Rashba spin-orbit coupling effects in quasiparticle interference of non-centrosymmetric superconductors
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Rashba spin-orbit coupling effects in quasiparticle interference ofnon-centrosymmetric superconductors
Alireza Akbari and
Peter Thalmeier
Max-Planck Institute for the Chemical Physics of Solids, D-01187 Dresden, Germany
PACS – Pairing symmetries
PACS – Tunneling phenomena: single particle tunneling and STM
PACS – Heavy-fermion superconductors
Abstract – The theory of quasiparticle interference (QPI) for non-centrosymmetric (NCS) super-conductors with Rashba spin-orbit coupling is developed using T-matrix theory in Born approx-imation. We show that qualitatively new effects in the QPI pattern originate from the Rashbaspin-orbit coupling: The resulting spin coherence factors lead to a distinct difference of charge-and spin- QPI and to an induced spin anisotropy in the latter even for isotropic magnetic impurityscattering. In particular a cross - QPI appears describing the spin oscillation pattern due to non-magnetic impurity scattering which is directly related to the Rashba vector. We apply our theoryto a 2D model for the NCS heavy fermion unconventional superconductor CePt Si and discuss thenew QPI features for a gap model with accidental node lines due to its composite singlet-tripletnature.
The determination of gap symmetry in unconventionalsuperconductors (SC) is a persistent problem. Most usefulmethods for its investigation are ARPES experiments [1],specific heat and thermal transport measurements underrotating field [2] and the use of STM-based quasiparti-cle interference technique (QPI) which utilizes the ripplesin electronic density generated by random surface impu-rities [3–11]. In the cuprate and iron pnictides the formeris readily applicable but sofar not for heavy fermion un-conventional superconductors where SC gaps are only inthe meV range. In this case the last two methods aremore powerful. QPI technique has recently been proposedto discriminate between different d-wave pairing states in115 systems [12] and has been successfully demonstratedfor CeCoIn [13]. Before it was also used to investigatethe hidden order state of URu Si [14–16].Here we propose the application of QPI to inversionsymmetry-breaking, non-centrosymmetric (NCS) super-conductors, notably the tetragonal heavy fermion 131 and113 compounds [17] like CePt Si [18] and CeRhSi [19].We develop the QPI theory for the case of mixed singlet-triplet gap under the presence of Rashba-type spin-orbitcoupling. We show that a wealth of new QPI features isto be expected: i) Distinct differences in the charge- andspin- QPI due to the effect of Rashba coherence factors. ii) Likewise Rashba - induced anisotropies in the spin-QPI even for isotropic magnetic impurity scattering. iii) Anew kind of cross- QPI where scattering by nonmagnetic (charge) impurities leads to spin density pattern directlyrelated to the non-zero Rashba vector (and vice versa).Finally the expected gap nodes in NCS superconductorsare generally not on symmetry positions but determinedby the ratio of singlet and triplet amplitudes. QPI cangive important information on the position of these acci-dental nodes. We use a weak coupling BCS theory for theNCS superconductor with a 2D model Fermi surface ap-propriate for 131 compounds. We employ Nambu Green’sfunction technique and T-matrix theory in Born approxi-mation to calculate the quasiparticle interference spectraand discuss their new features as compared to centrosym-metric unconventional superconductors.The present BCS model is given by [20–22] H SC = (cid:88) k σσ (cid:48) (cid:2) ( ε k − µ ) σ + g k · σ (cid:3) σσ (cid:48) c † k σ c k σ (cid:48) +12 (cid:88) k σσ (cid:48) (∆ σσ (cid:48) k c †− k σ c † k σ (cid:48) + H.c. ) . (1)Here ε k = 2 t (cos k x + cos k y ) + 4 t cos k x cos k y is the ki-netic energy with respect to chemical potential µ where t p-1 a r X i v : . [ c ond - m a t . s up r- c on ] J u l . Akbari et al. - - H t L DO S H (cid:144) t L G X M G H a L Ξ = Ξ = - H b L M Ω = H c L Ω = t H d L Ω = t H e L q q q q q q M Ω = t H f L Ω = t H g L Ω = t H h L Ω = t Fig. 1: a) Electronic density of states (DOS) of Rashba bands( ξ = ±
