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-mediated indirect condensation through antiferromagnetic insulators

Citation for published version (APA): Johansen, Ø., Kamra, A., Ulloa, C., Brataas, A., & Duine, R. A. (2019). Magnon-mediated indirect exciton condensation through antiferromagnetic insulators. Physical Review Letters, 123(16), [167203]. https://doi.org/10.1103/PhysRevLett.123.167203

DOI: 10.1103/PhysRevLett.123.167203

Document status and date: Published: 18/10/2019

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Download date: 02. Oct. 2021 PHYSICAL REVIEW LETTERS 123, 167203 (2019)

Magnon-Mediated Indirect Exciton Condensation through Antiferromagnetic Insulators

Øyvind Johansen ,1,* Akashdeep Kamra,1 Camilo Ulloa,2 Arne Brataas,1 and Rembert A. Duine1,2,3 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, Netherlands 3Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, Netherlands (Received 30 April 2019; published 18 October 2019)

Electrons and holes residing on the opposing sides of an insulating barrier and experiencing an attractive Coulomb interaction can spontaneously form a coherent state known as an indirect exciton condensate. We study a trilayer system where the barrier is an antiferromagnetic . The and holes here additionally interact via interfacial coupling to the antiferromagnetic . We show that by employing magnetically uncompensated interfaces, we can design the magnon-mediated interaction to be attractive or repulsive by varying the thickness of the antiferromagnetic insulator by a single atomic layer. We derive an analytical expression for the critical Tc of the indirect exciton condensation. Within our model, anisotropy is found to be crucial for achieving a finite Tc, which increases with the strength of the in the antiferromagnetic bulk. For realistic material parameters, we estimate Tc to be around 7 K, the same order of magnitude as the current experimentally achievable exciton condensation where the attraction is solely due to the Coulomb interaction. The magnon-mediated interaction is expected to cooperate with the Coulomb interaction for condensation of indirect , thereby providing a means to significantly increase the exciton condensation temperature range.

DOI: 10.1103/PhysRevLett.123.167203

Introduction.—Interactions between result in fluctuations near an antiferromagnetic Mott insulating exotic states of . is a prime [30,31].Thusitiscrucialtoachieveagood example, where the negatively charged electrons can have understanding of antiferromagnetic magnon- an overall attractive coupling mediated by individual interactions, as well as electron-electron interactions couplings to the vibrations, known as , of the mediated by antiferromagnetic magnons. positively charged lattice. In addition to charge, the electron In this Letter, we theoretically demonstrate the applica- also has a degree of freedom. The electron spin can tion of antiferromagnetic insulators to condensation of interact with localized magnetic moments through an indirect excitons. An exciton is a consisting exchange interaction exciting the by of an electron and a hole. The excitons interact attractively transfer of angular momentum. These excitations are through the Coulomb interaction due to their opposite known as magnons. Theoretical predictions charges [32]. Initially predicted many decades ago [33,34], of electron-magnon interactions have shown that these can the exciton condensate has been surprisingly elusive. A also induce effects such as superconductivity [1–10]. challenge is that the exciton lifetime is too short to form a Research interest in antiferromagnetic materials is condensate due to exciton-exciton annihilation processes surging [11,12]. This enthusiasm is due to the promising such as Auger recombination [35–38]. The problem of properties of antiferromagnets such as high resonance short exciton lifetimes can be solved by having a spatial frequencies in the THz regime and a vanishing net separation between the electrons and holes in a trilayer magnetic moment. Much of this research focuses on system, where the electrons and holes are separated by an interactions involving magnons or spin at magnetic insulating barrier [39–41] to drastically lower the recombi- interfaces in hybrid structures. Examples of this are spin nation rate. Excitons in such systems are often referred to as pumping [13–19], spin transfer [15,20–22],andspinHall (spatially) indirect excitons, and these are ideal to observe magnetoresistance [23–28] at normal metal interfaces, the exciton condensate. Herein, we consider a system where and magnon-mediated superconductivity [9,10]. Recently, the insulating barrier is an antiferromagnetic insulator, as an experiment has also demonstrated spin transport in an shown in Fig. 1. The insulating barrier can then serve a dual antiferromagnetic insulator over distances up to 80 μm purpose: in addition to increasing the exciton lifetime, the [29]. Moreover, antiferromagnetic materials are also of spin fluctuations in the antiferromagnet mediate an addi- interest since it is believed that high-temperature super- tional attractive interaction between the electrons and the conductivity in cuprates is intricately linked to magnetic holes. This magnon-mediated interaction cooperates with

