitatively similar, both displaying a monotonic decay of the supercurrent with superimposed small-scale oscillations, but without any sign-change of the critical current. As a result of this, we show that it is possible to obtain pressure-induced transitions in antiferromagnetic Josephson junctions. Namely, applying pressure alters the Fermi level by moving it away from the van Hove singularity of the system, which simultaneously destroys the Fermi surface nesting properties giving antiferromagnetism in the first place. In this way, the pressure controls the magnitude of the staggered order parameter which triggers a transition once it drops below the aforementioned threshold value. This effect has no counterpart in conventional SFS junctions, since ferromagnetism does not rely on Fermi-surface nesting. The antiferromagnetic order thus offers a novel mechanism for controlling the quantum mechanical ground state of a Josephson junction.

Theory. Our approach closely follows that of Ref. (10). The system in question consists of an itinerant antiferromagnet sandwiched between two conventional -wave superconductors. The interfaces are cut in the (110) direction and are considered transparent. We model the antiferromagnetic region as consisting of two square sublattices shown in Fig. 1, where the A- and B-lattice have oppositely preferred spin directions. The mean-field Hamiltonian then reads (see for instance Ref. (4))

 ^H =−t∑⟨ij⟩σ^c†iσ^cjσ+∑i(Δi^c†i↑^c†i↓+H.c.)−μ∑iσ^c†iσ^ciσ +∑imi(^c†i↑^ci↑−^c†i↓^ci↓), (1)

where creates an electron with spin on lattice site , is the chemical potential, denotes the nearest-neighbor hopping integral, and and are the magnetic and superconducting order parameters, respectively.

We use the Bogoliubov-Valatin transformations , where , to obtain the standard Bogoliubov-de Gennes equations (BdG) (10)

 ∑j(HijσΔijσΔ∗jiσ−H∗ijσ)(unσ(j)vn¯σ(j))=En(unσ(i)vn¯σ(i)). (2)

Here, where for spin up and down, and and are the Kroenecker deltas for onsite and nearest-neighbor sites, respectively. The off-diagonal block for -wave symmetry.

Due to the crystal periodicity along the interface, we can Fourier transform the BdG-equations to make the problem effectively one-dimensional. Using the method described in Ref. (10), we get the one-dimensional BdG-equations for the (110)-direction (10)

 −μuA(B)i,σ(ky)−2tcosky√2e±iky/√2[uB(A)i,σ(ky)+uB(A)i∓1,σ(ky)] +σmA(B)iuA(B)i,σ(ky)−ΔA(B)jvA(B)i,¯σ(ky)=EuA(B)i,σ(ky), (3) μvA(B)i,¯σ(ky)+2tcosky√2e±iky/√2[vB(A)i,¯σ(ky)+vB(A)i∓1,¯σ(ky)] +σmA(B)ivA(B)i,¯σ(ky)−ΔA(B)j∗uA(B)i,σ(ky)=EvA(B)i,¯σ(ky). (4)

The denotes the A(B) sublattice while the order parameters are defined as and . These are determined self-consistently through (10)

 nA(B)iσ =∑n,ky[|uA(B)n,i,σ(ky)|2f(En,ky,σ) +|vA(B)n,i,σ(ky)|2f(−En,ky,σ)], (5) ΔA(B)i =−Vi∑n,ky,σun,i,σ(ky)v∗n,i,¯σ(ky)tanh(βEn,ky,σ2), (6)

where is the Fermi-Dirac distribution. Here, and denote the spatially dependent coupling constants for the magnetization and superconducting pairing, respectively. We model the coupling contants as , , where is the Heaviside step function, and denote the left and right interface coordinates, respectively.

The dc Josephson current is obtained from (4); (5)

 J =−8teℏNk∑n,ky,σcos(ky√2)[Im(unσ(x)u∗n,σ(x′))f(Enσ) +Im(vnσ(x)v∗n,σ(x′))f(−Enσ)], (7)

where is the electron charge, and and denote neighboring points in the antiferromagnet.

We fix the phase at the end of each superconductor, obtain self-consistent solutions from the above equations, and use Eq. (7) to calculate the Josephson current for that particular phase difference. We set throughout, and is used as the unit of energy in all calculations.