1) in the SC state. (inset: 2D band structure alongΓXMΓ with t = 0 . t , g = 0 . t , and µ = − t ). Totalband width is W (cid:39) t ≡ T ∗ (cid:39)
14 K (exp. Kondo tempera-ture from [18]). b) Normal state electron Fermi surface aroundM point. Dashed lines indicate nodes of gap functions ∆ k ξ with parameters: ψ = 2 t , ψ = t , and φ = 0 . t (alsoin subsequent figures). c) and d) show spectral functions inthe normal state for different energies ω . e) Spectral functionfor the superconducting state around M point. q i are promi-nent scattering vectors defining the QPI patterns (c.f. Figs.3e,4f). f-h) Spectral function for the superconducting state fordifferent ω . are nearest and t next nearest neighbor hopping. Further-more g k = − g − k defines the antisymmetric Rashba termdue to broken inversion symmetry [23]. Therefore the su-perconducting 2 × k = [ ψ k σ + d k · σ ] iσ y has an even singlet ( ψ k ) as well as odd triplet ( d k ) com-ponents. The latter must be aligned with the Rashba vec-tor d k = φ g k to avoid destruction by pairbreaking [24].Here σ = ( σ x , σ y , σ z ) denotes the Pauli matrices. Diag-onalization of the model leads to an effective two bandsuperconductor on the Rashba split bands ( ξ = ±
1) givenby (cid:15) k ξ = ε k − µ + ξ | g k | that have split Fermi surfaces (FS)and opposite helical spin polarizations as well as differentsuperconducting gaps ∆ k ξ = ψ k + ξ | d k | . For concretenesswe employ a 2D model for the tetragonal Ce- based 131compounds [25, 26]. The possible influence of magneticorder [27] is not discussed here.In this class g k = g (sin k y , − sin k x ,
0) is in the tetrag-onal plane. The resulting Rashba-split bands (inset) anddensity of states (DOS) (in the SC state) are shown inFig. 1a and the electron-type Fermi surface around theM( π, π ) points in Fig. 1b. The parameters are chosen [25](caption of Fig. 1) to obtain a realistic M-point Fermisurface sheet appropriate for CePt Si. The split constant-
Fig. 2: The individual charge- QPI (Λ q ) contributions forintra- or inter- band scattering with ξ, ξ (cid:48) = ±
1, from normal(non-magnetic) impurities at the energy ω = 0 . t : a-d) in thenormal state e-h) in the superconducting state. (Summationover each of the rows is shown in Fig.3.b and Fig.3.e). energy surfaces ( ω >
0) are presented in Fig. 1b,c. It isknown from thermal conductivity [28] that the supercon-ducting gap has line nodes. To achieve nodes on the M-point Fermi surface we use an extended s-wave [24] formfor ψ k and d k as before. This leads to the two distinctgaps∆ k ξ = ψ + ψ (cos k x + cos k y ) + ξφ g (sin k x + sin k y ) (2)on the Rashba-split Fermi surfaces (cid:15) k ξ = µ with quasi-particle energies E k ξ = [ (cid:15) k ξ + ∆ k ξ ] . The gap zeroes areshown as dashed lines in Fig. 1b-d. Nodes (node linesin 3D) appear on the FS for the E k − quasiparticle sheetbut not for E k + . Their position is accidental, i.e. deter-mined by the fine tuning of singlet and triplet amplitudes ψ and φ . It was shown in Refs. [29–31] that the nodalcase is also topologically nontrivial with possible forma-tion of Andreev bound states (ABS) appearing as surfaceor edge states. They exist when the surface normal isperpendicular to the Rashba or d -vector, i.e., lies withinthe tetragonal plane and then they lead to a zero-biasconductance peak in Andreev tunneling current. Here weinvestigate the opposite case of a tetragonal surface planewith normal parallel to the Rashba vector. Then ABS willnot appear and have no signature in the tunneling current[29] or QPI. We also will not discuss the nodeless gap casewith dominating singlet part. It does not correspond toCePt Si and because both Rashba FS sheets are fullygapped have no interesting QPI features in the SC state.For the nodal gap situation the evolution of constant en-ergy surfaces in the SC state is shown in Fig. 1e-h. A fewwave vectors q i connecting high curvature points close tonodal positions that will appear prominently in QPI spec-tra for small ω are shown in Fig. 1e. For larger ω (Fig. 1h)connected FS sheets reappear. For the calculation of theQPI pattern we use the normal and anomalous 2 × G = G + σ + G − (ˆ g k · σ ) and F = [ F + σ + F − (ˆ g k · σ )] iσ y ,respectively, where the scalar Green’s functions are givenby (ˆ g