0031-9007=19=123(16)=167203(7) 167203-1 © 2019 American Physical Society PHYSICAL REVIEW LETTERS 123, 167203 (2019) the Coulomb interaction thereby enabling an increase of the temperature range for observing exciton condensation in (a) (b) experiments. The exciton lifetimes achieved via antiferro- magnetic insulators will be comparable to their nonmagnetic counterparts (∼10 ns [42]), leaving the spin-independent physics unaltered. The indirect exciton condensate has two main exper- imental signatures. The first is a dissipationless counterflow of electric currents in the two layers [43–45]. When the exciton condensate moves in one direction, the resulting charge currents in the individual layers are antiparallel FIG. 1. (a) An antiferromagnetic insulator (AFI) sandwiched due to the oppositely charged carriers in the two layers. The between two separate reservoirs, denoted by L and R.We second signature is a large enhancement of the zero-bias let the spins on sublattice A (illustrated in blue) be down, and the tunneling conductance between the layers [46,47], remi- spins on sublattice B (illustrated in red) be up. (b) The fermions in niscent of the in superconductors. the two reservoirs can interact through emission and absorption Comparing the critical condensation in tri- of magnons. For the process in the figure we have that either a L S ℏ layers with magnetic and nonmagnetic insulating barriers, spin-up fermion in emits a z ¼þ magnon (red arrow) which R that otherwise have similar properties and dimensions, is absorbed by a spin-down fermion in , or a spin-down fermion in R emits a Sz ¼ −ℏ magnon (blue dashed arrow) absorbed by a should allow us to isolate the role of magnons in mediating spin-up fermion in L. the condensation. The exciton condensate is expected to exist when the number of electrons in one layer equals the number of holes reservoirs, as illustrated in Fig. 1(a). We will then later in the other. Thus far, to the best of our knowledge, consider the case where one of these reservoirs is populated experiments with an unequivocal detection of the exciton by electrons, and the other by holes. This system can be H H H H condensate have utilized quantum Hall systems with a half described by the Hamiltonian sys ¼ el þ mag þ int, H filling of the lowest Landau level to satisfy this criterion where el describes the electronic part of the system in the H [48–52]. Such systems rely on high external magnetic fermion reservoirs, mag describes the spins in the anti- H fields. A recent experiment studying double-bilayer gra- ferromagnetic insulator, and int describes the interfacial phene systems has, however, been able to detect the interaction between the fermions and magnons. We assume enhanced zero-bias tunneling conductance signature of all three layers to be atomically thin, and thus two-dimen- indirect exciton condensation without any magnetic field, sional, for simplicity. by controlling the electron and hole populations through We consider a uniaxial easy-axis antiferromagnetic gate voltages [53]. This is an indication of the possible insulator described by the Hamiltonian existence of an exciton condensate, and shows promise for X K X H J S S − S2 : finding a magnetic-field free exciton condensate. mag ¼ i · j 2 iz ð1Þ i In this Letter, we show that the magnon-mediated inter- hi;ji action between the electrons and holes can be attractive or Here J>0 is the strength of the nearest-neighbor exchange repulsive depending on whether the two magnetic interfaces interaction between the spins which have a magnitude are with the same or opposite magnetic sublattices. In turn, jSij¼ℏS for all i, and K>0 is the easy-axis anisotropy this enables an unprecedented control over the interaction constant. Next, we perform a Holstein-Primakoff trans- nature via the variation of the antiferromagnetic insulator formation (HPT) [54] of the spin operators on each thickness by a single atomic layer. Consequently, when the sublattice, denoted by sublattices A and B, as defined in † magnon-mediated interaction is paired with the Coulomb Fig. 1. From the HPT, we have that the operator að Þ interaction, this can be used to control the favored spin i annihilates (creates) a magnon at ri when ri ∈ A, and structure of the excitons. In our model, we find that the bð†Þ r critical temperature for condensation is enhanced by the equivalently i annihilates (creates) a magnon at i when r ∈ B exchange interaction in the antiferromagnet, and that a finite i . The magnetic Hamiltonian can be diagonalized magnetic anisotropy is needed to have an attractive inter- through FourierP and Bogoliubov transformations to the H ε μ†μ ν†ν action around the Fermi level. Our results suggest that if one form mag ¼ qk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð k k þ k kÞ. The magnon is lets the insulating barrier in indirect exciton condensation ε ℏ 1 − γ2 ω2 ω 2ω ω k given by k ¼ ð kÞ E þ kð E þ kÞ, where is experiments be an antiferromagnetic insulator, the magnon- P γ z−1 ik δ δ mediated interactions can significantly strengthen the corre- the magnon momentum, k ¼ δ exp ð · Þ, a set z lations between the electrons and holes. of vectors to each nearest neighbor, the number of nearest ω ℏJSz ω ℏKS Model.—We consider a trilayer system where an anti- neighbors, E ¼ , and k ¼ . The eigenmagnon ð†Þ ð†Þ ferromagnetic insulator is sandwiched between two fermion operators μk and νk are related to the HPT magnon