Results and Discussion. In the right panel of Fig. 1 we show results for the critical current as a function of the number of AF-layers in the junction, for various values of the magnetization coupling constant . The main point to note is that the even-odd effect (sign-change for the current) appears only provided that exceeds a threshold value. Below this threshold, the critical current displays no change of sign with increasing , although there are small oscillations in the quantity superimposed on a monotonic decay. However, increasing sufficiently will induce -transitions for odd junctions, while only decreasing the effective interface transparency for even junctions, as seen in Fig. 1. Note that for , the threshold value for observing a -transition in the critical current is above , while for , the threshold value is below . Since the magnitude of the staggered magnetic order parameter increases monotonically with , this indicates that longer odd junctions require a weaker magnetization to undergo transitions. This result has some resemblance to the case of a ferromagnetic junction, where such transitions can occur regardless of the magnitude of the exchange field provided that the junction is sufficiently long. We will discuss this point in more detail below.

In order to understand the physics behind these results, note that even junctions have a vanishing total magnetic moment as there is an equal number of spins with opposite direction. An even junction is essentially an SNS junction with the extra effect that increased staggering increases the resistance of the junction. This may be understood in simple terms by noting that an increasing staggering provides an increasing spin-dependent potential that scatters the current-carrying states in the AF-junction for low values of .

Odd junctions, on the other hand, have a finite magnetic moment due to the uncompensated spin at one of the edges of the AF region. The layers adjacent to the interfaces have parallel spins for odd junctions, while even junctions have antiparallel spins. This resembles the situation in, for instance, SFIFS junctions with parallel or anti-parallel alignment of the exchange fields of the two ferromagnets, (8) or SIFIS junctions where the interfaces are spin-active. (9) In odd junctions, the subgap states are spin-split, while for even junctions, the states are spin degenerate. Naively, odd junctions are thus equivalent to ferromagnetic junctions, as there is a net magnetic moment (albeit weak) if one averages over the junction length. One notable difference is, however, that the average magnetic moment density in the SAFS-case scales with , while it is constant in the SFS-case. Moreover, it is known in the SFS case that, provided the junctions are wide enough, any finite amount of magnetic moment density suffices to induce a transition. For the SAFS case, it is far from obvious that one should get the same behavior, due to the effective -scaling of the average magnetic moment density. It is therefore of some importance to investigate the scaling with of the critical moment density required to induce oscillations in the SAFS-case, and compare it with the behavior of the SFS-case.

To do so, it is instructive to consider an analogy with a ferromagnetic junction, where the Andreev-bound state energy in the ballistic limit reads (11): Here, where is the ferromagnetic exchange energy, and is the Fermi velocity. We have assumed that the interface transparency is perfect in order to use a simple analytical expression for the ensuing discussion. Now, for even the total magnetic moment of the antiferromagnet is zero, hence we get no transition in even junctions. For the odd junction in question we can relate the finite magnetic moment to an effective ferromagnetic exchange field. The two factors contributing to this extra phase is the magnetic moment and the width of the junction, by analogy to an effective ferromagnet for odd . A long junction can display a transition for a smaller magnetic moment magnitude than a short junction, as seen in the right panel of Fig. 1. For , has not undergone the transition yet, while it has undergone a transition for and longer junctions. This, superficially at least, seems to indicate that the behavior is similar to that of an SFS-junction. Note however, that in the SFS-case, the parameter increases linearly with and will exceed any prescribed threshold value, provided is large enough. For an SAFS-junction, however, where one naively expects such that , a threshold value for the magnitude of the staggered order parameter would be required in order to observe a oscillation.

To investigate this further, we have numerically compared an SFS-junction with an odd SAFS-junction. Fig. 2 shows the critical magnetic moment as a function of junction width for both kinds of junctions, SFS and SAFS. Here, the critical magnetic moment is defined as the value where the critical current changes sign for a given value of the junction length . Although the fits appear to be quite similar on a linear scale, they are significantly different when viewed on a log-log scale. The results for the SFS-junction follows the intuition one gets based on the bound-state energy and phase-shifts shown above, namely that the critical value of the magnetic moment density, which is proportional to a threshold value of in order to get a transition, scales as . This is not so for the SAFS-case. Rather, since in this case, one expects the critical value of the magnetic moment density to be non-zero also for long junctions. This is indeed seen in the log-log plot, where the results for the SFS-junction falls on a linear curve, while the results for the SAFS-junction reaches a constant value asymptotically.

