- Email: [email protected]
Interplay of magnetism and superconductivity at the nanometer scale: the case of complex oxide heterostructures
Physica C 387 (2003) 17–25 www.elsevier.com/locate/physc
Interplay of magnetism and superconductivity at the nanometer scale: the case of complex oxide heterostructures C.A.R. S a de Melo School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Abstract It is suggested that magnetism and superconductivity can coexist at the nanometer scale in heterostructures consisting of complex ferromagnetic and superconducting oxides. The geometry discussed is that of a trilayer where the superconductor is a copper oxide and is sandwiched by two layers of ferromagnets consisting of manganese oxides. The superconductor is assumed to be layered and have a d-wave order parameter. It is shown that the ground state of such nanometer scale heterostructures alternates between ferromagnetic and antiferromagnetic depending on the thickness of the superconductor. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Manganite/cuprate multilayers; Ferromagnetic-superconducting heterostructures; Nanometer scale; Ferromagnetic superconductors
1. Introduction Very recently several experimental groups [1–5] were able to grow, characterize and measure a few properties of complex oxide heterostructures consisting of manganese oxides (colossal magnetoresistance materials) and copper oxides (high Tc superconductors). A nice review about these complex oxide heterostructures can be found in the literature [6], where effects such as spin injection properties, critical currents and critical temperatures are discussed. Part of these experimental efforts were inspired in earlier theoretical work [7,8] were the coexistence and interplay of magnetism and superconductivity in multilayers con-
E-mail address: [email protected] (C.A.R. S a de Melo).
sisting of manganese oxides and copper oxides was suggested, but not fully analysed. In the present work, the coexistence and interplay of magnetism and superconductivity at the nanometer scale is considered for d-wave superconductors. Situations where the proximity to a ferromagnet completely destroys superconductivity or where the proximity to a superconductor completely destroys ferromagnetism are not interesting for the purpose of this paper. The main conditions for the coexistence and interplay of these two phenomena is reflected on the magnitude of the proximity effects between the superconductors and ferromagnets, when the thicknesses of different layers are in the nanometer range. For coexistence of these two phases it is essential from the superconductorÕs perspective that (a) magnetic pair breaking effects are not so large, and (it is very helpful) that the superconductor has a high critical
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00635-X
18
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
temperature. From the ferromagnetÕs perspective it is essential that the superconductor does not destroy magnetism via the proximity effect, and (it is very helpful) that the ferromagnet has a high Curie temperature [7,8]. These conditions are very difficult to be satisfied simultaneously by most ferromagnets and superconductors in the regime of nanometer thicknesses, and if we add the inevitable experimental restriction of good epitaxial growth, only very special classes of materials seem to satisfy the conditions mentioned above. Two classes of materials (a) manganese oxides and copper oxides; and (b) ruthenium oxides and copper oxides were suggested as possible candidates for coexistence and interplay of magnetism and superconductivity, given that the conditions outlined above seemed to be theoretically satisfied [7,8]. In this manuscript the focus is on the existence of a three-dimensional magnetic phase mediated by coexisting d-wave superconductivity at T ¼ 0, and on the derivation of the effective magnetic coupling responsible for the stabilization of these magnetic phases. Thus, the main interest here is on the magnitude and sign of the magnetic coupling between neighboring ferromagnetic layers spaced by a d-wave superconducting layer, which dictates the fate of the ground state of the heterostructure. For definiteness and calculational purposes it is necessary to chose a geometry, which is indicated in Fig. 1. However, the physical mechanism is largely independent of the choice of geometry, and is discussed next. One of the possible ways of describing theoretically magnetic coupling through metallic multilayers is to use an RRKY-type of argument, where the electronic spins are polarized by the ferro-
FO
LSMO
SO
YBCO (NCCO)
FO
LSMO
Fig. 1. Trilayer geometry for ferromagnetic oxide/superconducting oxide (FO/SO) multilayers. The FO is assumed to be La1y Sry MnO3 (LSMO), while the SO is assumed to be either YBa2 Cu3 O7d (YBCO), or Nd2x Cex CuO4 (NCCO).
magnet, creating an effective field felt by the other ferromagnet a certain distance away. This spin polarization involves a summation over virtual electronic states and has an oscillatory nature in 1 real space characterized by ‘p ¼ ð2kF Þ , where kF is the relevant Fermi momentum of the metallic spacer. The magnetic coupling that results from this analysis is very sensitive to the shape of the Fermi surface of the metal [9,10], given that nesting plays an important role in the final form of the magnetic coupling at T ¼ 0. A similar type of argument can be used for magnetic coupling across a singlet (s-wave or dwave) superconductor at T ¼ 0. The essential difference now is that the magnetic coupling is mediated by virtual quasiparticles states. When the superconductor is s-wave, a full gap opens up in the excitation spectrum. The appearance of this gap makes the polarization of virtual quasiparticle states more difficult, and reduces both the range and the magnitude of the magnetic coupling at T ¼ 0 in comparison to the expected coupling for a normal spacer layer at T ¼ 0. Thus, for s-wave superconductors there is a large spin polarization stiffness controlled by the gap to quasiparticle excitations. When the superconductor is d-wave, the gap in the excitation spectrum is highly anisotropic, and have nodes along special directions. This leads to the existence of zero and lowenergy quasiparticles in d-wave superconductors. For wave vectors connecting nodal regions it is possible to carry more spin information (at longer length scales) across d-wave superconductors than across s-wave superconductors. Therefore the detrimental effects to magnetic coupling caused by a full gap in the excitation spectrum (as in the swave case) are less effective in d-wave systems, because of the existence of zero-energy quasiparticles. Therefore, d-wave high Tc superconductors are ideal spacer systems to study magnetic coupling and coexistence of magnetism and superconductivity. In order to study this coexistence and magnetic coupling, the rest of the paper is organized as follows. In Section 2, the Hamiltonian used for the trilayer geometry is discussed. In Section 3, the functional integral method, used to analyse coexistence and magnetic coupling, is outlined. In
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
Section 4, the main results are presented and analysed. In Section 5, comments are made regarding the approximations used and other important questions of experimental relevance. Lastly, in Section 6, the main conclusions are summarized.
2. Hamiltonian To study the coexistence and interplay of magnetism and superconductivity in nanometer scale complex oxide heterostructures consisting of ferromagnetic oxides (FO) and superconducting oxides (SO) the trilayer structure shown in Fig. 1 is chosen. The trilayer consists of two manganese oxide layers as the ferromagnets separated by a dwave copper oxide superconductor. In the FO/SO multilayered heterostructures the FO can be, for instance, La1y Sry MnO3 (LSMO), La1y Cay MnO3 (LCMO) or Nd1y Sry MnO3 (NSMO) and the SO can be YBa2 Cu3 O7d (YBCO), La2x Srx CuO4 (LSCO) (d-wave) or Nd2x Cex CuO4 (NCCO) (swave). It is assumed that the Curie temperature Tf of the ferromagnetic oxide layers is larger than the critical temperature Tc of the superconducting copper oxide layer. To describe this extremely complex heterostructure the following Hamiltonian: H ¼ Hf 1 þ Hf 2 þ Hs þ Hsf 1 þ Hsf 2 ;
ð1Þ
is used, where repeated greek indices are used to indicate summation. The first two terms (Hf 1 and Hf 2 ) on the right hand side of Eq. (1) are contributions due to the ferromagnets. Hf j ¼
Z
drfjay ðrÞfðKf j Þac þ ðHcj Þac gfjc ðrÞ:
j
which characterizes the existence of two semi-infinite ferromagnets for z 6 ds =2 and z P ds =2. The intrinsic assumption is that df ds , which corresponds to the worst case scenario for suppression of superconductivity. Additional comments regarding this non-essential choice will be made in Section 4. The magnitude U0 controls if the ferromagnet is half-metallic (when U0 is smaller than the bandwidth of the copper oxide) or insulating (when U0 is bigger than the bandwidth of the copper oxide), see Fig. 2. The double exchange mechanism is implicitly assumed, but not explicitly taken into account in the continuum description of the ferromagnet provided here. The second term on the right hand side of Eq. (2) corresponds to the exchange R energy ðHcj Þac ¼ ðrl Þac hlj ðrÞ; where hlj ðrÞ ¼ dr0 Jlj ðr; r0 ÞSlj ðr0 Þ Fj ðrÞ; with Slj ðr0 Þ being the total spin associated
SINGLE PARTICLE POTENTIALS
Ef
eg
eg U0
t2g
ds
j
Fj ðrÞ ¼ H½ðÞ z ds =2 H½ðÞ z ds =2 df ; ð3Þ
t2g
PAIR POTENTIAL
ð2Þ
The first term on the left hand side of Eq. (2) is ðKf j Þac ¼ ½Kf j þ Uj ðrÞ dac ; and contains the kinetic 2 energy Kf j ¼ Fj ðrÞ½ðirÞ =2mf lf ; associated with the itinerant (eg ) electrons of the CMR-material, and the background potential energy Uj ðrÞ ¼ U0 Fj ðrÞ with the function
19
P0
ds Fig. 2. Top: single particle potentials for FO/SO trilayer, indicating eg and t2g states in the FO, and the dx2 y 2 states in the SO. Bottom: shape of projection function P ðrÞ that appears in the pair potential for FO/SO trilayer. EF is the Fermi level, ds is the thickness of the SO layer.
