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Charge, spin and orbitals in oxides

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- 1. Aspects of the eory of Oxide Interfaces Thilo Kopp Center for Electronic Correlations and Magnetism Universität AugsburgInternational Summer School on Surfaces and Interfaces in Correlated Oxides Vancouver 2011
- 2. Aspects of the eory of Oxide Interfaces Thilo Kopp Center for Electronic Correlations and Magnetism Universität Augsburg *»Electronic reconstruction« at interfaces of correlated electron systems *) coined by R. Hesper, L.H. Tjeng, A. Heeres & G.A. Sawatzky, PRB (2000)
- 3. coauthors and coworkersNatalia Pavlenko Lviv, UkraineJochen Mannhart MPI for Solid State Research, StuttgartGeorge Sawatzky UBC VancouverPeter Hirschfeld University of Florida, GainesvilleEvgeny Tsymbal University of Nebraska, LincolnFlorian Loder EKM, Universität AugsburgArno Kampf EKM, Universität AugsburgCyril Stephanos, Kevin Steffen EKM, Universität Augsburg
- 4. topics• electronic structure at LaAlO3/SrTiO3 interfaces• magnetism at LaAlO3/SrTiO3 interfaces and surfaces• superconductivity at transition metal oxide interfaces• negative compressibility of the 2-dimensional electron system
- 5. Charge Transport @ Interfaces of Oxidesparallel to interface perpendicular to interface vacuum YBCO LaAlO3 SrTiO3 YBCO YBCO Liao et al. (PRB, 2010) Schneider et al. (PRL, 2004)
- 6. Charge Transport @ Interfaces of Oxidesparallel to interface perpendicular to interface vacuum YBCO LaAlO3 SrTiO3 YBCO ? ? YBCO Liao et al. (PRB, 2011) Schneider et al. (PRL, 2004)
- 7. LaAlO3/SrTiO3 interface vacuum this talk LaAlO3 metallic interface SrTiO3MIT @ nc ∼ 10−13 /cm2 Y.C. Liao, T.K., C. Richter, A. Rosch, J. Mannhart PRB 83, 075402 (2011)
- 8. Electronic structure of LaAlO3/SrTiO3
- 9. LaAlO3/SrTiO3 interface stack of alternating subunit cell layers LaAlO3: … band insulator Δ = 5.6 eV AlO2 LaO TiO2 metallic interface SrO SrTiO3: … band insulator Δ = 3.2 eV quantum paraelectric (001)A. Ohtomo and H. Hwang, Nature 427, 423 (2004) high mobility electron gas formed at interface
- 10. The polar catastrophe ρ E V ρ E V AlO2- 0.5 + AlO2-1- 1-1+ LaO+ 1+ LaO+1- AlO2- 1- AlO2-1+ LaO+ 1+ LaO+ 0 TiO20 0.5 - 0 TiO20 0 SrO0 0 SrO0 0 TiO20 0 TiO20 0 SrO0 0 SrO0 critical thickness of LaAlO3 layer? N. Nakagawa, H.Y. Hwang, D.A. Muller, Nature Materials 5, 204–209 (2006).
- 11. The polar catastrophe ρ E V ρ E V AlO2- 0.5 + AlO2-1- 1-1+ LaO+ 1+ LaO+1- AlO2- 1- AlO2-1+ LaO+ 1+ LaO+ 0 TiO20 0.5 - 0 TiO20 0 SrO0 0 SrO0 0 TiO20 0 TiO20 0 SrO0 0 SrO0 critical thickness of LaAlO3 layer? N. Nakagawa, H.Y. Hwang, D.A. Muller, Nature Materials 5, 204–209 (2006).
- 12. Critical thickness S. Thiel, G. Hammerl, A. Schmehl, C. W. Schneider, J. Mannhart Science 313, 1942 (2006)critical thickness dc of LaAlO3 layer !dc = 4 unit cells
- 13. Gate field across SrTiO3 substrate DS-channel with 3 unit cells 3 unit cells 70 V relative resistance change >107
- 14. Electronic structure in LDA 1- AlO2- 1+ LaO+ 1- AlO2- 1+ LaO+ 0 TiO20 0 SrO0 0 TiO20 0 SrO0
- 15. LDA for SrTiO3 bulk SrTiO3 10 Ti 3d t2g Ti 3d eg 8 O 2pDOS (1/eV) 6 4 2 0 -4 -2 0 2 4 6 (E - EV) (eV) V. Eyert
- 16. LDA for SrTiO3 single layer SrTiO3 3.5 Ti1 3dxy 3 Ti1 3dxz,yz Ti1 3d3z2-r2 2.5 Ti1 3dx -y 2 2DOS (1/eV) 2 1.5 1 0.5 0 -4 -2 0 2 4 6 (E - EV) (eV) V. Eyert 0.1
- 17. LDA for SrTiO3 single layer SrTiO3 3.5 Ti1 3dxy 3 Ti1 3dxz,yz Ti1 3d3z2-r2 2.5 Ti1 3dx -y 2 2DOS (1/eV) 2 3 dxy band 1.5 1 2D-like DOS: step at band edge 0.5 (+ van Hove) 0 -4 -2 0 2 4 6 (E - EV) (eV) V. Eyert 0.1
