Comment on "Surface Pair-Density-Wave Superconducting and Superfluid States"
CComment on “Surface Pair-Density-Wave Superconducting and Superfluid States”
Anton B. Vorontsov
Department of Physics, Montana State University, Bozeman, Montana 59717, USA (Dated: July 28, 2020)
The claim in [1] that pair-density wave (PDW) super-conductivity in magnetic field is more stable near surfacethan in bulk, is not supported by microscopic theory.The Ginsburg-Landau (GL) analysis presented in [1]is flawed in several respects: (i) it uses GL coefficients[2] that are valid only in the vicinity of the tricriticalpoint (TCP), where H F F LO ( T ) ≈ H ( T ), Eqs. (2-4) inref.[1]; (ii) the truncated-gradient GL expansion is gener-ally inadequate, since at low temperatures and high fieldsthe modulation wave-vectors q are large and the gradi-ent and magnetic energies are comparable µH ≈ v F q/ ξ = v F / πT c (cid:29) k − F , and is a tested tool for studyinginhomogeneous FFLO states [3, 4, 7].A singlet superconducting order parameter ∆( R ) isself-consistently determined from∆( R ) (cid:104)|Y ˆ k | (cid:105) ln TT c = 2 πT (cid:88) ε m > (cid:28) Y ∗ ˆ k (cid:20) f ↑ + f ↓ − ∆ Y ˆ k ε m (cid:21)(cid:29) ˆ k (1)where Y ˆ k is a symmetry basis function (s-wave, d-wave,etc), and (cid:104) . . . (cid:105) ˆ k denotes Fermi surface average. The in-stability is given by linearisation of (1), correspondingto ∆ -terms (with arbitrary gradients) in the free en-ergy expansion. Linearised Eilenberger equations for theanomalous propagators f ↑ , ↓ ( R ; ˆ k, ε m > (cid:20) v F (ˆ k ) · ∇ R + ( ε m ± iµH ) (cid:21) f ↑ , ↓ ( x, y ; ˆ k, ε m ) = ∆( x, z ) Y ˆ k (2)is supplemented with the boundary conditions at the z = 0 interface: for atomically smooth surface f α ( x,
0; ˆ p, ε m ) = f α ( x,
0; ˆ p, ε m ), while for diffuse surfaceI use scattering matrix method [8].I expand the order parameter amplitude∆( x, z ) = n − (cid:88) m =0 ∆ m ϕ m ( z ) η ( Q x x ) (3) c µ H c / π T c TCPH
FFLObulk n=5n=10n=40z x ABC SC H (T) (a) p^_p^ Q x =0.4/ ξ ξ ∆ ( z ) Order Parameter profile12 ξ A BC at the N-SC instability (b)
FIG. 1. (a) The critical Zeeman field of a 2D s-wave su-perconductor with cylindrical Fermi surface, and atomicallysmooth boundary (ˆ p = ˆ p − z (ˆ z · ˆ p )). H c for surface FFLOstate (solid and dotted lines) never exceeds the bulk criticalfield (open circles). (b) profiles of the order parameter emerg-ing at the transition points A,B,C in panel (a). in the orthonormal basis ϕ m ( z ) = e − z/ ξ L m ( z/ξ ) on z ∈ [0 , ∞ ), where L m are Laguerre polynomials. Thecut-off, n , limits the extent of the order parameter along z axis to about 4 n ξ . The x -dependence η ( Q x x ) = { exp( iQ x x ) or cos( Q x x ) } . Then one solves (2) analyt-ically for f ↑ , ↓ along classical trajectories starting withnormal state value f ↑ , ↓ ( z → ∞ ; ˆ p ) = 0 and connectingˆ p → ˆ p via the boundary conditions at z = 0. Substi-tuting f ↑ , ↓ into (1) and projecting out ϕ m -modes givesan eigenvalue problem that I solve to find the highest H c transition and the corresponding eigenvector ∆ m =0 ...n − .The results for the critical field H c are shown inFig. 1(a). A superconducting state, localized within 20 ξ of the surface with n = 5 components, is suppressed byfields lower than the bulk critical field. As number ofthe basis components increases, the most favorable su-perconducting state expands away from the surface, andthe H c approaches H bulkF F LO . For n = 40 we basically re-gain the bulk behavior; the profile of the order parameter a r X i v : . [ c ond - m a t . s up r- c on ] J u l along the transition line, shown in Fig. 1(b), is uniformeverywhere above the TCP, long modulations appear justbelow TCP and they get shorter at lower temperatures.The point C period of 12 ξ exactly corresponds to thevalue found in bulk system [3].The solution with finite modulation Q x along the sur-face also never exceeds the bulk transition. This indi-cates that the bulk instability is the most favorable oneand there is no special more robust surface PDW state.This conclusion is also reproduced for d -wave symme-try, for atomically rough (diffuse) boundaries, for Fermisurfaces of different shape (tight binding), in 1, 2, or 3dimensions. By evaluating free energy ∆ terms alongthe transition I confirm that in all cases the transitionremains 2-nd order, with exception of the 3-dimensionalcase, which is of the first order below the TCP [2] butbecoming again second order at low temperatures [4].While quasiclassical theory is strictly invalid for ξ ∼ k − F , it still gives qualitatively correct answer for sur-face states [9], which may indicate that the surface PDWstates obtained by the same authors in a BdG treatmentin [10] is a feature of extreme parameters when the spin-splitting of bands is comparable to the bandwidth. It isalso worth pointing out that oscillations of the order pa-rameter in tight-binding BdG approach may also appear as a result of surface and geometry effects [9, 11]. [1] M. Barkman, A. Samoilenka, and E. Babaev, PhysicalReview Letters , 165302 (2019), arXiv:1811.09590.[2] A. I. Buzdin and H. Kachkachi, Physics Letters A ,341 (1997).[3] H. Burkhardt and D. Rainer, Annalen der Physik ,181 (1994).[4] S. Matsuo, S. Higashitani, Y. Nagato, and K. Nagai,Journal Of The Physical Society Of Japan , 280 (1998).[5] M. Sigrist and K. Ueda, Reviews of Modern Physics ,239 (1991).[6] V. Ambegaokar, DeGennes, D. Rainer, P. G. De Gennes,and D. Rainer, Physical Review A , 2676 (1974).[7] A. B. Vorontsov, J. A. Sauls, and M. J. Graf, PhysicalReview B , 184501 (2005).[8] Y. Nagato, S. Higashitani, K. Yamada, and K. Na-gai, Journal of Low Temperature Physics , 1 (1996);S. Higashitani and N. Miyawaki, Journal Of The PhysicalSociety Of Japan , 33708 (2015).[9] N. Wall Wennerdal, A. Ask, P. Holmvall, T. L¨ofwander, and M. Fogelstr¨om, arXiv e-prints , arXiv:2006.00456(2020), arXiv:2006.00456 [cond-mat.supr-con].[10] A. Samoilenka, M. Barkman, A. Benfenati, andE. Babaev, Physical Review B , 054506 (2020).[11] L.-F. Zhang, L. Covaci, M. V. Miloˇsevi´c, G. R. Berdiy-orov, and F. M. Peeters, Phys. Rev. Lett. , 107001(2012); Phys. Rev. B88