167203-2 PHYSICAL REVIEW LETTERS 123, 167203 (2019) operators through the Bogoliubov transformation μk ¼ interactions will dominate over the intralayer interactions, u a v b† ν u b v a† k k þ k −k, k ¼ k k þ k −k. The Bogoliubovpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coef- which is the case for a system designed for indirect exciton u v u Γ 1 =2 condensation. Next, we integrate out the magnon fields μðÞ ficients kpandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik are given by k ¼ ð k þ Þ v Γ − 1 =2 and νðÞ, and writeR the path integral over the fermion and k ¼ ð k Þ , where we have introduced 2 2 2 −1=2 reservoirs as Z ≈ D ψ LD ψ R expð−S =ℏÞ. In the Γk ¼f1 − ½ωEγk=ðωE þ ω Þ g . eff k momentum and Matsubara-frequency bases, the effective The interfacial exchange interaction between the fer- S mions and magnons at the two magnetic interfaces is action eff is given by [61] modeled by the s-d interaction [55,56] X XX 0 X X X S ¼S þℏβ Uσðq;iωnÞψ σ;Lðk þq;iνl þiωnÞ j eff el H − J r ρˆ r S r ; σ ↑;↓ lmn kk0q int ¼ kð iÞ jð iÞ · ð iÞ ð2Þ ¼ j¼L;R k¼A;B i∈Aj 0 k ×ψ −σ;Lðk ;iνlÞψ −σ;Rðk−q;iνm −iωnÞψ σ;Rðk;iνmÞ; which has been successfully applied to describe inter- ð5Þ actions at magnetic interfaces in similar systems [19, LðRÞ 57–60]. Here Ak is the interface section between the where we have here introduced the fermionic and bosonic left (right) fermion reservoir and the kth (k ¼ A, B) Matsubara frequencies, νn ¼ð2n þ 1Þπ=ðℏβÞ and ωn ¼ 2πn= ℏβ S sublattice of the antiferromagnetic insulator. The interfacial ð Þ, respectively. The action el describes the con- j tribution of the fermionic fields to the action in Eq. (4), exchange coupling constants J ðriÞ are defined so that they k H H j j j except for the contributions from int. The latter term, int take on the value J ðriÞ¼J if ri ∈ A , and zero other- k k k is instead described by the contribution of the magnon- wise. We have also defined the electronic spin density mediated interlayer-fermion potential X 1 †   ρˆjðriÞ¼ ψ σ;jðriÞσσσ0 ψ σ0;jðriÞð3Þ 2 L R L R 2 ℏ S Jμ ðqÞJμ ðqÞ Jν ðqÞJν ðqÞ σ;σ0 Uσðq;iωnÞ ≡ − þ ð6Þ N −σiℏωn þ εq σiℏωn þ εq ð†Þ with ψ σ;j annihilating (creating) a fermion with spin σ in N the jth (j ¼ L, R) fermion reservoir, and σ ¼ðσx; σy; σzÞ to the effective action, where is the total number of spin being a vector of Pauli matrices. sites in the antiferromagnet. We assume the two magnetic Effective magnon potential.