- 18. LDA for SrTiO3 single layer SrTiO3 3.5 Ti1 3dxy 3 Ti1 3dxz,yz Ti1 3d3z2-r2 2.5 Ti1 3dx -y 2 2DOS (1/eV) 2 3 dxz,yz band 1.5 1 1D-like DOS: singular at band edge 0.5 0 -4 -2 0 2 4 6 (E - EV) (eV) V. Eyert 0.1
- 19. LDA for SrTiO3 single layer SrTiO3 3.5 Ti1 3dxy 3 Ti1 3dxz,yz Ti1 3d3z2-r2 2.5 Ti1 3dx -y 2 2DOS (1/eV) 2 3d 3z2-r2 band 1.5 1 0D-like DOS: peak 0.5 0 -4 -2 0 2 4 6 (E - EV) (eV) V. Eyert 0.1
- 20. LDA+U for LaAlO3/SrTiO3 interfaces N. Pavlenko, T.K., Surf. Sci., 605 1114 (2011) R. Pentcheva, W. Pickett, PRL 102, 107602 (2009) U. Schwingenschlögl, C. Schuster, CPL 467, 354 (2009)
- 21. LDA+U for LaAlO3/SrTiO3 interfaces 100 4 LAO unit cells on 1 STO 50 02 uc LaAlO3 on SrTiO3 100 -5 0 total DOS 3 LAO unit cells on 1 STO 50 0 -5 0 100 total DOS 80 2 LAO unit cells on 1 STO 60 40 20 0 -5 0 100 total DOS 80 1 LAO unit cell on 1 STO 60 40 20 0 -5 0 Energy (eV)
- 22. LDA+U for LaAlO3/SrTiO3 interfacesdipolar distortion of LaO plane + displacement of AlO2 plane versus polar catastrophe 1.5 eV
- 23. LDA+U for LaAlO3/SrTiO3 interfacesdipolar distortion of LaO + displacement of AlO2 planeplane versus polar catastrophe
- 24. Scanning Tunneling Spectroscopy at LaAlO3/SrTiO3 M. Breitschaft, V. Tinkl, N. Pavlenko, S. Paetel, C. Richter, J. R. Kirtley, Y. C. Liao, G. Hammerl, V. Eyert, T. K., J. Mannhart STM PRB 81, 153414 (2010) It tip Vs 4 unit cells LaAlO3 2-DEG SrTiO3
- 25. DFT-evaluation: LDA+U Supercell: Coulomb repulsion • supercell of LDA on Ti 3d and La 5d calculations orbitals: • in z direction structure UTi 3d = 2 eV fully relaxed ULa 5d = 8 eVU values from T. Bandyopadhyay, D. D. Sarma, Phys. Rev. B 39, 3517 (1989)
- 26. Tunneling spectra compared to LDA+U DOSTunneling spectra compared to LDA+U DOS Experiment: NDC Theory: 3d-DOS interface Ti atom
- 27. The shape of the 2-DEG quantum wellIII–V semiconductor: LaAlO3/SrTiO3:The quantum well conﬁning electrons at the LaAlO3/SrTiO3 interface is the potential of the Ti ions superimposed with band bending.
- 28. to a small variation of p except in the limit d ( , imply- ing that the parameter range for p and d can be narrowed effectively by comparison with experiment. This is illustrated in Fig. 3, where we show the angle de- pendence of the Ið3þÞ=Ið4þÞ ratio for several LAO/STO samples, as obtained by a standard ﬁtting procedure. The shaded areas mark the array of curves according to Eq. (1) falling within the error bars (Æ20%) of the experimental Ið3þÞ=Ið4þÞ ratios. The corresponding parameter ranges for p and d are indicated in Fig. 3 and listed in Table I for all samples. Also drawn are best ﬁt curves (solid lines). The Phys. Rev. Lett. 102, 176805 (2009) ˚ electron escape depth in STO was ﬁxed to 40 A accord- PRL 102, 176805 (2009) PHYSICAL REVIEW ing to the NIST database [18] and experimental ﬁndings on other insulating oxide compounds [19–21]. As can be seen 5uc LAO, PSI 0.012 4uc LAO, Augsburg 0.07 best fit best fit (p=0.28, d=1uc) (p=0.05, d=1uc) Intensity (arb. units) Intensity (arb. units) 0.06 0.010 I(3+)/I(4+) 0.05 p=0.10...0.28 p=0.02...0.06 d=1uc...3uc 0.008 d=1uc...4uc 0.04 458 457 456 458 457 456 0.006 0.03 0.02 0.004 FIG 0 20 40 60 0 20 40 60 ang 468 466 464 462 460 458 456 468 466 464 462 460 458 456 Emission angle (degree) Emission angle (degree) Binding energy (eV) Binding energy (eV) the FIG. 3 (color online). Experimental Ið3þÞ=Ið4þÞ ratios for FIG. 1 (color online). Ti 2p spectra of two different LAO/STO rat two LAO/STO samples as a function of angle. samples for various emission angles . the6805-2 p is fraction of Ti3+ ions wrt Ti4+ ions 2DEG conﬁned to one or at most few STO uc ! photons amounted to %500 meV. for 2 STO uc p is ﬁnite already Binding energies were calibrated with reference to the Au 4f core level at 84.0 eV.