20
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
with the localized t2g orbitals in the jth ferromagnet. The total spin projection along the quantization axis (chosen to be parallel to the FO/SO interface along the y-direction) is Sy ¼ 3=2 corresponding to three localized electrons in the t2g orbitals. Here, the exchange coupling between delocalized eg -electrons and localized t2g -electrons is represented by the point contact (local) interaction Jij ðr; r0 Þ ¼ Jij dðr r0 Þ, where in addition, the exchange coupling field Jij points preferably along the y-direction (parallel to the FO/SO interfaces). The third term on the left hand side of Eq. (1) is Z Hs ¼ drwya ðrÞ½Ks dac wc ðrÞ þ V ; ð4Þ
of the superconductor. Given that the ferromagnets are half-metallic or insulating, it is assumed that the superconductors do not produce a proximity effect in the ferromagnetic. Furthermore, the coupling terms Hsf 1 and Hsf 2 involving the superconductor and the two ferromagnets are of the form:
and corresponds to the Hamiltonian describing the superconductor. The first term on the right hand side of Eq. (4) is the kinetic energy term: " # 2 ðirk Þ Ks ¼ P ðrÞ þ tz cosðci@z Þ ls ; ð5Þ 2ms
with local interaction Jsf j ðr; r0 Þ ¼ Jj dðr r0 Þ. This term involves localized spins of the t2g orbitals of the ferromagnet and spins of the charge carriers in the The second term Ht ðeg ; R Rsuperconductor. y 0 0 sÞ ¼ dr dr0 ½tab ðr; r Þf ðr Þwb ðrÞ þ C:C: , involves ja sf j charge transfer between the ferromagnet (when it is half-metallic) and the superconductor. For the purposes of this paper, this term will be neglected in comparison to the first. However, this additional term is pair breaking since it allows spinpolarized charged carriers (possibly over long length scales) inside a singlet superconductor. The presence of this term is responsible for an additional reduction in the critical temperature of the superconductor, when sandwiched by a halfmetallic instead of an insulating manganese oxide ferromagnet. The maximal Curie temperature of bulk CMRferromagnet LaMnO is Tf 300 K, and the maximal critical temperature of bulk YBCO (LSCO) is Tc 90 K (Tc 40 K). However, the pair breaking ðiÞ parameter apb ¼ Ji hSzðiÞ i=2pTc seems to be small as suggested earlier [7,8] producing a finite Tc down to very few monolayers (nanometer lengths) of the superconducting copper oxides. Currently there is strong experimental evidence for this since the coexistence of magnetism and superconductivity in manganese oxide and copper oxide heterostructures down to a few monolayers of copper oxides has been firmly established [1–5]. The suppression of Tc in these systems can be from a few percent to complete (100%) depending on the thicknesses and compositions of the magnetic and superconducting
containing the chemical potential ls of the superconductor. Note the presence of the periodic operator cosðci@z Þ used to mimic the layered structure of the superconducting oxide. The projection function P ðrÞ ¼ Hðz þ ds =2Þ Hðz þ ds =2Þ just identifies the boundaries of the superconductor of thickness ds , i.e., the spacer extends from ds =2 to þds =2. The second term on the right hand side of Eq. (4) is the interaction term Z 1 dr0 gdcab ðr; r0 Þwyd ðrÞwyc ðr0 Þwa ðr0 Þwb ðrÞ Pe ðr; r0 Þ; V ¼ 2 ð6Þ where gdcab ðr; r0 Þ ¼ V ðjrk r0k jÞdd;c da;b dðz z0 Þ; is the interaction tensor which simulates singlet pairing only in the planes parallel to the xy-plane. The term V ðjrk r0k jÞ represents a two-dimensional interaction of the type V ðRÞ ¼ V1 HðR1 RÞ þ V0 HðR R1 ÞHðR0 RÞ; ð7Þ r0k j.
with R ¼ jrk Notice that V ðRÞ has a finite range, which allows for d-wave pairing within the xy-plane. Furthermore, the projection operator Pe ðr; r0 Þ ¼ P ðrÞ P ðr0 Þ enforces that the pairing interaction exists only within the spatial boundaries
Hsf i ¼ Hex ðt2g ; sÞ þ Ht ðeg ; sÞ; where Hex ðt2g ; sÞ ¼
XZ Z
ð8Þ
dr dr0 Jsf j ðr; r0 Þ
j
wya ðrÞðrl Þab wb ðrÞSlj ðr0 Þ;
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
oxides. Since here the interest is only in the T ¼ 0 (ground state) limit, the subject of Tc suppression will not be discussed any further.
leads to the Bogoliubov–deGennes (BdG) equations Z EN uN a ðrÞ ¼ He ðrÞuN a ðrÞ þ dr0 Dab ðr; r0 ÞvN b ðr0 Þ; ð11Þ
3. Method In order to calculate the magnetic coupling across the d-wave superconductor several additional assumptions are made. First, it is assumed that the magnetic coupling between the localized t2g spins in the half-metallic or insulating ferromagnet and the spin of the charge carriers in the d-wave superconductor does not completely destroy superconductivity. This means that the pair breaking parameter apb is reasonably small. Second, it is assumed that the momentum parallel to the FO/SO interface is conserved, i.e., the FO/SO interface is translationally invariant. Such assumption implies that the FO/SO interface is very well lattice matched and atomically flat. Third, it is assumed that the normal state of the d-wave superconductor is not the pseudogap phase, thus the results derived here are strictly valid only for the overdoped regime of cuprate superconductors. The functional integral method is used to calculate perturbatively the magnetic coupling across the d-wave superconductor. Within this approach the action corresponding to the Hamiltonian Hs of the superconductor is Ss ¼
Z
b
Z
21
dr wya ðrÞ@s dac wc ðrÞ Hs :
ð9Þ
0
Introduction of the matrix field Dab ðr; r0 Þ ¼ of the fermionic variables as wa ðrÞ ¼ N ½uN a ðrÞcN þ vN a ðrÞcyN and integration over the residual fermionic variables cN and cyN leads to an action that depends only on the variables Dab , uN a , and vN ;a and Hermitian conjugates. Subsequent minimization with respect to Dyab leads to the order parameter equation X KN Dab ðr; r0 Þ ¼ gabdc ðr; r0 Þ Wcd ðr; r0 Þ Wdc ðr0 ; rÞ ; 4 N 1 g ðr; r0 Þwd ðr0 Þwc ðrÞ;Protation 2 abdc
ð10Þ 0
where KN ¼ ð1 2f ðEN ÞÞ, and Wcd ðr; r Þ ¼ vN c ðrÞ uN d ðr0 Þ: Minimization with respect to uN a , and vN ;a
EN vN a ðrÞ ¼ He ðrÞvN a ðrÞ þ
Z
dr0 Dab ðr; r0 ÞuN b ðr0 Þ: ð12Þ
The magnetic coupling across the superconductor is mediated by quasiparticles which carry spin information, thus it is necessary to calculate the quasiparticle energies from the BdG equations (11) and (12). Since the momentum parallel to the FO/ SO interface is assumed to be conserved the BdG amplitudes can be written as uN a ¼ u~N a ðzÞ exp½ikk rk and vN a ¼ v~N a ðzÞ exp½ikk rk . Furthermore, the local approximation is used where the order parameter Dab ðr; r0 Þ becomes Dab ðkk ; zÞ upon Fourier transformation of the parallel ðxyÞ coordinates. This leads to simplified BdG equations EN ~hN ðzÞ ¼ H~e ðkk ; zÞr3 ~hN ðzÞ þ Dðkk ; zÞr1 ~hN ðzÞ; ð13Þ where ~hN ðzÞ is a vector with components ½~ uN ðzÞ; v~N ðzÞ , and ri are the standard Pauli matrices. This equation, however, is strictly valid only in the limit where the phase coherence length nz is large ðkz nz 1Þ. This local equation can be a quite good approximation even in the regime where the BCS theory is not truly applicable, i.e., when the size of the Cooper pairs are very small, provided that the phase coherence length is large! In the BCS approximation, the size of the Cooper pair and the phase coherence length are large and nearly identical, however towards local pairing theories (Bose–Einstein condensation regime) the size of the Cooper pair is small, but the phase coherence is still very large [11,12]. Thus, even substantially away from the BCS limit the local theory can be used for the calculation of the magnetic coupling across the superconductor in the asymptotic limit of thickness kz ds 1. In this limit, Andreev bound states in the superconductor [13–15], which are localized near the interfaces, cannot carry efficiently spin information across the superconducting spacer and do
22
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
not contribute substantially to the magnetic coupling. Therefore, the contribution from Andreev bound states is not included here.