—We will now use a path interfaces are uncompensated; i.e., each interface is only integral approach where we treat the magnon-fermion with one of the antiferromagnetic sublattices [25,63,64] as shown in Fig. 1. We compute that the coupling constants interaction as a perturbation, and integrate out the mag- L;R nonic fields that give rise to processes as illustrated in Jμ;ν ðqÞ describing the effective exchange coupling strength Fig. 1(b) to express the interaction as an effective potential between the spin of the fermions in reservoirs L, R to L=R between the fermion reservoirs.R We consider the coherent- the spin of the eigenmagnons μq, νq are Jμ ðqÞ¼ 2 2 2 2 state path integral Z ¼ D ψ LD ψ RD μD ν exp ð−S=ℏÞ L=R L=R L=R L=R vqJB ðrL=RÞ − uqJA ðrL=RÞ and Jν ðqÞ¼vqJA ðrL=RÞ− D2μ ≡ DμDμ S in imaginary time, where , etc. The action u JL=R r is given by q B ð L=RÞ. Since each interface is with only one sub- JL q −u JL   lattice, μ ð Þ¼ q A if the left interface is with sublattice Z X X X L L ℏβ A, and Jμ ðqÞ¼vqJ if the left interface is with sublattice S dτ ℏ ψ r ; τ ∂ ψ r ; τ B ¼ σ;jð i Þ τ σ;jð i Þ B 0 . We get analogous results for the right interface. We see i σ¼↑;↓ j¼L;R L;R   that the effective coupling constants Jμ;ν ðqÞ can have the X L;R η r ; τ ∂ η r ; τ H τ ; J þ ð i Þ τ ð i Þ þ sysð Þ ð4Þ same or opposite sign as the coupling constants A;B η¼μ;ν depending on which sublattice is at the interface. This has to do with the spin projection of the eigenmagnon where τ ¼ it is imaginary time, and β ¼ 1=ðkBTÞ with kB relative to the equilibrium spin direction of the sublattice at being the and T the temperature. Note L;R the interface. The effective coupling constants Jμ;ν ðqÞ are that in the coherent-state path integral we can replace u v † also enhanced by a Bogoliubov coefficient q or q with fermion operators by Grassman numbers (e.g., ψ → ψ ) L;R respect to the coupling constants J . These are typically and operators by complex numbers (e.g., η† → η). A;B q 0 u ≈ v ≈ 2−3=4 We now treat H as a perturbation, and keep terms up to large numbers. For ¼ we have 0 0 × int ω =ω 1=4 ω =ω second order. We discard any terms that only contribute to ð E kÞ to lowest order in the small ratio k E. intralayer interactions, as we are interested in the interlayer The enhancement is due to large spin fluctuations at each potential between the fermion reservoirs. By discarding the sublattice of the antiferromagnet per eigenmagnon in the intralayer terms, we effectively assume that the interlayer system, since the eigenmagnons are squeezed states [9,65].

