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Aspects
                               of the
              eory of Oxide Interfaces


                              Thilo Kopp

               Center for Electronic Correlations and Magnetism
                            Universität Augsburg




International Summer School on Surfaces and Interfaces in Correlated Oxides
                               Vancouver 2011
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)
coauthors and coworkers


Natalia Pavlenko     Lviv, Ukraine

Jochen Mannhart      MPI for Solid State Research, Stuttgart

George Sawatzky      UBC Vancouver

Peter Hirschfeld     University of Florida, Gainesville

Evgeny Tsymbal      University of Nebraska, Lincoln

Florian Loder        EKM, Universität Augsburg

Arno Kampf           EKM, Universität Augsburg

Cyril Stephanos, Kevin Steffen       EKM, Universität Augsburg
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
Charge Transport @ Interfaces of Oxides




parallel to interface                   perpendicular to interface

                               vacuum                                         YBCO


                               LaAlO3



                               SrTiO3   YBCO



                                                                              YBCO




     Liao et al. (PRB, 2010)                   Schneider et al. (PRL, 2004)
Charge Transport @ Interfaces of Oxides




parallel to interface                   perpendicular to interface

                               vacuum                                         YBCO


                               LaAlO3



                               SrTiO3   YBCO




                 ?                                         ?                  YBCO




     Liao et al. (PRB, 2011)                   Schneider et al. (PRL, 2004)
LaAlO3/SrTiO3 interface
                                                                                              vacuum

                          this talk                                                           LaAlO3
                                                                                      metallic interface


                                                                                              SrTiO3




MIT @   nc ∼ 10−13 /cm2   Y.C. Liao, T.K., C. Richter, A. Rosch, J. Mannhart   PRB 83, 075402 (2011)
Electronic structure
        of
  LaAlO3/SrTiO3
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
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).
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).
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
Gate field across SrTiO3 substrate

         DS-channel with 3 unit cells



           3 unit cells           70 V




     relative resistance change >107
Electronic structure in LDA


       1-      AlO2-

       1+      LaO+

       1-      AlO2-

       1+      LaO+

        0      TiO20

        0      SrO0

        0      TiO20

        0      SrO0
LDA for SrTiO3

                         bulk SrTiO3

             10
                                        Ti 3d t2g
                                        Ti 3d eg
              8                             O 2p
DOS (1/eV)




              6


              4


              2


              0
                  -4    -2     0        2      4    6
                             (E - EV) (eV)
                                                        V. Eyert
LDA for SrTiO3

                         single layer SrTiO3

             3.5
                    Ti1 3dxy
               3 Ti1 3dxz,yz
                 Ti1 3d3z2-r2
             2.5 Ti1 3dx -y
                         2 2
DOS (1/eV)




               2

             1.5

               1

             0.5

               0
                    -4          -2     0        2     4    6
                                     (E - EV) (eV)
                                                               V. Eyert
                                                     0.1
LDA for SrTiO3

                         single layer SrTiO3

             3.5
                    Ti1 3dxy
               3 Ti1 3dxz,yz
                 Ti1 3d3z2-r2
             2.5 Ti1 3dx -y
                         2 2
DOS (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
LDA for SrTiO3

                         single layer SrTiO3

             3.5
                    Ti1 3dxy
               3 Ti1 3dxz,yz
                 Ti1 3d3z2-r2
             2.5 Ti1 3dx -y
                         2 2
DOS (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
LDA for SrTiO3

                         single layer SrTiO3

             3.5
                    Ti1 3dxy
               3 Ti1 3dxz,yz
                 Ti1 3d3z2-r2
             2.5 Ti1 3dx -y
                         2 2
DOS (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
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)
LDA+U for LaAlO3/SrTiO3 interfaces
                                    100
                                                     4 LAO unit cells on 1 STO
                                    50

                                     0
2 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)
LDA+U for LaAlO3/SrTiO3 interfaces

dipolar distortion of LaO plane   + displacement of AlO2 plane
                       versus
                   polar catastrophe




                                                                 1.5 eV
LDA+U for LaAlO3/SrTiO3 interfaces

dipolar distortion of LaO        + displacement of AlO2 plane
plane
                        versus
                   polar catastrophe
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
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 eV




U values from T. Bandyopadhyay, D. D. Sarma, Phys. Rev. B 39, 3517 (1989)
Tunneling spectra compared to LDA+U DOS
Tunneling spectra compared to LDA+U DOS

 Experiment:
 NDC




 Theory:
 3d-DOS
 interface
 Ti atom
The shape of the 2-DEG quantum well
III–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.
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 .                                                                         the