4. Results and discussion ðiÞ
Provided that the pair breaking parameter apb is small, the Tc of the superconducting oxide film is not strongly suppressed by the ferromagnetic oxide boundaries, and the exchange coupling Hex ðt2g ; sÞ appearing in (8) may be treated as a perturbation. Upon integration of the t2g (local) fermionic degrees of freedom, and effective action is obtained from which the effective magnetic Hamiltonian Heff containing only the magnetic coupling across the superconductor is extracted Heff ¼ Jeff ðds ÞSy1 Sy2 :
ð14Þ
Here Jeff ðds Þ is given by the correlation function X Z Z Jeff ðds Þ ¼ J1 J2 dz1 dz2 UNMkk ðz1 ; z2 Þ; N ;Mkk
oX
ð15Þ under the assumption that the momentum parallel to the FO/SO interface is conserved, i.e., the FO/ SO interface is translationally invariant. Such an assumption implies that the FO/SO interface is very well lattice matched and atomically flat. The matrix element UNM kk ðz1 ; z2 Þ ¼ 2DNM ðz1 ; z2 ÞQNM ðkk Þ defines the non-local susceptibility of the superconductor to the magnetic perturbation near its boundaries. The weighting factor DNM ðz1 ; z2 Þ ¼ wN kk ðz1 ÞwN kk ðz2 ÞwMkk ðz1 ÞwMkk ðz2 Þ; ð16Þ contains the eigenstates wN kk ðzÞ, i.e., He ðz; kk ÞwN kk ðzÞ ¼ nn ðkk ÞwN kk ðzÞ of the one-dimensional Hamiltonian He ðz; kk Þ ¼ tz cosðci@z Þ leff þ U ðzÞ; where the effective chemical potential leff ¼ l kk2 =2m with kk2 ¼ kx2 þ ky2 . The additional term QNM ðkk Þ ¼ CTNM ðkk ÞTNM ðkk Þ þ CPNM ðkk ÞPNM ðkk Þ; ð17Þ CTNM ðkk Þ
contains the coherence factor ¼ ½puN puM þ 2 pvN pvM ; the thermal factor TNM ¼ ½f ðEM Þ f ðEN Þ = ½EM EN in the quasiparticle–quasihole chan-
nel, and the coherence factor CPNM ðkk Þ ¼ ½puN pvM 2 puN pvM ; the thermal factor PNM ¼ ½f ðEM Þ þ f ðEN Þ 1 =½EM þ EN in the quasiparticle–quasiparticle channel. The coefficients puN and pvN de2 fined respectively by jpuN j ¼ ½1 þ nN =EN =2 and 2 jpvN j ¼ ½1 nN =EN =2, reflect the simplification hDðkk ; zÞi Dav ðkk Þ thus leading to the approximate eigenenergies EN2 ðkk Þ n2N ðkk Þ þ D2av ðkk Þ. The magnetic coupling Jeff ðds Þ (Eq. (15)) across the superconducting oxide layer is calculated numerically in order to establish the coexistence of three-dimensional ferromagnetism or antiferromagnetism with d-wave superconductivity at T ¼ 0. The magnetic coupling is calculated within the approximations outlined in Sections 2 and 3 with parameters U0 ¼ 1, Ht ðeg ; sÞ ¼ 0 corresponding to the case where the ferromagnetic oxide is insulating, and where only the coupling with localized t2g spins at the FO/SO interfaces is important. In addition, the parameters for the superconducting oxide are tz ¼ 10 K, and EF ¼ 104 K, within the assumption that a nearly two-dimensional version of the BCS theory applies, therefore constraining any quantitative analysis to the overdoped limit of copper oxides. Under all these considerations the following results are obtained. A plot for Jeff ðds Þ is presented in Fig. 3, for growth along the (0 0 1) direction (c-axis) of the copper oxide superconductor. For a d-wave superconducting oxide (e.g. YBCO) the main contribution to Jeff ðds Þ comes from wave vectors connecting points at the Fermi surface along the (0 0 1) direction (c-axis) and for kk lying along the directions of nodes of the order parameter Dav ðkk Þ ¼ Dd kk2 cosð2/Þ; where / is an angle in the parallel ðxyÞ plane. In this case, virtual quasiparticles have zero-energy threshold, and thus can mediate more effectively magnetic coupling. The optimal connecting vectors ð0; 0; QFz Þ along the is (0 0 1) direction have QFz ¼ p=c, where c 12 A the unit cell size along the (0 0 1) (z-direction), connecting states of fixed parallel momentum kFk at angles / ¼ p=4; 3p=4. In the case of an swave superconducting oxide (possibly NCCO) the order parameter Dav ðkk Þ ¼ Ds is independent of kk at the corresponding Fermi energy and all quasiparticle states are gapped. This implies that the virtual quasiparticles have to pay an energy cost
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
2e-05 1.5e-05 1e-05 5e-06 0 -5e-06 -1e-05 -1.5e-05 -2e-05 -2.5e-05 10
23
+
+ -
Qz
+
+ -
(0,0,1) ds
15
20
25
30
35
40
z
Lz
y x
Fig. 3. Indirect magnetic coupling Y ¼ Jeff ðds Þ=EJ dependence on thickness ds of superconducting layer, for FO/SO trilayer grown along (0 0 1) direction. Here, EJ ¼ J1 J2 Ns ðEF Þ, where Ns ðEF Þ is the density of states at EF for the normal state of the SO. The dotted line corresponds to the case where the order parameter amplitude D ¼ 0 (normal state). The solid (dashed) line corresponds to a d-wave (s-wave) superconductor.