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By studying Eq. (6), we note that we have and its Hermitian conjugate, and perform a Hubbard- Re½Uσðq;iωnÞ < 0 for identical uncompensated interfaces, Stratonovich transformation of the effective action. By doing whereas for a system where one of the interfaces is with a saddle-point approximation and integrating over the sublattice A and the other with sublattice B,wehave fermionic fields [61], we then obtain the gap equation Re½Uσðq;iωnÞ > 0. Consequentially, this allows us to X X 0 −1 0 control whether the magnon-mediated interlayer-fermion Δ−σðk ;iνnÞ¼ β Uσðk − k ;iνm − iνnÞ m k potential Uσðq;iωnÞ is attractive or repulsive by designing Δ k;iν the interfaces. Whether this potential is attractive or σð mÞ : × 2 2 2 ð9Þ repulsive can depend on a single atomic layer. This allows jΔσðk;iνmÞj þ ϵðkÞ þðℏνmÞ for an unprecedented high degree of control and tunability of the interlayer-fermion interactions. The sign difference We note that the magnon-mediated potential is attractive of the potential can be explained by how the two fermions when Uσðq;iωnÞ > 0 in the case of indirect exciton con- coupled by the magnon interact with the eigenmagnon densation, which can be seen from rearranging the fermionic spin. For Re½Uσðq;iωnÞ < 0 we have processes where fields in Eq. (5). the fermions couple symmetrically to the magnon spin; We now use Eq. (9) to find an analytical expression for i.e., both fermions couple either ferromagnetically or anti- the critical temperature Tc below which the excitons ferromagnetically to its spin. On the other hand, for spontaneously form a condensate. To obtain an analytical Re½Uσðq;iωnÞ > 0 we have an asymmetric coupling, where solution, we focus on the case when the gap functions and one fermion couples ferromagnetically to the eigenmagnon the magnon-mediated potential are independent of momen- spin and the other fermion couples antiferromagnetically. tum and frequency. This corresponds to an instantaneous Indirect exciton condensation.—We will now study contact interaction, and we therefore assume that the gap spontaneous condensation of spatially indirect excitons functions have an s- pairing. Moreover, we see that the where the attraction is mediated by the antiferromagnetic gap equation in Eq. (9) only has a solution when Δσ and magnons. We consider the left (right) reservoir to be an n- Δ−σ have the same sign. In the case where spin degeneracy doped (p-doped) . We describe the semi- is unbroken, it is fair to assume that Δσ ¼ Δ−σ, indicating conductors by the Hamiltonian tripletlike pairing. In superconductivity, s-wave and triplet X X X pairing are mutually exclusive for even frequency order H τ ϵ k ψ † k; τ ψ k; τ ; elð Þ¼ jð Þ σ;jð Þ σ;jð Þ ð7Þ parameters, but in the case of indirect excitons the same j¼L;R k σ¼↑;↓ symmetry restrictions on the order parameter do not apply, as the composite boson does not consist of identical ϵ k −ϵ k ℏ2k2= 2m − ϵ ≡ ϵ k m with Lð Þ¼ Rð Þ¼ ð Þ F ð Þ. Here is . In other words, for indirect excitons the sym- the effective electron and hole mass, which we assume to be metries in momentum space and spin space are decoupled ϵ ψ † equal, and F is the Fermi level. While the operator σ;L=R from one another. As both the magnon-mediated potential creates an electron with spin σ in the left or right layer, we and the Coulomb potential are in the s-wave channel and note that due to the negative dispersion in the right layer the the Coulomb potential is independent of spin, the magnon- excitations in this layer are effectively described by electron mediated potential works together with the Coulomb holes. We also note that we have not included a Coulomb potential enhancing the attractive exciton pairing interac- interaction between the electron and the holes in our model. tion. The fact that we can design whether the magnon- The effect of the Coulomb potential on indirect exciton mediated potential is attractive or repulsive allows us to condensation has been widely studied in previous literature control which spin channel is the most favorable for the [66]. We will later argue why the magnon-mediated excitons to condensate. potential is expected to cooperate with the Coulomb To determine Tc we perform a BCS-like calculation potential in the case of indirect exciton condensation. [61,68] and restrict the sum over Matsubara frequencies to The interaction in Eq. (5) is too complicated for us to solve a thin shell around the Fermi level (jℏνmj < ε0), where the for the exciton condensation. We then do an approximation magnon-mediated potential is attractive. The analytical similar to the Bardeen-Cooper-Schrieffer (BCS) theory of expression for Tc is found to be   superconductivity [67,68], and assume that the dominant γ 2e EM ε 2πε contribution to the interaction arises when the excitons have 0 0 Tc exp − ; 10 0 ¼ 2 L R ð Þ zero net momentum (k þ k ¼ q), and similarly for the πkB Su0v0ma JAJB iν iν iω Matsubara frequencies ( l þ m ¼ n). Next, we intro- γ ≈ 0 577 a duce the order parameter where EM . is the Euler-Mascheroni constant and the lattice constant of the . Here we have X X 0 assumed that the left and right magnetic interfaces consist Δσðk;iνmÞ ≡ − Uσðk − k ;iνm − iνnÞ n k0 of opposite sublattices. This leads to an attractive exciton interaction. If we assume the exchange energy among the ψ k0;iν ψ k0;iν × σ;Rð nÞ σ;Lð nÞð8Þ spins in the bulk is much larger than the interface coupling