6805-2             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.
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
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 systems
ons result in conducting 2DEGs at the inserted
ns 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-laser
O3 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 that
th 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 critically
in “Metallic and insulating oxide interfaces controlled by electronic correlations”
 r-                                                                                  Fig. 1. (A) Schematic
he                                                                                   representation of on a3 TiO
                                              on the rare-earth ion, and that this dependence a SrTiO / 2-terminated SrTiO3 substrate, followed            tron
a-                                            arises from strong electronic correlations.RO/SrTiO hetero-
                                                                                     1-ML                 by deposition of a SrTiO3 overlayer of varying   Typ
                                                                                                      3
of                                                We grew epitaxial SrTiO3 heterostructures thickness (20). A thick SrTiO3 overlayer approx-
                                                                                     structure. The atomic struc-                                          a 1
he                                            containing a symmetric TiO2/RO/TiO2near the interface is a single RO monolayer embedded in an
                                                                                     ture interface imates                                                 SrT
8)                                            (Fig. 1A), resulting in RTiO3-like structure at+1 valent SrTiO3 matrix. Thicknesses of inserted
                                                                                     enlarged. The        infinite                                         forc
O3                                                                                   RO layer donates elec-
                                              the interface. Using pulsed-laser deposition, 1-monolayer (ML)–thick RO and 1-unit-cell (uc)–                of a
in                                                                                   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-ML
a–                                                                                   LaO and 10-uc SrTiO3 lay-                                             focu
als                                                                                  ers in sequence on a TiO2-
or,                                                                                  terminated SrTiO3 substrate.                                          elec
 te                                                                                  (C) AFM image of a 10-uc
nd                                                                                                                                                         the
 a.                                                                                  SrTiO3/1-ML LaO/SrTiO3                                                the
 al                                                                                  heterostructure showing an                                            10−
vi-                                                                                  atomically smooth surface.
A.
                                                                                                                                                           opti
 il:
                                                                                                                                                           to f
                                                                                                                                                           inse
                                                                                                                                                           and
                                                                                                                                                           con
RUARY 2011       VOL 331      SCIENCE       www.sciencemag.org                                                                                             as a
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%5
w. 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 structural
cts 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 ordered
nd                                                                                 tial variations. The work presented here is impor-
 e-                                                                                tant in elucidating correlation effects in systems
nd                                                                                  metal                               ULa,Y = 8 external
                                                                                   with atomic-scale perturbations (32) andeV
al-                                                                                perturbation-induced changes in oxidespurious mixing with
                                                                                                                        to avoid 2DEG sys-
 ty                                                                                tems (8, 15–17). The abilityTi-states and grow
                                                                                                                          to design
nd                                                                                 heterostructures with atomic-scale variations, and
 n.                                                                                the demonstrated strong dependence of correlated
es                                                                                 2DEGs on these variations, open1 eV   J = new directions
mi
                                                                                   for oxide 2DEG heterostructures. independent of material
                                                                                                                        rather
 te
or                                                                                      References and Notes
ML                                                                                  UTiH.= 4 eV al., Science 305, 646 (2004). with optical band gap
                                                                                     1.    Yamada et                   in agreement
ce                                                                                   2. A. Ohtomo, D. A. Muller, J. L. for bulk YTiO3 (exp. + theor.)
                                                                                                                         Grazul, H. Y. Hwang,
or
ML
                                                                                    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
Magnetism
     at
LaAlO3/SrTiO3
Ferromagnetism
Tutorial: Ferromagnetism

    electron gas: exchange hole                    triplet pair-correlation functiong↑↓ (r)
                                                   singlet pair-correlation function g↑↑ (r)
                             g↑↓ (r)
1
2
                            g↑↑ (r)
                                           electrons with parallel spins avoid each other
                                           through fermionic statistics!
                                  rkf /π
          1     2     3
Tutorial: Ferromagnetism

    electron gas: exchange hole                    triplet pair-correlation functiong↑↓ (r)
                                                   singlet pair-correlation function g↑↑ (r)
                             g↑↓ (r)
1
2
                            g↑↑ (r)
                                           electrons with parallel spins avoid each other
                                           through fermionic statistics!
                                  rkf /π
          1     2     3
Tutorial: Ferromagnetism

    electron gas: exchange hole                           triplet pair-correlation function g↑↓ (r)
                                                          singlet pair-correlation function g↑↑ (r)
                             g↑↓ (r)
1
2
                             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)
1
2
                                                  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
Tutorial: Ferromagnetism

F. 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
Tutorial: Ferromagnetism

F. 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
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 )
Magnetotransport at LaAlO3/SrTiO3 interfaces

A. 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
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                  no
sample 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 where
imensions of 50ϫ 750 microns              superconductivity shows up.
align perpendicular or parallel to           In Fig. 3͑b͒ we show the temperature dependence of the
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 phase
ected using a wire bonder. One of
using 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 SC
align perpendicular or parallel to        In Fig. 3͑b͒ we show the temperature dependence of the
Magnetism at LaAlO3/SrTiO3 interfaces

J.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”
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
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
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

                                                                                                 0

z          vacuum                                                                             -0.02
                                                                                                      3         2                      1          0
                                                                                                                    TiO2 layer index
    x
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