Fig. 4. Top left: vector ð0; 0; Qz Þ along the (0 0 1) direction and the Fermi surface of the normal state of the superconductor. Top right: vector ð0; 0; Qz Þ and the nodal structure of the dwave order parameter. Bottom left: schematic of the trilayer heterostructure showing the superconductor thickness ds and the oscillation period Lz . Bottom right: coordinate system and (0 0 1) orientation.
equal to Ds , and thus the magnetic coupling is dramatically reduced. In comparison, when Dd ¼ 0 or Ds ¼ 0, the system is normal, and the virtual quasiparticle states become just virtual electronic states which mediate the magnetic interaction. The magnitude of Jeff ðds Þ in the normal state, for the Fourier component ð0; 0; Qz Þ, is indicated by the dotted line in Fig. 3. Notice that the suppression of magnetic coupling is much stronger in the s-wave than in the d-wave case, and that no new periods are generated by the appearance of superconductivity. These results indicate that the ground state of the FO/SO/FO trilayer changes from antialigned (antiferromagnetic) to aligned (ferromagnetic) with coexisting superconductivity. The optimal Fermi vector along the (0 0 1) zdirection is kFz ¼ p=2c, which means that when (approximately six unit kFz ds ¼ 10, ds ¼ 76:4 A cells). The oscillation period is predicted to be , where the ground state of the Lz ¼ 2c ¼ 24 A system oscillates between either coexisting ferromagnetism and d-wave superconductivity or coexisting antiferromagnetism and d-wave superconductivity (Fig. 4). If the superconducting oxide spacer were triplet (p-wave or f-wave) the magnetic coupling would be more effective given that the proximity to ferromagnets is not necessarily pair breaking, which is definitely the case for s-wave
and d-wave systems, where some reduction of Tc is inevitable. However, there are no known high Tc triplet superconductors, and thus this question is presently only of academic interest.
5. Final comments The calculations outlined in this paper provide a first approximate answer to the problem of coexistence of magnetism and superconductivity in FO/ SO heterostructures, where the SO is a d-wave superconductor. However, there are several important comments regarding the limitations of the approach which are in order. First, the SO is assumed to be in the BCS limit, which restricts our results to the overdoped limit of the cuprate oxides. Second, the calculation is perturbative in nature and relies heavily on the assumption that the ferromagnetic oxides are weak pair breakers. Third, the effects of Andreev bound states are neglected given that only the asymptotic limit of kFz ds 1 is considered here. Fourth, the inevitable effects of interfacial roughness are not taken into account in the present calculation. Fifth, the results are only qualitative in nature given the local approximation for the Bogoliubov–deGennes equations and the spatial averaging of the order
24
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
parameter. However, these calculations can provide guidance to experimentalists regarding the observation of the coexistence of magnetism and superconductivity in nanometer scale complex oxide heterostructures. Furthermore, in the case of multilayers with several repeat units, this calculation provides the qualitative behavior of the effective coupling between two consecutive ferromagnetic layers. It is also important to comment that the nonessential assumption that the ferromagnetic layer thickness df ds can be relaxed quite easily. In the case of an insulating ferromagnetic oxide, the superconducting state of the copper oxides should be largely insensitive to df because there is no superconducting proximity effect on the ferromagnetic side, and the magnetic proximity effect on the superconducting side is concentrated at the FO/SO interface. However, when the ferromagnetic oxide is half-metallic, there is still a negligible superconducting proximity effect on the ferromagnetic side (unless somehow triplet or LOFF-type superconductivity could be induced), but there is a magnetic proximity effect on the superconducting side due to the itinerant nature of spin-polarized conduction electrons of the ferromagnet. These spin-polarized electrons can penetrate inside the superconductor creating an electronic magnetic wind that is pair breaking, and thus should suppress Tc more than in the insulating ferromagnetic case. For a more standard metallic ferromagnetic oxide both the superconducting and magnetic proximity effects may be important and should be taken into account. In this last case, it also becomes harder to separate the two proximity effects. Thus, FO/SO heterostructures consisting of insulating or halfmetallic manganese oxides and d-wave copper oxide superconductors could be used to study specifically the magnetic proximity effect discussed above. In addition, generalizations of these calculations can be performed for growth along the (1 1 0) and (1 0 0) directions of the superconductor, in which cases the magnetic coupling for d-wave superconductors is qualitatively modified, while for s-wave superconductors it remains qualitatively unchanged. For a d-wave superconductor coupling along the (1 1 0) direction should be stronger
than along the (0 0 1) or (1 0 0) directions. This occurs because the nodes of the order parameter occur exactly along (1 1 0), which means that a large number of zero and low-energy virtual quasiparticles can mediate the spin information across the superconductor more efficiently. However, growing these heterostructures along the (1 1 0) or (1 0 0) directions presents a serious experimental challenge. Lastly, in FO/SO multilayers, superconducting oxide layers can couple via ferromagnetic oxide layers and stabilize a three-dimensional superconducting phase, provided that the ferromagnetic oxide is thin enough. This study is particularly interesting for the case of half-metallic or insulating ferromagnetic oxides, where careful monitoring of the Tc variation as a function of thickness of different layers may be now experimentally possible.
6. Summary The magnetic coupling Jeff ðds Þ between ferromagnetic oxides separated by d-wave superconducting oxides has been calculated for a trilayer geometry. The specific systems considered were nanometer scale heterostructures of ferromagnetic manganese oxides (or secondarily ruthenium oxides) and superconducting copper oxides. For simplicity, the ferromagnetic oxide was chosen to be insulating, by considering only the contribution of localized t2g electrons. The ferromagnetic oxides are assumed to have a high Curie temperature Tf , but to be weak pair breakers [7,8], such that the Tc of the superconducting oxide is not completely suppressed down to the nanometer scale. These assumptions seem to be verified experimentally where coexistence of superconductivity and ferromagnetism in these systems exist down to the nanometer scale (a few unit cells) [1–5]. At T ¼ 0, it was found that Jeff ðds Þ along the (0 0 1) z-direction oscillates as a function of the thickness ds of the superconducting oxide, and that the magnitude of this coupling is larger when the superconducting oxide is d-wave than when it is an s-wave. These calculations lead to the conclusion that ferromagnetism (aligned magnetizations) or antiferromagnetism (antialigned magnetizations) coexist
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
with d-wave superconductivity in nanometer scale ferromagnetic oxide and superconducting oxide heterostructures. Furthermore, it was predicted that the ground state of the heterostructure in the asymptotic limit of kFz ds 1 changes from ferromagnetic to antiferromagnetic with a period of two unit cells. This result, if confirmed experimentally is quite remarkable, since the effect is expected to be observable only at the nanometer which is less than 11 unit scale, i.e., for ds < 130 A cells for most copper oxides (e.g. YBCO).
References [1] P. Przyslupski, T. Nishizaki, N. Kobayashi, S. Kolesnik, T. Skoskiewicz, E. Dynowwska, Physica B 259 (1999) 820. [2] P. Przyslupski, I. Komissarov, E. Dynowska, S. Kolesnik, T. Skoskiewicz, J. Wosik, T. Nishizaki, N. Kobayashi, Physica C 341 (2000) 767.
25
[3] H.U. Habermeier, G. Cristiani, R.K. Kremer, O. Lebedev, G. van Tendello, Physica C 364 (2001) 298. [4] Z. Sefrioui, M. Varela, V. Pena, D. Arias, J.E. Villegas, W. Saldarriaga, P. Prieto, C. Leon, J.L. Martinez, J. Santamaria, cond-mat/0207164, 2002. [5] P. Przyslupski, I. Komissarov, P. Dluzewski, J. Pelka, E. Dynowska, M. Sawicki, pre-print, 2002. [6] A.M. Goldman, V. VasÕko, P. Kraus, K. Nikolaev, V.A. Larkin, J. Magn. Magn. Mater. 200 (1999) 69. [7] C.A.R. Sa de Melo, Phys. Rev. Lett. 79 (1997) 1933. [8] C.A.R. Sa de Melo, Phys. Rev. B (2000) 12303. [9] D.M. Edwards, J. Mathon, R.B. Muniz, M.S. Phan, Phys. Rev. Lett. 67 (1991) 493. [10] P. Bruno, C. Chappert, Phys. Rev. Lett. 67 (1991) 3598. [11] C.A.R. Sa de Melo, M. Randeria, J.R. Engelbrecht, Phys. Rev. Lett. 71 (1993) 3202. [12] J.R. Engelbrecht, M. Randeria, C.A.R. Sa de Melo, Phys. Rev. B 55 (1997) 15153. [13] L.J. Buchholtz, G. Zwicknagl, Phys. Rev. B 23 (1981) 5788. [14] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [15] T. Lofwander, V.S. Shumeiko, G. Wendin, Supercond. Sci. Technol. 14 (2001) R53.