- 29. Electronic structure‣ Polar catastrophe versus distortion polarization of LaAlO3 determines critical thickness of LaAlO3 film‣ Dimensionality of electronic system: electrons confined to a conducting sheet of 1–3 uc of SrTiO3 whereas in III-V semiconductors separated interface bands: inelastic scattering rate between subbands smaller than gaps‣ Correlations: U ~ bandwidth (intermediate regime) close to a charge ordered state ? ferromagnetism
- 30. c layer of a rare-earth oxide (RO) [(R is lanthanum of their structure and composition, to deliberately 70#8)9%9!:5!0001234%#3% marium (Sm), or yttrium (Y)] into an epitaxial manipulate the 2DEG electronic properties. pulsed-laser deposition with atomic layer control. We studied the effect of strong electron cor- Electron liquid – correlated electronic systemsons result in conducting 2DEGs at the insertedns are insulating. Our local spectroscopic and relations on an oxide 2DEG by inserting a single atomic layer of RO (R is La, Pr, Nd, Sm, or Y)l conductivity is dependent on electronic into an epitaxial SrTiO3 matrix using pulsed-laserO3 matrix. Such correlation effects can lead to deposition with atomic layer control. The RO es. H. W. Jang, D. A. Felker, C. W. Bark, Y. Wang,donates electrons to the conduction band of layer M. K. Niranjan, C. T. Nelson, Y. Zhang, D. Su, C. M. Folkman, S. H. Baek, S. Lee,1 K. Janicka, Y. Zhu, X.SrTiO3. These D. Fong, E. Y.near the inserted S. Rzchowski, C. B. Eom Q. Pan, D. electrons remain Tsymbal, M.he which the 2DEG is confined near the LaO/TiO2 RO layer due to Coulomb attraction. We find thatth Science 886, and superconducting ground the transport properties of these electrons range interface. Magnetic 331 (2011)as states of the 2DEG have been identified (12–14), from metallic to insulating, depending criticallyin “Metallic and insulating oxide interfaces controlled by electronic correlations” r- Fig. 1. (A) Schematiche representation of on a3 TiO on the rare-earth ion, and that this dependence a SrTiO / 2-terminated SrTiO3 substrate, followed trona- arises from strong electronic correlations.RO/SrTiO hetero- 1-ML by deposition of a SrTiO3 overlayer of varying Typ 3of We grew epitaxial SrTiO3 heterostructures thickness (20). A thick SrTiO3 overlayer approx- structure. The atomic struc- a 1he containing a symmetric TiO2/RO/TiO2near the interface is a single RO monolayer embedded in an ture interface imates SrT8) (Fig. 1A), resulting in RTiO3-like structure at+1 valent SrTiO3 matrix. Thicknesses of inserted enlarged. The infinite forcO3 RO layer donates elec- the interface. Using pulsed-laser deposition, 1-monolayer (ML)–thick RO and 1-unit-cell (uc)– of ain trons to neighboring TiO2 the heterostructures were fabricated by depos- thick RTiO3 layers were accurately controlled step iting either a RO monolayer or a RTiO3 unit cell to the planes, leading by monitoring in situ reflection high-energy elec- face larger electron density ne ty ture near the interface. (B) Typ- nt ical RHEED oscillations RO/WI stru ka for the growth of 1-MLa– LaO and 10-uc SrTiO3 lay- focuals ers in sequence on a TiO2-or, terminated SrTiO3 substrate. elec te (C) AFM image of a 10-ucnd the a. SrTiO3/1-ML LaO/SrTiO3 the al heterostructure showing an 10−vi- atomically smooth surface.A. opti il: to f inse and conRUARY 2011 VOL 331 SCIENCE www.sciencemag.org as a
- 31. density functional calcula- 4B (3.5-uc SrTiO3/1-ML YO). For the LaO- tational distortions, and rare-earth ion effects on 70#8)9%9!:5!0001234%#3%5w. The octahedral rotations based heterostructure, the Fermi energy lies in the the band structure. Indications of electron cor- interfacial plane, with typi-region of nonzero density of states, consistent relations have also been recently reported in iving an in-plane domain with the previous calculations (27, 28), whereas LaIO3/SrTiO3 heterostructures (30). Electron liquid – correlated electronic systems breadths of the half-order ne direction are consistent for the YO heterostructure the Fermi energy Strong correlations in 2DEGs at oxide inter- lies between the split-off lower Hubbard band faces have been shown to result from electronic tions at the RTiO3 layer and the higher energy density of states. This in- properties of different RO inserted layers, as well the SrTiO3 matrix. These dicates that the LaO-based interface is metallic, as the structural and electronic modification of otations lead to Jang et H. W. a spatial al., Science 886, interface is insulating, nearby layers. Quantitatively exploring the under- whereas the YO-based 331 (2011). onic structure, influencing supporting our experimental observations. Our lying physics of the experimental data presented “Metallic and insulatingpredict that the ground state of here is complex and challenging, because strong calculations oxide interfaces controlled by electronic correlations” ial strain in the interfacial the SrTiO3/LaO heterostructure