k = g k / | g k | ) G ± ( k , iω n ) = 12 (cid:88) ξ (cid:26) ξ (cid:27) ( iω n + (cid:15) k ξ )[( iω n ) − E k ξ ] ,F ± ( k , iω n ) = 12 (cid:88) ξ (cid:26) ξ (cid:27) ∆ k ξ [( iω n ) − E k ξ ] . (3)The QPI density oscillations are obtained from the fullGreen’s function that is determined by the effect of scat-tering from random charge and spin impurities at the sur-face. The total scattering Hamiltonian in compact formreads H imp = (cid:88) kq α V α ( q ) S α Ψ † k + q ˆ ρ α Ψ k . (4)Here we use the Nambu 4-component spinor representa-tion Ψ † = ( c † k ↑ c † k ↓ c − k ↑ c − k ↓ ). In the Nambu space the4 × ρ α ( α = (0 , i ) = (0 , x, y, z )) are given by { ˆ ρ α } = (ˆ ρ , ˆ ρ ) = ( τ σ , τ σ x , τ σ y , τ σ z ). The τ α - and σ α - Pauli matrices (with τ = σ ≡
1) act on Nambuand spin indices, respectively. Furthermore we define { S α } = (1 , S ). The first index α = 0 corresponds to non-magnetic impurity scattering V ( q ) and entries i = x, y, z to isotropic magnetic exchange scattering V i ( q ) = V ex ( q )from impurity spins S . Their components S i are treatedas frozen, i.e. polarized in a given direction by a smallmagnetic field. An important consequence of the Rashbaterm is that spin and charge channels for impurity scat-tering are not decoupled, this also holds true for spin andcharge response functions [25].We calculate the change in STM tunneling conductancein charge or spin channel α (0 , x, y, z ) due to impurity scat-tering in charge or spin channel β (0 , x, y, z ). For magneticimpurity the scattering channel β is fixed by applying asmall field H (cid:28) H c along the x,y,z axis. The conduc-tance channel α is selected by using either a nonmagnetic( α = 0) or a half-metallic (fully spin polarized) tunnel-ing tip with moment polarized along α = x, y, z and anexchange splitting larger than the heavy fermion quasi-particle band width. Such configuration in principle wouldallow to determine all elements of the QPI differential con-ductance tensor. It is given by [3] dδI α ( r , V ) dV ∼ δN αβ ( r , ω = V )= − π Im[Tr σ ˆ ρ α δ ˆ G β ( r , r , ω )] . (5)Here δ ˆ G β is the change of the 4 × β (0 , x, y, z ). Furthermore matrix index (11) refersto the Nambu space which results from the trace with re-spect to τ including the projector (1 + τ z ). The remain-ing trace refers to spin space only. In this work we focus on the spatial oscillations or momentum dependence byweak scattering and ignore the possibility of bound stateformation [32] in the strong scattering limit. Thereforewe treat the former in Born approximation which leads tothe Fourier transform of differential conductances given by( k (cid:48) = k − q ) δN αβ ( q , ω ) = − π V β ( q )Im (cid:104) Λ αβ ( q , iω n ) (cid:105) iω n → ω + iδ , Λ αβ ( q , iω n ) = 1 N (cid:88) k Tr σ (cid:104) ˆ ρ α ˆ G ( k , iω n )ˆ ρ β ˆ G ( k (cid:48) , iω n ) (cid:105) , (6)where N = L is the number of grid points. We assumehere that the tunneling happens out of the coherent heavyquasiparticle states. This is justified for temperature T and frequency ω much smaller than the Kondo tempera-ture T ∗ [15] which is of the order 14 K for CePt Si [18].We therefore restrict to frequencies (Fig. 3) of the orderof the SC gap and we do not intend to describe the Fanoresonance shape that appears for higher frequencies of theorder of the effective quasiparticle bandwidth T ∗ [15]. TheQPI function Λ αβ ( q , iω n ) is evaluated using Eq. (3), per-forming the σ − trace we obtain per spinΛ q ( iω n ) = 14 N (cid:88) k ξξ (cid:48) (cid:2) ξξ (cid:48) (ˆ g k · ˆ g k (cid:48) ) (cid:3) K kq ξξ (cid:48) ( iω n ) , Λ q ii ( iω n ) = 14 N (cid:88) k ξξ (cid:48) (cid:2) − ξξ (cid:48) (ˆ g k · ˆ g k (cid:48) − i k ˆg i k (cid:48) ) (cid:3) K kq ξξ (cid:48) ( iω n ) , Λ q i0 ( iω n ) = 12 N (cid:88) k ξξ (cid:48) ξ ˆg i k K kq ξξ (cid:48) ( iω n ) , (7)where the integration kernel for intra-band ( ξ = ξ (cid:48) ) andinter-band ( ξ (cid:54) = ξ (cid:48) ) processes is given by K kq ξξ (cid:48) ( iω n ) = ( iω n + (cid:15) k ξ )( iω n + (cid:15) k − q ξ (cid:48) ) − ∆ k ξ ∆ k − q ξ (cid:48) [( iω n ) − E k ξ ][( iω n ) − E k − q ξ (cid:48) ] . These equations give the QPI patterns for the NCS super-conductors. Before presenting numerical results we pointout the salient