Interplay of magnetism and superconductivity at the nanometer scale: the case of complex oxide heterostructures C.A.R. S a de Melo School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Abstract It is suggested that magnetism and superconductivity can coexist at the nanometer scale in heterostructures consisting of complex ferromagnetic and superconducting oxides. The geometry discussed is that of a trilayer where the superconductor is a copper oxide and is sandwiched by two layers of ferromagnets consisting of manganese oxides. The superconductor is assumed to be layered and have a d-wave order parameter. It is shown that the ground state of such nanometer scale heterostructures alternates between ferromagnetic and antiferromagnetic depending on the thickness of the superconductor. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Manganite/cuprate multilayers; Ferromagnetic-superconducting heterostructures; Nanometer scale; Ferromagnetic superconductors
1. Introduction Very recently several experimental groups [1–5] were able to grow, characterize and measure a few properties of complex oxide heterostructures consisting of manganese oxides (colossal magnetoresistance materials) and copper oxides (high Tc superconductors). A nice review about these complex oxide heterostructures can be found in the literature [6], where effects such as spin injection properties, critical currents and critical temperatures are discussed. Part of these experimental efforts were inspired in earlier theoretical work [7,8] were the coexistence and interplay of magnetism and superconductivity in multilayers con-
E-mail address: [email protected] (C.A.R. S a de Melo).
sisting of manganese oxides and copper oxides was suggested, but not fully analysed. In the present work, the coexistence and interplay of magnetism and superconductivity at the nanometer scale is considered for d-wave superconductors. Situations where the proximity to a ferromagnet completely destroys superconductivity or where the proximity to a superconductor completely destroys ferromagnetism are not interesting for the purpose of this paper. The main conditions for the coexistence and interplay of these two phenomena is reflected on the magnitude of the proximity effects between the superconductors and ferromagnets, when the thicknesses of different layers are in the nanometer range. For coexistence of these two phases it is essential from the superconductorÕs perspective that (a) magnetic pair breaking effects are not so large, and (it is very helpful) that the superconductor has a high critical
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00635-X
18
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
temperature. From the ferromagnetÕs perspective it is essential that the superconductor does not destroy magnetism via the proximity effect, and (it is very helpful) that the ferromagnet has a high Curie temperature [7,8]. These conditions are very difficult to be satisfied simultaneously by most ferromagnets and superconductors in the regime of nanometer thicknesses, and if we add the inevitable experimental restriction of good epitaxial growth, only very special classes of materials seem to satisfy the conditions mentioned above. Two classes of materials (a) manganese oxides and copper oxides; and (b) ruthenium oxides and copper oxides were suggested as possible candidates for coexistence and interplay of magnetism and superconductivity, given that the conditions outlined above seemed to be theoretically satisfied [7,8]. In this manuscript the focus is on the existence of a three-dimensional magnetic phase mediated by coexisting d-wave superconductivity at T ¼ 0, and on the derivation of the effective magnetic coupling responsible for the stabilization of these magnetic phases. Thus, the main interest here is on the magnitude and sign of the magnetic coupling between neighboring ferromagnetic layers spaced by a d-wave superconducting layer, which dictates the fate of the ground state of the heterostructure. For definiteness and calculational purposes it is necessary to chose a geometry, which is indicated in Fig. 1. However, the physical mechanism is largely independent of the choice of geometry, and is discussed next. One of the possible ways of describing theoretically magnetic coupling through metallic multilayers is to use an RRKY-type of argument, where the electronic spins are polarized by the ferro-
FO
LSMO
SO
YBCO (NCCO)
FO
LSMO
Fig. 1. Trilayer geometry for ferromagnetic oxide/superconducting oxide (FO/SO) multilayers. The FO is assumed to be La1y Sry MnO3 (LSMO), while the SO is assumed to be either YBa2 Cu3 O7d (YBCO), or Nd2x Cex CuO4 (NCCO).
magnet, creating an effective field felt by the other ferromagnet a certain distance away. This spin polarization involves a summation over virtual electronic states and has an oscillatory nature in 1 real space characterized by ‘p ¼ ð2kF Þ , where kF is the relevant Fermi momentum of the metallic spacer. The magnetic coupling that results from this analysis is very sensitive to the shape of the Fermi surface of the metal [9,10], given that nesting plays an important role in the final form of the magnetic coupling at T ¼ 0. A similar type of argument can be used for magnetic coupling across a singlet (s-wave or dwave) superconductor at T ¼ 0. The essential difference now is that the magnetic coupling is mediated by virtual quasiparticles states. When the superconductor is s-wave, a full gap opens up in the excitation spectrum. The appearance of this gap makes the polarization of virtual quasiparticle states more difficult, and reduces both the range and the magnitude of the magnetic coupling at T ¼ 0 in comparison to the expected coupling for a normal spacer layer at T ¼ 0. Thus, for s-wave superconductors there is a large spin polarization stiffness controlled by the gap to quasiparticle excitations. When the superconductor is d-wave, the gap in the excitation spectrum is highly anisotropic, and have nodes along special directions. This leads to the existence of zero and lowenergy quasiparticles in d-wave superconductors. For wave vectors connecting nodal regions it is possible to carry more spin information (at longer length scales) across d-wave superconductors than across s-wave superconductors. Therefore the detrimental effects to magnetic coupling caused by a full gap in the excitation spectrum (as in the swave case) are less effective in d-wave systems, because of the existence of zero-energy quasiparticles. Therefore, d-wave high Tc superconductors are ideal spacer systems to study magnetic coupling and coexistence of magnetism and superconductivity. In order to study this coexistence and magnetic coupling, the rest of the paper is organized as follows. In Section 2, the Hamiltonian used for the trilayer geometry is discussed. In Section 3, the functional integral method, used to analyse coexistence and magnetic coupling, is outlined. In
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
Section 4, the main results are presented and analysed. In Section 5, comments are made regarding the approximations used and other important questions of experimental relevance. Lastly, in Section 6, the main conclusions are summarized.
2. Hamiltonian To study the coexistence and interplay of magnetism and superconductivity in nanometer scale complex oxide heterostructures consisting of ferromagnetic oxides (FO) and superconducting oxides (SO) the trilayer structure shown in Fig. 1 is chosen. The trilayer consists of two manganese oxide layers as the ferromagnets separated by a dwave copper oxide superconductor. In the FO/SO multilayered heterostructures the FO can be, for instance, La1y Sry MnO3 (LSMO), La1y Cay MnO3 (LCMO) or Nd1y Sry MnO3 (NSMO) and the SO can be YBa2 Cu3 O7d (YBCO), La2x Srx CuO4 (LSCO) (d-wave) or Nd2x Cex CuO4 (NCCO) (swave). It is assumed that the Curie temperature Tf of the ferromagnetic oxide layers is larger than the critical temperature Tc of the superconducting copper oxide layer. To describe this extremely complex heterostructure the following Hamiltonian: H ¼ Hf 1 þ Hf 2 þ Hs þ Hsf 1 þ Hsf 2 ;
ð1Þ
is used, where repeated greek indices are used to indicate summation. The first two terms (Hf 1 and Hf 2 ) on the right hand side of Eq. (1) are contributions due to the ferromagnets. Hf j ¼
Z
drfjay ðrÞfðKf j Þac þ ðHcj Þac gfjc ðrÞ:
j
which characterizes the existence of two semi-infinite ferromagnets for z 6 ds =2 and z P ds =2. The intrinsic assumption is that df ds , which corresponds to the worst case scenario for suppression of superconductivity. Additional comments regarding this non-essential choice will be made in Section 4. The magnitude U0 controls if the ferromagnet is half-metallic (when U0 is smaller than the bandwidth of the copper oxide) or insulating (when U0 is bigger than the bandwidth of the copper oxide), see Fig. 2. The double exchange mechanism is implicitly assumed, but not explicitly taken into account in the continuum description of the ferromagnet provided here. The second term on the right hand side of Eq. (2) corresponds to the exchange R energy ðHcj Þac ¼ ðrl Þac hlj ðrÞ; where hlj ðrÞ ¼ dr0 Jlj ðr; r0 ÞSlj ðr0 Þ Fj ðrÞ; with Slj ðr0 Þ being the total spin associated
SINGLE PARTICLE POTENTIALS
Ef
eg
eg U0
t2g
ds
j
Fj ðrÞ ¼ H½ðÞ z ds =2 H½ðÞ z ds =2 df ; ð3Þ
t2g
PAIR POTENTIAL
ð2Þ
The first term on the left hand side of Eq. (2) is ðKf j Þac ¼ ½Kf j þ Uj ðrÞ dac ; and contains the kinetic 2 energy Kf j ¼ Fj ðrÞ½ðirÞ =2mf lf ; associated with the itinerant (eg ) electrons of the CMR-material, and the background potential energy Uj ðrÞ ¼ U0 Fj ðrÞ with the function
19
P0
ds Fig. 2. Top: single particle potentials for FO/SO trilayer, indicating eg and t2g states in the FO, and the dx2 y 2 states in the SO. Bottom: shape of projection function P ðrÞ that appears in the pair potential for FO/SO trilayer. EF is the Fermi level, ds is the thickness of the SO layer.