is not charge- correlations combined with atomic-scale structuralcts the interface conductiv- ordered, whereas the SrTiO3/YO heterostructure and chemical variations severely limit the effective- ﬁlling close to n = 0.5 ness of theoretical calculations. The details can- nt not be fully captured within the DFT+U calculations used = 3 eV more advanced approaches— UTi here, and in agreement with optical band gap c-uc for bulk LaTiO3 (exp. based on dynamical mean-field theory (31), for + theor.)nd example—are likely necessary to capture the spa- not charge orderednd tial variations. The work presented here is impor- e- tant in elucidating correlation effects in systemsnd metal ULa,Y = 8 external with atomic-scale perturbations (32) andeVal- perturbation-induced changes in oxidespurious mixing with to avoid 2DEG sys- ty tems (8, 15–17). The abilityTi-states and grow to designnd heterostructures with atomic-scale variations, and n. the demonstrated strong dependence of correlatedes 2DEGs on these variations, open1 eV J = new directionsmi for oxide 2DEG heterostructures. independent of material rather teor References and NotesML UTiH.= 4 eV al., Science 305, 646 (2004). with optical band gap 1. Yamada et in agreementce 2. A. Ohtomo, D. A. Muller, J. L. for bulk YTiO3 (exp. + theor.) Grazul, H. Y. Hwang,orML charge419, 378 (2002).V. Colla, J. N. Eckstein, Nature ordered 3. M. P. Warusawithana, E. M. B. Weissman, Phys. Rev. Lett. 90, 036802 e. (2003). ferromagnetic insulator 4. E. Bousquet et al., Nature 452, 732 (2008). 5. M. P. Warusawithana et al., Science 324, 367 Ti 3.05+ (2009). 0.90 µ B Ti 3.9+ 0.05 µB
- 32. Magnetism atLaAlO3/SrTiO3
- 33. Ferromagnetism
- 34. Tutorial: Ferromagnetism electron gas: exchange hole triplet pair-correlation functiong↑↓ (r) singlet pair-correlation function g↑↑ (r) g↑↓ (r)12 g↑↑ (r) electrons with parallel spins avoid each other through fermionic statistics! rkf /π 1 2 3
- 35. Tutorial: Ferromagnetism electron gas: exchange hole triplet pair-correlation functiong↑↓ (r) singlet pair-correlation function g↑↑ (r) g↑↓ (r)12 g↑↑ (r) electrons with parallel spins avoid each other through fermionic statistics! rkf /π 1 2 3
- 36. Tutorial: Ferromagnetism electron gas: exchange hole triplet pair-correlation function g↑↓ (r) singlet pair-correlation function g↑↑ (r) g↑↓ (r)12 g↑↑ (r) electrons with parallel spins avoid each other through fermionic statistics! rkf /π 1 2 3 Hund’s coupling with Coulomb interaction: ferromagnetic state favorable? for atomic states g↑↓ (r)12 energy gain: exchange energy I ≡ JH g↑↑ (r) ψa,↑ (r)ψb,↑ (r) ψb,↑ (r )ψa,↑ (r ) Iab = e2 dr dr |r − r | 1 2 3 rkf /π
- 37. Tutorial: FerromagnetismF. Bloch (1929): spontaneous spin polarization of dilute electron gas through exchange ? E = Ekin + Eex + Ec spin polarization 0≤ξ≤1 rs = 1/ πna2 B for 2D 2 1 + ξ 2 Ekin =N pay kinetic energy for spin polarization ∼ ξ2 2m a2 B 2 rs √ e 4 2 1 1 2 Eex = −N (1 + ξ) + (1 − ξ) 3/2 3/2 reduce Coulomb energy ∼ −ξ 2 2aB 3π eﬀ rs Ec { 2 + 3 ξ 2 + O(ξ 4 ) from quantum Monte Carlo; Tanatar Ceperley, PRB 39, 5005 (1989) for sufficiently large rs ∆Eex ∼ N m2 I m is magnetization Coulomb interaction will support ferromagnetism through exchange + correlation terms however other phases, such as Wigner crystallization, may preempt the ferromagnetism
- 38. Tutorial: FerromagnetismF. Bloch (1929): spontaneous spin polarization of dilute electron gas through exchange ? E = Ekin + Eex + Ec spin polarization 0≤ξ≤1 rs = 1/ πna2 B for 2D 2 1 + ξ 2 Ekin =N pay kinetic energy for spin polarization ∼ ξ2 2m a2 B 2 rs √ e 4 2 1 1 2 Eex = −N (1 + ξ) + (1 − ξ) 3/2 3/2 reduce Coulomb energy ∼ −ξ 2 2aB 3π eﬀ rs Ec { 2 + 3 ξ 2 + O(ξ 4 ) from quantum Monte Carlo; Tanatar Ceperley, PRB 39, 5005 (1989) Stoner criterion: I ρ(EF ) 1 ∆Eex ∼ N m2 I m is magnetization
- 39. Tutorial: Ferromagnetism• lattice models: different DOS but still I ρ(EF ) 1 U;• on-site Hubbard interaction Stoner criterion: U ρ(EF ) 1 E reduce Coulomb energy density in FM state by ∼ − m2 U pay kinetic energy density ∼ + m2 /ρ(EF ) strong exchange coupling I and on-site interaction U are favorable for ferromagnetism µ ∆• exchange splitting ∆ = I/m or ∆ = U/m with m = n↑ − n↓ ρ↓ (EF ) ρ↑ (EF )
- 40. Magnetotransport at LaAlO3/SrTiO3 interfacesA. Brinkman, M. Huijben, M. Van Zalk, J.Huijben, U. Zeitler, J.C. Maan, W.G.Van der Wiel,G. Rijnders, D.H.A. Blank, H. Hilgenkamp, Nature Mater. 6, 493 (2007)“Magnetic effects at the interface of nonmagnetic oxides” large negative magnetoresistance, independent of orientation at low T: interface-induced moments ! magnetoresistance hysteresis from ferromagnetic ordering? T = 0.3 K