new features as compared to centrosym-metric unconventional superconductors [33]. Here Λ de-scribes the charge- QPI pattern due to nonmagnetic im-purities, Λ ii the (diagonal) spin- QPI polarization patterndue to magnetic impurities and Λ i the spin polarizationgenerated by nonmagnetic impurities which we term ascross- QPI. It occurs only for finite Rashba coupling g k .The coupling of charge and spin degrees in cross- QPIappears because the Rashba band states are described byhelicity and not spin quantum numbers. Therefore in gen-eral a particle-hole (charge) excitation will also change thespin state. The diagonal QPI functions in Eq. (7) are fi-nite even for g k = 0. However they are strongly modifiedwhen inversion symmetry is broken due to the Rashba he-licity coherence factors in square brackets of Eq. (7) . Thelatter have different sign for pure charge and spin QPIfunctions and also change sign between intra- and inter-band contributions. The coherence factors can amplify orp-3. Akbari et al. Fig. 3: a-c) Total charge- QPI (Λ q ) in the normal state fordifferent ω and scattering from non-magnetic impurities. d-f) The same quantity for the superconducting state. Subset( q , q , q ) of characteristic QPI wave vectors correspond tothose in the spectral functions given in Fig.1.e ( ω = 0 . t ).The frequency satisfies ω < | ∆ k ξ | (cid:39) . t (cid:28) W = T ∗ = 8 t . annihilate the individual contributions in the sum depend-ing on the relative orientation of Rashba vectors and bandindices. Therefore one expects that charge- (Λ ( q )) andspin- (Λ ii ( q )) QPI pattern are profoundly different as op-posed to non-Rashba case where they should be identical.The spin- QPI pattern shows a further striking Rashba ef-fect: Although the exchange scattering itself is isotropic,Λ ii ( q ) is to be expected anisotropic in the present casewhen g k is confined to the tetragonal plane as seen fromEq. (7). Furthermore the Rashba term leads to nondiago-nal elements in the spin- QPI pattern, in the present caseto Λ q xy ( iω n ) = Λ q yx ( iω n ) which is given byΛ q xy ( iω n ) = 14 N (cid:88) k ξξ (cid:48) ξξ (cid:48) (ˆg x k ˆg y k − q + ˆg y k ˆg x k − q )K kq ξξ (cid:48) (i ω n ) . (8)The QPI functions Λ q and Λ q ij are even and Λ q i is odd in q . Therefore δN i ( r , ω ) is finite only when V ( q ) containsodd (p-wave) contributions and the net (area integrated)cross-QPI conductance always vanishes.We conclude that QPI for non-centrosymmetric super-conductors derived here exhibits a wealth of new effectsdue to the inversion symmetry breaking Rashba termwhich we present now for the 2D model of 131 compounds.All QPI spectra have four contributions from two intra-band and two (equivalent) interband scattering processes.They are shown individually in Fig.2a-d for charge-QPI inthe normal state and in Fig.2e-h for the SC state. In theformer the scattering across the Fermi surface of Fig. 1bmaps out the ’2k F ’ contours, partly folded back into thefirst BZ. They are slightly different due to the different di-ameter of Rashba split sheets. The diagonal center crossappears only for intraband contributions due to scattering Fig. 4: a-c) Total spin- QPI (Λ q ii ) for the normal state forexchange scattering from magnetic impurities at energy ω =0 . t . d-f) The same quantity for the superconducting state.Each panel corresponds to different orientation of the impurityspin S parallel to an applied field B →
0. Characteristic QPIwave vectors q i (Fig.1.e) are shown for B (cid:107) z . process parallel to each Rashba sheet. In the SC state forsmall ω the sheets break up into small pieces in BZ re-gions connecting the node points where the gap is small.Therefore the E k − Rashba band dominates because it isthe only one with nodes (Fig.1b,e) and the SC QPI spec-trum reflects the ( ξ, ξ (cid:48) ) = ( − , −
1) intraband transitionsin Fig.2e. A set of the typical wave vectors q i of Fig.1eselected by the coherence factors show up as prominentspots in Fig.2e. The other three contributions have loweramplitude due to the small E k + sheets at this energy. Thesum of all four contributions is the total charge-QPI spec-trum which is presented in Fig.3a-f for the normal and SCstate for three energies (bias voltages V): ω/t = 0 . , . .