Furthermore, we have considered the crossing of the energy-levels of the current-carrying states as the strength of the antiferromagnetic ordering, parametrized by , is increased. This is shown in Fig. 3. Level crossings of spin-up states and spin-down states, with opposite -dispersion, occur for large enough value of , i.e. as the magnitude of the magnetic ordering increases. Such level crossings tend to reverse the sign of the current, which is essentially determined by the -derivative of the levels. The levels that contribute most significantly to the currents are seen to be the levels close to zero energy, as the levels further away are -independent and therefore carry little current. The main difference between such spectra for the SAFS-junctions and the corresponding ones for SFS-junctions, is that complete level-reversion of near-zero energy states occurs much more easily in the SFS-case than the SAFS case, since the total magnetic moment scales with the width of the junction in the SFS-case, while it does not in the SAFS-case. Hence, the energy bands with a definite spin-content are much more susceptible to a Zeeman effect in the SFS-case, compared to the SAFS-case. In particular, one feature of the SAFS-spectra shown below is that there is essentially only one level-reversion between spin-up and spin-down subgap states before the bands flatten out, thus no longer contributing to the currents. For the SFS-case (not shown here), there are several level-inversions of subgap-states with only minor flattening of the bands as the magnetization increases, thus providing a much more efficient way of reversing the sign of the currents over the junction.

It is clear that antiferromagnetic order is not in itself sufficient to induce transitions in the supercurrent. Ref. (4) showed that for a sufficiently strong staggered order parameter, the current-phase relation of an antiferromagnetic Josephson junction revealed a - or -state depending on whether the antiferromagnet had an even or odd number of atomic layers. We have above explicitly studied the actual threshold value of the magnitude of the staggered order parameter where transitions cease to occur, regardless of whether the interlayer has an odd or even number of lattice sites. To observe the even-odd effect (and its absence as predicted here for weak antiferromagnets), it would be necessary to exert satisfactory control of the junction length (of order lattice spacing). It would certainly be an experimental challenge to tailor the junctions in this way (12). Nevertheless, previous studies have demonstrated that it is possible to fabricate ultrathin films with atomic-scale control over the thickness (13); (14), which would allow for a test of our predictions. It is difficult within the framework of our model to make quantitative predictions for actual parameter values in candidate systems. That would require a more refined approach including non-ideal effects such as disorder both in terms of non-magnetic impurities and with regard to the magnetic sublattices.

The threshold value predicted here has an interesting consequence, namely that it becomes possible to induce a transition in antiferromagnets via pressure. The reasoning is as follows. For an odd- junction with sufficiently strong AF order, the junction is in the -state. Were one to bring the magnitude of the AF order back down below the threshold value for the transition, one would effectively have reversed the dc Josephson-current across the junction. Suppression of the AF order may effectively be obtained by destroying the Fermi-surface nesting, something that can be achieved by applying pressure to the sample. The pressure may be induced via an electric mechanism, such that one effectively is using electric means to control magnetic order, and by that in turn current-switching. This effect has no counterpart in an SFS-junction, since ferromagnetism does not rely on Fermi surface nesting. The combination of a threshold value for the staggered order parameter at which one may have oscillations, and the possibility to destroy nesting and thus AFM order via pressure, amounts to electrically controlled transitions in a single sample. This sets the results for an SAFS-junction apart from what is found for the ferromagnetic SFS-case, where any amount of magnetic order in principle suffices to induce transitions. Such transitions typically occur by changing the junction length, requiring multiple samples (although temperature-dependent transitions are also possible). The AFM order thus offers a novel mechanism for controlling the quantum mechanical ground state of a Josephson junction.

H.E. thanks NTNU for financial support. JL and AS acknowledge support from the Norwegian Research Council, through Grants 205591/V20 and 216700/F20.

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