- 41. Magnetotransport at LaAlO3/SrTiO3 interfaces RAPID COMMUNICATIONS M. Ben Shalom, C. W. Tai, Y. Lereah, M. Sachs, E. Levy, D. Rakhmilevitch, A. Palevski, and Y. Dagan, PRB 80, 140403(R) (2009) “Anisotropic magnetotransport at the SrTiO3/LaAlO3 interface”B 80, 140403͑R͒ ͑2009͒ PHYSICAL REVIEW anisotropic magnetoresistance H⊥ is perpendicular to plane suggests magnetic ordering H is in plane MR is maximal negative for H J no hysteresis down to Tc = 135 mK MR is positive for H ⊥ J no long-range magnetic order?he sheet resistance as a function of FIG. 3. ͑Color online͒ Sample 1 ͑a͒ blue circles: the MR as a samples: sample1 ͑black squares͒, function of magnetic ﬁeld applied perpendicular to the interface.ple 3 ͑blue triangles͒, and the two Red squares are the MR data for ﬁeld applied along the interface tars, magenta crosses͒. Insert: sheet smaller negative current. ͑b͒ is perpendicular to plane function of i) 2D weak localization: much parallel to the MR with H The sheet resistance as a nosample 1. temperature at zero ﬁeld ͑black circles͒ and at 14 T applied parallel ii) magnetic impurities: usually isotropic no to the current ͑red squares͒ ected iii) magnetic material One of using a wire bonder. (magnetic order at the interface) yes?using reactive ion etch ͑RIE͒ into lar ﬁelds no hysteresis is observed down to 130 mK whereimensions of 50ϫ 750 microns superconductivity shows up.align perpendicular or parallel to In Fig. 3͑b͒ we show the temperature dependence of the
- 42. Magnetotransport at LaAlO3/SrTiO3 interfaces RAPID COMMUNICATIONS M. Ben Shalom, C. W. Tai, Y. Lereah, M. Sachs, E. Levy, D. Rakhmilevitch, A. Palevski, and Y. Dagan, PRB 80, 140403(R) (2009) “Anisotropic magnetotransport at the SrTiO3/LaAlO3 interface”B 80, 140403͑R͒ ͑2009͒ PHYSICAL REVIEW anisotropic magnetoresistance H⊥ is perpendicular to plane suggests magnetic ordering H is in plane MR is maximal negative for H J no hysteresis down to Tc = 135 mK MR is positive for H ⊥ J no long-range magnetic order?he sheet resistance as a function of FIG. 3. ͑Color online͒ Sample 1 ͑a͒ blue circles: the MR as a samples: sample1 ͑black squares͒, function of magnetic ﬁeld applied perpendicular to the interface.ple 3 ͑blue triangles͒, and the two D.A. Dikin, M. Mehta, C.W. Bark, Red squares are the MR dataand ﬁeld applied along the interface C.M. Folkman, C.B. Eom, for V. Chandrasekhar, arXiv:1103.4006 (2011) tars, magenta crosses͒. Insert: sheet parallel to the current. ͑b͒ The sheet resistance as a function of “Coexistence of superconductivity and ferromagnetism in two dimensions”sample 1. temperature at zero ﬁeld ͑black circles͒ and at 14 T applied parallel to the current ͑red squares͒ hysteretic magnetoresistance behavior in the superconducting phaseected using a wire bonder. One ofusing reactive ion etch ͑RIE͒description: one with Tihysteresis is observed down to 130 mK where they suggest a two-band into lar ﬁelds no ions responsible for ferromagnetism,imensions of 50ϫ 750 microns superconductivity shows up. a second associated with oxygen vacancies in STO responsible for SCalign perpendicular or parallel to In Fig. 3͑b͒ we show the temperature dependence of the
- 43. Magnetism at LaAlO3/SrTiO3 interfacesJ.A. Bert, B. Kalisky, C. Bell, M. Kim, Y. Hikita, H. Y. Hwang, and K. A. Moler, Nature Physics (2011)“Direct imaging of the coexistence of FM and SC at the LaAlO3/SrTiO3 interface” scanning SQUID device with micron-scale spatial resolution submicron patches of ferromagnetism in superconducting background “landscape of ferromagnetism, paramagnetism, and superconductivity”
- 44. Magnetism at LaAlO3/SrTiO3 interfaces Lu Li, C. Richter, J. Mannhart R. Ashoori, Nature Physics 7 (20111) magnetic torque magnetometry“Coexistence of magnetic order and 2D SC at LaAlO3/SrTiO3 interfaces” directly determines the magnetic moment m of a H sample by measuring the torque on a cantilever when the sample is placed in an external field H, T = m × H , so this method detects m perpendicular to H; great sensitivity! ~ 0.3 µB per interface unit cell ∆H ~ mT H-independent magnetic moment up to ~ 0.5 T magnetism in the superconducting state in-plane magnetic moment either phase separation or coexistence between magnetic and SC state
- 45. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum N. Pavlenko, T. Kopp, E.Y. Tsymbal, G.A. Sawatzky, J. Mannhart, arXiv:1105.1163 (2011) GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) interface DOS Ti 3dxy 1 exchange splitting for the 0 Ti 3dxy band ∆ -1 -1 0 1 2 3 E − EF (eV)z vacuum x
- 46. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) interface 0.08 ) 0.06 Ti magnetic moment ( Ti (0,0) small Ti magnetic moment ~ 0.07 µB Ti (0.5,0.5) 0.04 only at interface layer! 0.02 0z vacuum -0.02 3 2 1 0 TiO2 layer index x
- 47. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) ) interface magnetic moments in AlO 2 ( 0.05 O Al O magnetic moment ~ 0.07 µB at surface layer! 0 -0.05z vacuum 3 5 7 thickness of SrTiO 3 layer (in unit cells) x
- 48. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) interface 0.25 total magnetic moment ~ 0.23 µB ) B magnetization ( total magnetization per unit cell of LAO/STO interface absolute magnetization 0 -0.125 3 5 7z vacuum STO layer thickness (unit cells) x