20
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
with the localized t2g orbitals in the jth ferromagnet. The total spin projection along the quantization axis (chosen to be parallel to the FO/SO interface along the y-direction) is Sy ¼ 3=2 corresponding to three localized electrons in the t2g orbitals. Here, the exchange coupling between delocalized eg -electrons and localized t2g -electrons is represented by the point contact (local) interaction Jij ðr; r0 Þ ¼ Jij dðr r0 Þ, where in addition, the exchange coupling field Jij points preferably along the y-direction (parallel to the FO/SO interfaces). The third term on the left hand side of Eq. (1) is Z Hs ¼ drwya ðrÞ½Ks dac wc ðrÞ þ V ; ð4Þ
of the superconductor. Given that the ferromagnets are half-metallic or insulating, it is assumed that the superconductors do not produce a proximity effect in the ferromagnetic. Furthermore, the coupling terms Hsf 1 and Hsf 2 involving the superconductor and the two ferromagnets are of the form:
and corresponds to the Hamiltonian describing the superconductor. The first term on the right hand side of Eq. (4) is the kinetic energy term: " # 2 ðirk Þ Ks ¼ P ðrÞ þ tz cosðci@z Þ ls ; ð5Þ 2ms
with local interaction Jsf j ðr; r0 Þ ¼ Jj dðr r0 Þ. This term involves localized spins of the t2g orbitals of the ferromagnet and spins of the charge carriers in the The second term Ht ðeg ; R Rsuperconductor. y 0 0 sÞ ¼ dr dr0 ½tab ðr; r Þf ðr Þwb ðrÞ þ C:C: , involves ja sf j charge transfer between the ferromagnet (when it is half-metallic) and the superconductor. For the purposes of this paper, this term will be neglected in comparison to the first. However, this additional term is pair breaking since it allows spinpolarized charged carriers (possibly over long length scales) inside a singlet superconductor. The presence of this term is responsible for an additional reduction in the critical temperature of the superconductor, when sandwiched by a halfmetallic instead of an insulating manganese oxide ferromagnet. The maximal Curie temperature of bulk CMRferromagnet LaMnO is Tf 300 K, and the maximal critical temperature of bulk YBCO (LSCO) is Tc 90 K (Tc 40 K). However, the pair breaking ðiÞ parameter apb ¼ Ji hSzðiÞ i=2pTc seems to be small as suggested earlier [7,8] producing a finite Tc down to very few monolayers (nanometer lengths) of the superconducting copper oxides. Currently there is strong experimental evidence for this since the coexistence of magnetism and superconductivity in manganese oxide and copper oxide heterostructures down to a few monolayers of copper oxides has been firmly established [1–5]. The suppression of Tc in these systems can be from a few percent to complete (100%) depending on the thicknesses and compositions of the magnetic and superconducting
containing the chemical potential ls of the superconductor. Note the presence of the periodic operator cosðci@z Þ used to mimic the layered structure of the superconducting oxide. The projection function P ðrÞ ¼ Hðz þ ds =2Þ Hðz þ ds =2Þ just identifies the boundaries of the superconductor of thickness ds , i.e., the spacer extends from ds =2 to þds =2. The second term on the right hand side of Eq. (4) is the interaction term Z 1 dr0 gdcab ðr; r0 Þwyd ðrÞwyc ðr0 Þwa ðr0 Þwb ðrÞ Pe ðr; r0 Þ; V ¼ 2 ð6Þ where gdcab ðr; r0 Þ ¼ V ðjrk r0k jÞdd;c da;b dðz z0 Þ; is the interaction tensor which simulates singlet pairing only in the planes parallel to the xy-plane. The term V ðjrk r0k jÞ represents a two-dimensional interaction of the type V ðRÞ ¼ V1 HðR1 RÞ þ V0 HðR R1 ÞHðR0 RÞ; ð7Þ r0k j.
with R ¼ jrk Notice that V ðRÞ has a finite range, which allows for d-wave pairing within the xy-plane. Furthermore, the projection operator Pe ðr; r0 Þ ¼ P ðrÞ P ðr0 Þ enforces that the pairing interaction exists only within the spatial boundaries
Hsf i ¼ Hex ðt2g ; sÞ þ Ht ðeg ; sÞ; where Hex ðt2g ; sÞ ¼
XZ Z
ð8Þ
dr dr0 Jsf j ðr; r0 Þ
j
wya ðrÞðrl Þab wb ðrÞSlj ðr0 Þ;
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
oxides. Since here the interest is only in the T ¼ 0 (ground state) limit, the subject of Tc suppression will not be discussed any further.
leads to the Bogoliubov–deGennes (BdG) equations Z EN uN a ðrÞ ¼ He ðrÞuN a ðrÞ þ dr0 Dab ðr; r0 ÞvN b ðr0 Þ; ð11Þ
3. Method In order to calculate the magnetic coupling across the d-wave superconductor several additional assumptions are made. First, it is assumed that the magnetic coupling between the localized t2g spins in the half-metallic or insulating ferromagnet and the spin of the charge carriers in the d-wave superconductor does not completely destroy superconductivity. This means that the pair breaking parameter apb is reasonably small. Second, it is assumed that the momentum parallel to the FO/SO interface is conserved, i.e., the FO/SO interface is translationally invariant. Such assumption implies that the FO/SO interface is very well lattice matched and atomically flat. Third, it is assumed that the normal state of the d-wave superconductor is not the pseudogap phase, thus the results derived here are strictly valid only for the overdoped regime of cuprate superconductors. The functional integral method is used to calculate perturbatively the magnetic coupling across the d-wave superconductor. Within this approach the action corresponding to the Hamiltonian Hs of the superconductor is Ss ¼
Z
b
Z
21
dr wya ðrÞ@s dac wc ðrÞ Hs :
ð9Þ
0
Introduction of the matrix field Dab ðr; r0 Þ ¼ of the fermionic variables as wa ðrÞ ¼ N ½uN a ðrÞcN þ vN a ðrÞcyN and integration over the residual fermionic variables cN and cyN leads to an action that depends only on the variables Dab , uN a , and vN ;a and Hermitian conjugates. Subsequent minimization with respect to Dyab leads to the order parameter equation X KN Dab ðr; r0 Þ ¼ gabdc ðr; r0 Þ Wcd ðr; r0 Þ Wdc ðr0 ; rÞ ; 4 N 1 g ðr; r0 Þwd ðr0 Þwc ðrÞ;Protation 2 abdc
ð10Þ 0
where KN ¼ ð1 2f ðEN ÞÞ, and Wcd ðr; r Þ ¼ vN c ðrÞ uN d ðr0 Þ: Minimization with respect to uN a , and vN ;a
EN vN a ðrÞ ¼ He ðrÞvN a ðrÞ þ
Z
dr0 Dab ðr; r0 ÞuN b ðr0 Þ: ð12Þ
The magnetic coupling across the superconductor is mediated by quasiparticles which carry spin information, thus it is necessary to calculate the quasiparticle energies from the BdG equations (11) and (12). Since the momentum parallel to the FO/ SO interface is assumed to be conserved the BdG amplitudes can be written as uN a ¼ u~N a ðzÞ exp½ikk rk and vN a ¼ v~N a ðzÞ exp½ikk rk . Furthermore, the local approximation is used where the order parameter Dab ðr; r0 Þ becomes Dab ðkk ; zÞ upon Fourier transformation of the parallel ðxyÞ coordinates. This leads to simplified BdG equations EN ~hN ðzÞ ¼ H~e ðkk ; zÞr3 ~hN ðzÞ þ Dðkk ; zÞr1 ~hN ðzÞ; ð13Þ where ~hN ðzÞ is a vector with components ½~ uN ðzÞ; v~N ðzÞ , and ri are the standard Pauli matrices. This equation, however, is strictly valid only in the limit where the phase coherence length nz is large ðkz nz 1Þ. This local equation can be a quite good approximation even in the regime where the BCS theory is not truly applicable, i.e., when the size of the Cooper pairs are very small, provided that the phase coherence length is large! In the BCS approximation, the size of the Cooper pair and the phase coherence length are large and nearly identical, however towards local pairing theories (Bose–Einstein condensation regime) the size of the Cooper pair is small, but the phase coherence is still very large [11,12]. Thus, even substantially away from the BCS limit the local theory can be used for the calculation of the magnetic coupling across the superconductor in the asymptotic limit of thickness kz ds 1. In this limit, Andreev bound states in the superconductor [13–15], which are localized near the interfaces, cannot carry efficiently spin information across the superconducting spacer and do
22
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
not contribute substantially to the magnetic coupling. Therefore, the contribution from Andreev bound states is not included here.