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To show how high the Tc of indirect exciton condensa- tion in our model can be using only the magnon-mediated interaction, we give a numerical estimate for realistic material parameters. Using the parameters S ¼ 1, m equal L R the electron mass, a ¼ 5 Å, ℏJA ¼ ℏJB ¼ 10 meV [7], 13 −1 ωE ¼ 8.6 × 10 s [72], and assuming the magnetic ωðoptÞ 9 9 anisotropy takes on its optimal value k ¼ . × 9 −1 ðoptÞ 10 s , we obtain a Tc of approximately 7 K. Antiferromagnetic insulators that can be suitable for the proposed experiment are Cr2O3 [63], α-Fe2O3 [29], and NiO [72]. A possible emergence of a strong electric field across the barrier could in principle alter the magnetic properties in, e.g., Cr2O3 [63,73] and NiO [74].We FIG. 2. Dependence of the normalized critical temperature on estimate the upper limit of a potential electric field to be the strength of the normalized magnetic anisotropy. around 47 V=mm, based on a “stress-test” scenario where 1% of the charge carriers have leaked through the 2 L R (ℏωE ≫ Sma JAJB), the value of the anisotropy that insulating barrier, assuming a charge carrier density of 10 −2 maximizes Tc is 2.6 × 10 cm [47]. This estimate is considerably weaker than the requirements for influencing typical magnetic 2 L R Sma JAJB ℏωðoptÞ ≡ : ð11Þ insulators [63,73,74]. In comparison to the critical temper- k 16π ature above, a recent experiment studying double bilayer in the quantum Hall regime found the Coulomb- The full dependence of Tc on the magnetic anisotropy is mediated exciton condensation to have an activation shown in Fig. 2. The critical temperature for indirect ∼8 exciton condensation is largest for a nonzero and finite energy of K [52], which was 10 times higher than ω → 0 what was found in an experiment using GaAs [75].This magnetic anisotropy. This is because in the limit k the magnon gap in the antiferromagnetic insulator vanishes, demonstrates that the potential mediated by the antiferro- and consequentially so does the thin shell around the Fermi magnetic magnons is capable of creating strong correlations level where the magnon-mediated potential is attractive. In between the electrons and holes that could significantly opt increase the critical temperature for condensation compared the case of a large anisotropy, ω ≫ ωð Þ, the enhance- k k to when the excitons are just bound through the Coulomb ment of the magnon-mediated potential due to magnon interaction. squeezing is lost [65]. When the anisotropy takes on its optimal value, the critical temperature becomes This work was supported by the Research Council of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Norway through its Centres of Excellence funding scheme, 2 L R “ ” ðoptÞ ℏωESma JAJB γ −1=2 Project No. 262633 QuSpin and Grant No. 239926 T ≡ ffiffiffi e EM : c p 3=2 ð12Þ “Super Insulator Spintronics,” the European Research 2π kB Council via Advanced Grant No. 669442 “Insulatronics,” Notably, we see that the critical temperature increases as well as the Stichting voor Fundamenteel Onderzoek der monotonically with increasing strength of the exchange Materie (FOM), which is part of the Netherlands interaction ℏωE. The optimal choice of an antiferromag- Organisation for Scientific Research (NWO). netic insulator would then be a material with a magnetic ℏω anisotropy ( k) on an energy scale proportional to the L;R exchange coupling at the interface (ℏJA;B), and a very strong *[email protected] exchange interaction in the bulk of the antiferromagnetic [1] H. Suhl, Simultaneous Onset of and Super- insulator (ℏωE). As discussed in the Supplemental Material conductivity, Phys. Rev. Lett. 87, 167007 (2001). [61], inclusion of retardation and renormaliza- [2] N. Karchev, Magnon exchange mechanism of ferromagnetic tion effects [69–71] via the Eliashberg method is expected superconductivity, Phys. Rev. B 67, 054416 (2003). pffiffiffi [3] H. Funaki and H. Shimahara, Odd- and even-frequency to reduce the Tc estimated here by a factor between e e3=2 superconductivities mediated by ferromagnetic magnons, and . At the same time, accounting for the proper magnon J. Phys. Soc. Jpn. 83, 123704 (2014). dispersion leads to a similar increase [70] in Tc thereby [4] R. Kar, T. Goswami, B. C. Paul, and A. Misra, On magnon leaving our estimate essentially unchanged after including mediated formation in ferromagnetic super- these complications. conductors, AIP Adv. 4, 087126 (2014).

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