- 49. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) interface calculated magnetic moment ~ 0.23 µB close to experimental value of ~ 0.3 µBz vacuum x
- 50. Magnetism at LaAlO3/SrTiO3 : stoichiometric state vacuum GGA with 50 } ρ↑ (E) 7 unit cells SrTiO3 4 unit cells LaAlO3 each 0 13 Å vacuum interface ρ↓ (E) + structural relaxation along z 50 -6 -4 -2 0 2 4 E − EF (eV) interface Ti interface moment too small to guarantee a robust magnetic state O surface moments may depend on surface reconstruction ?z vacuum x
- 51. Magnetism at LaAlO3/SrTiO3 : oxygen vacancies• introduce oxygen vacancies at the interface TiO2 layer N. Pavlenko, T. Kopp, E.Y. Tsymbal, G.A. Sawatzky, J. Mannhart, arXiv:1105.1163 (2011) Ti O Ti O Ti Ti O Ti O Ti O O O O O y Ti O Ti O Ti Ti O Ti O Ti cf. I.S. Elfimov, S. Yunoki, G.A. Sawatzky,PRL 89, 216403 (2002) x rules for the generation of magnetic states through• keep charge neutrality 2 electrons are introduced vacancies in CaO charge density increased density of states ρ(EF ) raised exchange splitting of the spin bands stabilization of ferromagnetic state ? remember the Stoner criterion: I ρ(EF ) 1
- 52. Magnetism at LaAlO3/SrTiO3 : oxygen vacanciesTi O Ti O TiO OTi O Ti O Ti 1 DOS Ti 3dxy 0 pure system -1 -1 0 1 2 3 E-EF (eV)
- 53. Magnetism at LaAlO3/SrTiO3 : oxygen vacancies Ti O Ti O Ti O O Ti O Ti O Ti 1 DOS Ti 3dxy 0 pure system -1 1 } DOS Ti 3dxy 0.5enhanced DOS ρ(EF ) 0enlarged exchange splitting ∆ -0.5 with O-vacancy -1 -1 0 1 2 3 E-EF (eV)
- 54. Magnetism at LaAlO3/SrTiO3 : oxygen vacancies Ti O Ti O Ti O O Ti O Ti O Ti 1 DOS Ti 3dxy 0 pure system -1 1 DOS Ti 3dxy 0.5 0substantial amount of the excess charge -0.5 with O-vacancytransferred to t2g spin-up orbitals -1 1 3dxz DOS Ti 3dxz,yzdominant contribution from 3dxy 3dyz 0 with O-vacancy -1 -2 -1 0 1 2 3 4 Energy E-EF(eV)
- 55. Magnetism at LaAlO3/SrTiO3 : oxygen vacancies Ti O Ti O Ti 0.08 O O ) 0.06 Ti magnetic moment ( Ti O Ti O Ti Ti (0,0) Ti (0.5,0.5) 0.04 0.02 pure 0 -0.02 3 2 1 0 TiO2 layer indexstrong magnetic moment in theinterfacial plane: 0.47 µB at Ti (0,0) ) 0.4 Ti magnetic moment ( Ti (0,0)extended local magnetic moments: Ti (0.5,0.5)triplet state of the 2 extra electronson more than two Ti sites 0.2 with O vacancyidentify: mTi = 0.47 µB 0 ∆ = 0.5 eV 3 2 1 0 TiO2 layer index Iρ(EF ) = 1.9
- 56. Magnetism at LaAlO3/SrTiO3 : oxygen vacanciesscenario: areas with increased density of oxygen vacancies ferromagnetic puddles their collective magnetic moments align in an external field superparamagnetic behavior 1000 µB SC SC SC
- 57. Magnetism at LaAlO3/SrTiO3 : alternative scenarioscenario suggested by K. Michaeli, A.C. Potter, and P. A. Lee [arXiv:1107.4352 (2011)]“SC and FM in oxide interface structures: possibility of finite momentum pairing” interface layer is quarter-filled through polar catastrophe sufficiently strong on-site nearest-neighbor Coulomb interaction charge order non-conducting layer with magnetic moment of ~µB on every 2nd Ti-site additional mobile charge carriers in 2nd TiO2 layer – through impurity doping, back gate etc. exchange coupling between local moments and conduction electrons J ~ 0.65 eV will yield TC = 300 K AlO2- Zener kinetic exchange mechanism exchange splitting of conduction bands LaO+ localized moments TiO20 SrO0 mobile charge carriers TiO20 SrO0
- 58. Magnetism at LaAlO3/SrTiO3 : oxygen vacancies −JK t −JK Kondo lattice model i jHK = −t c† cjσ + JK iσ sj · Sj i,j σ j ˜ t2 ˜ t JK ∼ 2 U 1 † † † sz = (cj↑ cj↑ − c† cj↓ ) s+ = cj↑ cj↓ s− = cj↓ cj↑ j 2 j↓ j j
- 59. Magnetism at LaAlO3/SrTiO3 : alternative scenarioscenario suggested by K. Michaeli, A.C. Potter, and P. A. Lee [arXiv:1107.4352 (2011)]“SC and FM in oxide interface structures: possibility of finite momentum pairing” strong (Rashba) spin-orbit coupling may help singlet pairing ˆ HSO = α (E × k) · σ ; E = internal + external field; σ are the Pauli matrices (spin); α(E) ∼ E effective magnetic field (in the rest frame of the electrons) in the interface plane but perpendicular to wave vector ∆SO = 2αkf ≤ 10 meV A.D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri, J.-M. Triscone, PRL, 104, 126803 (2010) AlO2- dispersion relation: two branches separated by a splitting with ∆SO LaO+ spin is in the plane, perpendicular to k TiO20 SrO0 pairing with (k, −k) pairs ( , ) in the lower branch TiO20 SrO0
- 60. Magnetism at LaAlO3/SrTiO3‣ Ferromagnetic in-plane ordering, probably not long-range; in coexistence with superconducting state‣ LSDA electronic structure calculations support the FM; however the Ti-moments appear to be rather small – robust FM?‣ Oxygen vacancies provide two electrons: they are in a triplet state; puddles with high concentration of O-vacancies would support superparamagnetic behavior‣ Superconductivity in the presence of ferromagnetism: triplet state as in Helium-3 ? or rather finite momentum pairing?