3. In the former (a-c) the ’2 k F ’ contours whichincrease with ω are now blurred due contributions withslightly different dimensions. The diagonal cross due toinraband processes appears at all energies. In the SC statefor small ω the set of prominent spots at q i due to Fig.2esurvive and are shown explicitly in Fig.3e. The observa-tion of these features in QPI and their variation with ω would allow to confirm directly the gap model with nodestructure in Fig.1b,e. For larger ω instead of spots at q i arcs appear due to scattering on and between the againconnected constant- ω surfaces of both Rashba bands inFig.1h.In the centrosymmetric superconductor ( g k = 0) withinBorn approximation the charge- and spin- QPI functionsin Eq. (7) are identical. The presence of a Rashba termintroduces two new aspects due to the appearance of co-herence factors (square brackets in Eq. (7)): i) the charge-and spin-QPI becomes different due to different intra-bandp-4ashba spin-orbit coupling effects in quasiparticle interference of non-centrosymmetric superconductors Fig. 5: a) Total cross- QPI of odd spin density Λ q x in thenormal state from non-magnetic scattering, and b) same quan-tity in superconducting state. c) Total non-diagonal even spin-QPI Λ q xy for the normal state from exchange scattering bymagnetic impurities, and d) same quantity in superconduct-ing state. Here B → (cid:107) y and ω = 0 . t . contributions in the two cases ii) While the charge- QPIretains the fourfold symmetry as obvious from the firstof Eq. (7) and Fig. 3, the spin-QPI coherence factors inthe second Eq. (7) explicitly breaks rotational symmetrywith respect to the impurity moment direction. This isshown in Fig.4 for normal and SC phase where we assumethat a small magnetic field B polarizes the impurity spins S along one of the three symmetry directions. The spin-QPI pattern for B along x,y directions are still equiva-lent but rotated by 90 ◦ , however, both are quite differentfrom the case B (cid:107) z , in particular in the SC state. Thisis because in the present model the Rashba vector ful-fils g k · ˆ z = 0 which singles out the z direction. Thisanisotropy of spin- QPI is therefore characteristic for non-centrosymmetric superconductors. The spin-QPI patternis dominated by the full set of scattering vectors q i con-necting the high-curvature points of the constant- ω surface(Fig.4f).Sofar we have only considered diagonal charge- or spin-QPI, however it is a very peculiar feature of NCS Rashbasystems that charge-spin cross- QPI ( and the reverse) de-scribed by Λ q i as well as non-diagonal spin-QPI, e.g. Λ q xy exist. The former describes spin oscillations introducedby non-magnetic impurities (and vice versa) which areodd under inversion because Λ − q i = − Λ q i . It is shown inFig.5a,b for (x0) component with spin polarization alongx, again the (y0) case is just rotated in the BZ by 90 ◦ whileit vanishes for (z0). Finally we show in Fig.5c,d the non-diagonal (xy) spin pattern polarized perpendicular to theimpurity spins and described by Eq. (8). Only the (xy)component is non-zero because g k lies in the tetragonalplane. It also exhibits reflection symmetry with respect to diagonals (interchange of x,y directions). It would also beinteresting to investigate QPI for the tetragonal NCS in ageometry where the surface normal is within the tetrag-onal plane. Then Andreev bound states may appear [29]and lead to additonal contributions to QPI spectrum forsmall bias voltages, similar as in the zero bias conductancepeak. To treat this case one has to include the quasipar-ticle dispersion perpendicular to the tetragonal plane andextend the t-matrix formalism to include inhomogeneousstates.We have shown that QPI in noncentrosymmetric systemsexhibits a wealth of new features due to the presence ofRashba spin orbit coupling. They are important signa-tures of the helical spin texture and a powerful tool to in-vestigate the nodal structure of the superconducting gap,in particular because the nodal positions are not fixed bysymmetry for mixed singlet-triplet gaps.We thank T. Takimoto for helpful comments and I.Eremin for useful discussion. REFERENCES[1] K. Okazaki, Y. Ota, Y. Kotani, W. Malaeb, Y. Ishida,T. Shimojima, T. Kiss, S. Watanabe, C.-T. Chen, K. Ki-hou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa,Y. Kohori, K. Hashimoto, T. Shibauchi, . H. I. Y. Mat-suda, H. Miyahara, R. Arita, A. Chainani, and S. Shin,2012
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