4. Results and discussion ðiÞ
Provided that the pair breaking parameter apb is small, the Tc of the superconducting oxide film is not strongly suppressed by the ferromagnetic oxide boundaries, and the exchange coupling Hex ðt2g ; sÞ appearing in (8) may be treated as a perturbation. Upon integration of the t2g (local) fermionic degrees of freedom, and effective action is obtained from which the effective magnetic Hamiltonian Heff containing only the magnetic coupling across the superconductor is extracted Heff ¼ Jeff ðds ÞSy1 Sy2 :
ð14Þ
Here Jeff ðds Þ is given by the correlation function X Z Z Jeff ðds Þ ¼ J1 J2 dz1 dz2 UNMkk ðz1 ; z2 Þ; N ;Mkk
oX
ð15Þ under the assumption that the momentum parallel to the FO/SO interface is conserved, i.e., the FO/ SO interface is translationally invariant. Such an assumption implies that the FO/SO interface is very well lattice matched and atomically flat. The matrix element UNM kk ðz1 ; z2 Þ ¼ 2DNM ðz1 ; z2 ÞQNM ðkk Þ defines the non-local susceptibility of the superconductor to the magnetic perturbation near its boundaries. The weighting factor DNM ðz1 ; z2 Þ ¼ wN kk ðz1 ÞwN kk ðz2 ÞwMkk ðz1 ÞwMkk ðz2 Þ; ð16Þ contains the eigenstates wN kk ðzÞ, i.e., He ðz; kk ÞwN kk ðzÞ ¼ nn ðkk ÞwN kk ðzÞ of the one-dimensional Hamiltonian He ðz; kk Þ ¼ tz cosðci@z Þ leff þ U ðzÞ; where the effective chemical potential leff ¼ l kk2 =2m with kk2 ¼ kx2 þ ky2 . The additional term QNM ðkk Þ ¼ CTNM ðkk ÞTNM ðkk Þ þ CPNM ðkk ÞPNM ðkk Þ; ð17Þ CTNM ðkk Þ
contains the coherence factor ¼ ½puN puM þ 2 pvN pvM ; the thermal factor TNM ¼ ½f ðEM Þ f ðEN Þ = ½EM EN in the quasiparticle–quasihole chan-
nel, and the coherence factor CPNM ðkk Þ ¼ ½puN pvM 2 puN pvM ; the thermal factor PNM ¼ ½f ðEM Þ þ f ðEN Þ 1 =½EM þ EN in the quasiparticle–quasiparticle channel. The coefficients puN and pvN de2 fined respectively by jpuN j ¼ ½1 þ nN =EN =2 and 2 jpvN j ¼ ½1 nN =EN =2, reflect the simplification hDðkk ; zÞi Dav ðkk Þ thus leading to the approximate eigenenergies EN2 ðkk Þ n2N ðkk Þ þ D2av ðkk Þ. The magnetic coupling Jeff ðds Þ (Eq. (15)) across the superconducting oxide layer is calculated numerically in order to establish the coexistence of three-dimensional ferromagnetism or antiferromagnetism with d-wave superconductivity at T ¼ 0. The magnetic coupling is calculated within the approximations outlined in Sections 2 and 3 with parameters U0 ¼ 1, Ht ðeg ; sÞ ¼ 0 corresponding to the case where the ferromagnetic oxide is insulating, and where only the coupling with localized t2g spins at the FO/SO interfaces is important. In addition, the parameters for the superconducting oxide are tz ¼ 10 K, and EF ¼ 104 K, within the assumption that a nearly two-dimensional version of the BCS theory applies, therefore constraining any quantitative analysis to the overdoped limit of copper oxides. Under all these considerations the following results are obtained. A plot for Jeff ðds Þ is presented in Fig. 3, for growth along the (0 0 1) direction (c-axis) of the copper oxide superconductor. For a d-wave superconducting oxide (e.g. YBCO) the main contribution to Jeff ðds Þ comes from wave vectors connecting points at the Fermi surface along the (0 0 1) direction (c-axis) and for kk lying along the directions of nodes of the order parameter Dav ðkk Þ ¼ Dd kk2 cosð2/Þ; where / is an angle in the parallel ðxyÞ plane. In this case, virtual quasiparticles have zero-energy threshold, and thus can mediate more effectively magnetic coupling. The optimal connecting vectors ð0; 0; QFz Þ along the is (0 0 1) direction have QFz ¼ p=c, where c 12 A the unit cell size along the (0 0 1) (z-direction), connecting states of fixed parallel momentum kFk at angles / ¼ p=4; 3p=4. In the case of an swave superconducting oxide (possibly NCCO) the order parameter Dav ðkk Þ ¼ Ds is independent of kk at the corresponding Fermi energy and all quasiparticle states are gapped. This implies that the virtual quasiparticles have to pay an energy cost
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
2e-05 1.5e-05 1e-05 5e-06 0 -5e-06 -1e-05 -1.5e-05 -2e-05 -2.5e-05 10
23
+
+ -
Qz
+
+ -
(0,0,1) ds
15
20
25
30
35
40
z
Lz
y x
Fig. 3. Indirect magnetic coupling Y ¼ Jeff ðds Þ=EJ dependence on thickness ds of superconducting layer, for FO/SO trilayer grown along (0 0 1) direction. Here, EJ ¼ J1 J2 Ns ðEF Þ, where Ns ðEF Þ is the density of states at EF for the normal state of the SO. The dotted line corresponds to the case where the order parameter amplitude D ¼ 0 (normal state). The solid (dashed) line corresponds to a d-wave (s-wave) superconductor.