- 61. Superconductivityin planes at interfaces
- 62. Superconductivity at 200 mKin DS-channel with more than 3 unit cells R sheet (Ω / ⫽) 8 uc LaAlO3 T (mK) N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, J. Mannhart Science 317, 1196 (2007)
- 63. Measured Phase Diagram of the LaAlO3/SrTiO3 Interface TBKT ∝ (VG -VGc)2/3 weak localization R400 mK (kΩ /⃞) TBKT (mK) VG (V) ~1013 /cm2 large n ~ 4.5×1013 /cm2A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, J.-M. Triscone, Nature 456, 624 (2008)
- 64. SuperconductivityOrder parameter symmetry and nature of pairing ? d-wave, spin singlet? – ﬁnite momentum pairing?Microscopic mechanism for superconductivity ? phonons, spin ﬂuctuations, excitons?Superconductivity in two dimensions: BKT-transition conﬁrms the 2D behavior in the superconducting state ✓
- 65. Tutorial: ﬁnite momentum pairingSuperconductivity in the presence of ferromagnetismFinite momentum pairingP. Fulde R. A. Ferrell, Phys. Rev. 135, A550 (1964), A. I. Larkin Y. N. Ovchinnikov, ZETF 47, 1136 (1964) in bulk superconductors: Fulde-Ferrell Larkin-Ovchinnikov state also realized in ﬂux threaded loops fascinating for d-wave superconductors, see F. Loder et al., nature physics 4, 112 (2008); NJP 11, 075005 (2009)Generalized BCS mean-ﬁeld Hamiltonian:Spin singlet pairing amplitude:
- 66. Pairing InteractionSpin singlet component of a nearest neighbor interaction:Fourier transform to momentum space:withand Do ground state solutions exist with ∆(k,q) ≠ 0 and ∆(k,-q) ≠ 0 for a speciﬁc q ? Yes !
- 67. Energy at T = 0Here: q = (q, 0) along x-direction andmean charge density ρ = 0.8 and next-nearest neighbor hopping t‘ = 0.3t. F. Loder, A.P. Kampf, T. Kopp, PRB 81, 020511(R) (2010) Weak interaction: d-wave superconductor with q = 0 Intermediate interaction: Finite momentum pairing with q ≈ π/3
- 68. Charge DensityCharge density from Green‘s function:Charge-stripe order with wave number 2q. ρ(r) ∝ ∆ (r)∆Q (r) −Q ρ1/ρ ≈ 1%
- 69. Theoretical design of the interfaceEﬀective modelMechanism for superconductivity ?
- 70. Theoretical design of the interface L1 dpd electric fieldL2 a a
- 71. Theoretical design of the interface L2: metallic layer accumulation of charge at interface through field dopingkinetic energy: ntot = n0 + n(Ez ) 2D band: bandwidth 8tinteraction between charge carriers in L2: H e− e = U ∑ ni,↑ ni,↓ + V ∑ (1 − ni )( 1 − n j ) i 〈i, j 〉
- 72. Theoretical design of the interface L1: dielectric gate layertwo-level systems: levels p, d Ez electric field energy ε g = ed pd Ez dipoles (2-level systems) in SrTiO3 : soft TO1-mode external electric field: with 50─80 cm-1 1 H 2l = Δ pd ∑ ( pi pi − di di ) † † 2 i H phonon = ω TO ∑ b b † i i i H ext = ε g ∑ ( pi † di +di † pi ) i
- 73. Theoretical design of the interfaceinteraction between charge excitations in L1 and L2: V pd / 4t = 1.9 (r / a = 1.5) H exciton = V pd ∑ ni,σ ( pi † di + di † pi ) V pd / 4t = 4.3 (r / a = 1.0) i,σinteraction between charge in L2 and phonons in L1: ηγ = ω TO E p γ H polaron = − η ∑ (1 − n iσ )(b + bi ) † i iσ γ η 0.01 − 0.1 eV Ep is the polaron binding energy E p / ω TO 0.1 − 5
- 74. Induced pairing (at U=0) V pdsecond order perturbation theory for zero field: 2 V exciton Veff |zero field = −2 pd Δ pd V pd positive: attractive interactionPossibility of Synthesizing an Organic Superconductor(W. A. Little, 1964) Vspine-sc spine: metallic half-filled band εk (polyene chain) side-chains: charge oscillation with low-lying excited state Δsc side-chains (sc) spine
- 75. interaction between metallic charge carriers and (polarized) two-level systems e2 d pd H int = V pd ∑ ci,σ †ci,σ ( pi † di + di † pi ) V pd i,σ r2 → Vx ∑c c † + (S +S ) + Vz - ∑c c † S z with V pd Δ pd εg Vx = Vz = 2V pd 2 1 1 εg 2 + ( Δ pd )2 εg 2 + ( Δ pd )2 2 2 (virtual) transitions driven interaction of field induced dipoles by field of nearest charge carrier with the 2D charge carriers induces pairing repulsive term in pairing channel
- 76. 3 Steps towards an approximate solution1. bosonization (Holstein-Primakoff) not exact but correct for negligible inversion:2. generalized Lang-Firsov transformation H = U LF HU LF † purpose of unitary transformation:
- 77. LF transition in the presence static polarization of a charge carrier • fix γ, θ through variational scheme • renormalized splitting: 1 E2l = 2 2 + ( ∆pd )2 g 23. Feynman variational scheme • determine the bilinear Hamiltonian Htest through variation of the Bogoliubov inequality
- 78. Interface mediated pairing interaction between metallic charge carriers and (polarized) two-level systems maximum in Tc for intermediate fields Δ opt ≈ 2.5 pd 4t not strongly dependent on other parameters like V pd and ε field energy / 4t limited by repulsion between charge carriers saturation ofcarrier doping and field induced dipoles dipole moment V. Koerting, Q. Yuan, P. Hirschfeld, T.K., and J. Mannhart, PRB 71, 104510 (2005)
- 79. d-wave pairingExtension:nonlocal effective interaction (through 2 microscopic processes)attractive in d-wave channel
- 80. d-wave pairingestimates for the most important parameters: Interface-induced d-wave singlet pairing: on-site Coulomb repulsion avoided ∆dp /4t = 2.5 Vdp /t = 1.3 nl C. Stephanos, T.K., J. Mannhart, P.J. Hirschfeld, V /t = 0.5 J/t = 1.1 Rapid Comm. (2011); arXiv:1108.1942 Vdp /t = 3.1(a) J *+,!#-.//.,0 3(7) 3(8) 3(5) 3(9) L1 3(44 Eext nl ddp #$#%!#() Vdp Vdp ! L2 ! #$#%!#() #$#%!#(8) #1$2#$#%! a ! t 3(35 J(b) (c) nl 3(3) Vdp Vdp Vdp Vdp 63(3% 63(3 3 3(3 %:;#$#%! Veff, 1 Veff, 2
- 81. Compressibility of the electron gas at LaAlO3/SrTiO3 interfaces
- 82. Coulomb interactions in the weak density regimeremember Landau theory: ρ(Ef )/n2 χ0 compressibility κ= and spin suszeptibility χ= 1 + F0s 1 + F0 a s a where χ0 is the Pauli suszeptibility thermodynamic stability conditions: F0 , F0 −1 how are these quantities measured?homogeneous 2D electron gas – Jellium model: E = Ekin + Eex + Ec 2 1 + ξ 2 direct Coulomb term is compensated kinetic energy Ekin =N by electron–background interaction 2m a2 B 2 rs √ e 4 2 1 1 2 exchange energy Eex = −N (1 + ξ) + (1 − ξ) 3/2 3/2 2aB 3π eﬀ rs eﬀ : dielectric constant of the 2D sheet; eﬀ = 1 if screening from core electrons and interband transitions is respected; self-screening arises from the correlation term
- 83. Coulomb interactions in the weak density regime E = Ekin + Eex + Ec where E is a functional of n through rs = 1/ πna2 B correlation term from MC; Tanatar Ceperley, PRB 39, 5005 (1989) 1/2 e 1 2 1+ a1 (ξ)rs correlation energy Ec = −N a0 (ξ) 2aB eﬀ 1/2 3/2 1 + a1 (ξ)rs + a2 (ξ)rs + a3 (ξ)rs with a0 (ξ), a1 (ξ), a2 (ξ) positive coefficients e2 1 1 as Ec ∼ −N has the form of Eex for rs 1 , it basically enhances Eex 2aB eﬀ rs by about 20 %Wigner crystallization into a triangular electronic crystal at rs = 37 ± 5 here, we will not discuss this regime possibility, that the MIT at very low densities in the LaAlO3/SrTiO3-interfaces is a transition into a Wigner crystallized state, albeit disorder and polaron formation may influence the MIT
- 84. Electronic compressibilityCompressibility κ calculated from energy functional E = Ekin + Eex + Ec through d2 E/A κ−1 = n2 ∂µ/∂n = n2 for T →0 d n2 κ−1 = κ−1 + κ−1 + κ−1 kin ex c for unpolarized system: κ−1 /n2 = π2 /m = 1/ρ(Ef ) kin 1/2 2 1 e2 κ−1 /n2 = − ex √ π eﬀ n negative exchange term “wins” for sufficently small n: negative compressibility enhanced by correlation term! first observed in Si-MOSFETs and III-V heterostructures J.P. Eisenstein, L.N. Pfeiffer, K.W. West, PRL 68, 674 (1992) S.V. Kravchenko, V.M. Pudalov, S.G. Semenchinsky, Phys. Lett. A 141, 71 (1989)
- 85. Electronic compressibility Q −QHow do you measure the electronic compressibility κ? through the capacitance: r 1 2 d2 E for equivalent plates E(Q) + E(−Q) = 2 E(0) + + ··· 2 Q d Q2 A = 2 E(0) + 1 C −1 Q2 + · · · 2 d κ −1 d2 E/A expect A/C = 2 ? κ−1 = n2 ∂µ/∂n = n2 (en)2 d n2 however, there is now also a direct Coulomb term from charging the plates r A EHartree = Q /(2Cgeom ) 2 with Cgeom = 4πd κ−1 A/C = 2 + A/Cgeom negative compressibility: enhancement of C (en)2
- 86. Negative compressibility κ−1A/C = 2 + A/Cgeom (en)2 6 4 2 C 2DC0 0 rc ro classical capacitor 2 4 d/r aB 6 0 2 4 6 8 10 rs Cgeom ≡ C0 m /m = 1 eﬀ = 1 d/r = aBT. K, J.Mannhart, J. Appl. Phys. 106, 064504 (2009)
- 87. Negative compressibility κ−1A/C = 2 + A/Cgeom (en)2

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