Fig. 4. Top left: vector ð0; 0; Qz Þ along the (0 0 1) direction and the Fermi surface of the normal state of the superconductor. Top right: vector ð0; 0; Qz Þ and the nodal structure of the dwave order parameter. Bottom left: schematic of the trilayer heterostructure showing the superconductor thickness ds and the oscillation period Lz . Bottom right: coordinate system and (0 0 1) orientation.
equal to Ds , and thus the magnetic coupling is dramatically reduced. In comparison, when Dd ¼ 0 or Ds ¼ 0, the system is normal, and the virtual quasiparticle states become just virtual electronic states which mediate the magnetic interaction. The magnitude of Jeff ðds Þ in the normal state, for the Fourier component ð0; 0; Qz Þ, is indicated by the dotted line in Fig. 3. Notice that the suppression of magnetic coupling is much stronger in the s-wave than in the d-wave case, and that no new periods are generated by the appearance of superconductivity. These results indicate that the ground state of the FO/SO/FO trilayer changes from antialigned (antiferromagnetic) to aligned (ferromagnetic) with coexisting superconductivity. The optimal Fermi vector along the (0 0 1) zdirection is kFz ¼ p=2c, which means that when (approximately six unit kFz ds ¼ 10, ds ¼ 76:4 A cells). The oscillation period is predicted to be , where the ground state of the Lz ¼ 2c ¼ 24 A system oscillates between either coexisting ferromagnetism and d-wave superconductivity or coexisting antiferromagnetism and d-wave superconductivity (Fig. 4). If the superconducting oxide spacer were triplet (p-wave or f-wave) the magnetic coupling would be more effective given that the proximity to ferromagnets is not necessarily pair breaking, which is definitely the case for s-wave
and d-wave systems, where some reduction of Tc is inevitable. However, there are no known high Tc triplet superconductors, and thus this question is presently only of academic interest.
5. Final comments The calculations outlined in this paper provide a first approximate answer to the problem of coexistence of magnetism and superconductivity in FO/ SO heterostructures, where the SO is a d-wave superconductor. However, there are several important comments regarding the limitations of the approach which are in order. First, the SO is assumed to be in the BCS limit, which restricts our results to the overdoped limit of the cuprate oxides. Second, the calculation is perturbative in nature and relies heavily on the assumption that the ferromagnetic oxides are weak pair breakers. Third, the effects of Andreev bound states are neglected given that only the asymptotic limit of kFz ds 1 is considered here. Fourth, the inevitable effects of interfacial roughness are not taken into account in the present calculation. Fifth, the results are only qualitative in nature given the local approximation for the Bogoliubov–deGennes equations and the spatial averaging of the order
24
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
parameter. However, these calculations can provide guidance to experimentalists regarding the observation of the coexistence of magnetism and superconductivity in nanometer scale complex oxide heterostructures. Furthermore, in the case of multilayers with several repeat units, this calculation provides the qualitative behavior of the effective coupling between two consecutive ferromagnetic layers. It is also important to comment that the nonessential assumption that the ferromagnetic layer thickness df ds can be relaxed quite easily. In the case of an insulating ferromagnetic oxide, the superconducting state of the copper oxides should be largely insensitive to df because there is no superconducting proximity effect on the ferromagnetic side, and the magnetic proximity effect on the superconducting side is concentrated at the FO/SO interface. However, when the ferromagnetic oxide is half-metallic, there is still a negligible superconducting proximity effect on the ferromagnetic side (unless somehow triplet or LOFF-type superconductivity could be induced), but there is a magnetic proximity effect on the superconducting side due to the itinerant nature of spin-polarized conduction electrons of the ferromagnet. These spin-polarized electrons can penetrate inside the superconductor creating an electronic magnetic wind that is pair breaking, and thus should suppress Tc more than in the insulating ferromagnetic case. For a more standard metallic ferromagnetic oxide both the superconducting and magnetic proximity effects may be important and should be taken into account. In this last case, it also becomes harder to separate the two proximity effects. Thus, FO/SO heterostructures consisting of insulating or halfmetallic manganese oxides and d-wave copper oxide superconductors could be used to study specifically the magnetic proximity effect discussed above. In addition, generalizations of these calculations can be performed for growth along the (1 1 0) and (1 0 0) directions of the superconductor, in which cases the magnetic coupling for d-wave superconductors is qualitatively modified, while for s-wave superconductors it remains qualitatively unchanged. For a d-wave superconductor coupling along the (1 1 0) direction should be stronger
than along the (0 0 1) or (1 0 0) directions. This occurs because the nodes of the order parameter occur exactly along (1 1 0), which means that a large number of zero and low-energy virtual quasiparticles can mediate the spin information across the superconductor more efficiently. However, growing these heterostructures along the (1 1 0) or (1 0 0) directions presents a serious experimental challenge. Lastly, in FO/SO multilayers, superconducting oxide layers can couple via ferromagnetic oxide layers and stabilize a three-dimensional superconducting phase, provided that the ferromagnetic oxide is thin enough. This study is particularly interesting for the case of half-metallic or insulating ferromagnetic oxides, where careful monitoring of the Tc variation as a function of thickness of different layers may be now experimentally possible.
6. Summary The magnetic coupling Jeff ðds Þ between ferromagnetic oxides separated by d-wave superconducting oxides has been calculated for a trilayer geometry. The specific systems considered were nanometer scale heterostructures of ferromagnetic manganese oxides (or secondarily ruthenium oxides) and superconducting copper oxides. For simplicity, the ferromagnetic oxide was chosen to be insulating, by considering only the contribution of localized t2g electrons. The ferromagnetic oxides are assumed to have a high Curie temperature Tf , but to be weak pair breakers [7,8], such that the Tc of the superconducting oxide is not completely suppressed down to the nanometer scale. These assumptions seem to be verified experimentally where coexistence of superconductivity and ferromagnetism in these systems exist down to the nanometer scale (a few unit cells) [1–5]. At T ¼ 0, it was found that Jeff ðds Þ along the (0 0 1) z-direction oscillates as a function of the thickness ds of the superconducting oxide, and that the magnitude of this coupling is larger when the superconducting oxide is d-wave than when it is an s-wave. These calculations lead to the conclusion that ferromagnetism (aligned magnetizations) or antiferromagnetism (antialigned magnetizations) coexist
C.A.R. Sa de Melo / Physica C 387 (2003) 17–25
with d-wave superconductivity in nanometer scale ferromagnetic oxide and superconducting oxide heterostructures. Furthermore, it was predicted that the ground state of the heterostructure in the asymptotic limit of kFz ds 1 changes from ferromagnetic to antiferromagnetic with a period of two unit cells. This result, if confirmed experimentally is quite remarkable, since the effect is expected to be observable only at the nanometer which is less than 11 unit scale, i.e., for ds < 130 A cells for most copper oxides (e.g. YBCO).
References [1] P. Przyslupski, T. Nishizaki, N. Kobayashi, S. Kolesnik, T. Skoskiewicz, E. Dynowwska, Physica B 259 (1999) 820. [2] P. Przyslupski, I. Komissarov, E. Dynowska, S. Kolesnik, T. Skoskiewicz, J. Wosik, T. Nishizaki, N. Kobayashi, Physica C 341 (2000) 767.
25
[3] H.U. Habermeier, G. Cristiani, R.K. Kremer, O. Lebedev, G. van Tendello, Physica C 364 (2001) 298. [4] Z. Sefrioui, M. Varela, V. Pena, D. Arias, J.E. Villegas, W. Saldarriaga, P. Prieto, C. Leon, J.L. Martinez, J. Santamaria, cond-mat/0207164, 2002. [5] P. Przyslupski, I. Komissarov, P. Dluzewski, J. Pelka, E. Dynowska, M. Sawicki, pre-print, 2002. [6] A.M. Goldman, V. VasÕko, P. Kraus, K. Nikolaev, V.A. Larkin, J. Magn. Magn. Mater. 200 (1999) 69. [7] C.A.R. Sa de Melo, Phys. Rev. Lett. 79 (1997) 1933. [8] C.A.R. Sa de Melo, Phys. Rev. B (2000) 12303. [9] D.M. Edwards, J. Mathon, R.B. Muniz, M.S. Phan, Phys. Rev. Lett. 67 (1991) 493. [10] P. Bruno, C. Chappert, Phys. Rev. Lett. 67 (1991) 3598. [11] C.A.R. Sa de Melo, M. Randeria, J.R. Engelbrecht, Phys. Rev. Lett. 71 (1993) 3202. [12] J.R. Engelbrecht, M. Randeria, C.A.R. Sa de Melo, Phys. Rev. B 55 (1997) 15153. [13] L.J. Buchholtz, G. Zwicknagl, Phys. Rev. B 23 (1981) 5788. [14] C.R. Hu, Phys. Rev. Lett. 72 (1994) 1526. [15] T. Lofwander, V.S. Shumeiko, G. Wendin, Supercond. Sci. Technol. 14 (2001) R53.