                                                                                                -0.05


z          vacuum                                                                                               3                5                 7
                                                                                                             thickness of SrTiO 3 layer (in unit cells)
    x
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                           7
z          vacuum                                                                                          STO layer thickness (unit cells)

    x
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 µB



z          vacuum

    x
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
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
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        0       1       2   3
                                                                E-EF (eV)
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
enhanced DOS            ρ(EF )                               0
enlarged exchange splitting      ∆                     -0.5       with O-vacancy
                                                             -1
                                                                      -1        0       1       2   3
                                                                                    E-EF (eV)
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
                                                                 0
substantial amount of the excess charge                  -0.5          with O-vacancy
transferred to t2g spin-up orbitals
                                                                 -1
                                  


                                                                  1         3dxz
                                                DOS Ti 3dxz,yz



dominant contribution from 3dxy                                             3dyz
                                                                 0
                                                                       with O-vacancy
                                  




                                                                 -1
                                                                  -2       -1           0          1        2   3   4
                                                                                            Energy E-EF(eV)
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 index
strong magnetic moment in the
interfacial 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 electrons
on more than two Ti sites                                                        0.2                      with O vacancy

identify: mTi = 0.47 µB
                                                                                  0
            ∆ = 0.5 eV                                                                 3    2                      1       0
                                                                                                TiO2 layer index
            Iρ(EF ) = 1.9
Magnetism at LaAlO3/SrTiO3 : oxygen vacancies

scenario:

 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
Magnetism at LaAlO3/SrTiO3 : alternative scenario

scenario 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
Magnetism at LaAlO3/SrTiO3 : oxygen vacancies



                                           −JK                      t            −JK
   Kondo lattice model


                                                                i j
HK = −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
Magnetism at LaAlO3/SrTiO3 : alternative scenario

scenario 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
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?
Superconductivity
in planes  at interfaces
Superconductivity at 200 mK

in 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)
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 /cm2

A. 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)
Superconductivity


Order 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   ✓
Tutorial: finite momentum pairing
Superconductivity in the presence of ferromagnetism

Finite momentum pairing
P. 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:
Pairing Interaction

Spin singlet component of a nearest neighbor interaction:
Fourier transform to momentum space:




with

and




                  Do ground state solutions exist with
               ∆(k,q) ≠ 0 and ∆(k,-q) ≠ 0 for a specific q ?


                                   Yes !
Energy at T = 0
Here: q = (q, 0) along x-direction and
mean 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
Charge Density

Charge density from Green‘s function:




Charge-stripe order with wave number 2q.

        ρ(r) ∝ ∆ (r)∆Q (r)
                −Q




        ρ1/ρ ≈ 1%
Theoretical design of the interface



Effective model


Mechanism for superconductivity ?
Theoretical design of the interface




     L1
                                                dpd

                                                      electric field
L2
                                           a
                               a
Theoretical design of the interface

                                 L2: metallic layer



                                                           accumulation of charge
                                                                at interface

                                                            through field doping



kinetic energy:                                           ntot = n0 + n(Ez )

                                  2D band: bandwidth 8t


interaction between charge carriers in L2:


     H e− e = U ∑ ni,↑ ni,↓ + V ∑ (1 − ni )( 1 − n j )
                   i                    〈i, j 〉
Theoretical design of the interface

                                 L1: dielectric gate layer


two-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
Theoretical design of the interface




interaction 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,σ


interaction 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
Induced pairing (at U=0)
                                                          V pd
second order perturbation theory for zero field:
                                       2
                                   V                              exciton
         Veff |zero field = −2
                                       pd

                                   Δ pd
                                                           V pd
   positive: attractive interaction


Possibility 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
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
3 Steps towards an approximate solution

1. bosonization (Holstein-Primakoff)




       not exact but correct for negligible inversion:




2. generalized Lang-Firsov transformation

           H ' = U LF HU LF
                        †



         purpose of unitary transformation:
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
                                         2

3. Feynman variational scheme




     • determine the bilinear Hamiltonian Htest through
      variation of the Bogoliubov inequality
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 of
carrier doping       and field induced dipoles                dipole moment
                        V. Koerting, Q. Yuan, P. Hirschfeld, T.K., and J. Mannhart, PRB 71, 104510 (2005)
d-wave pairing

Extension:
nonlocal effective interaction (through 2 microscopic processes)
attractive in d-wave channel
d-wave pairing

estimates 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
Compressibility of the electron gas
                at
     LaAlO3/SrTiO3 interfaces
Coulomb interactions in the weak density regime

remember 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
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
Electronic compressibility

Compressibility    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)
Electronic compressibility

                                                                                Q −Q
How 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
Negative compressibility

         κ−1
A/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 = aB
T. K, J.Mannhart, J. Appl. Phys. 106, 064504 (2009)
Negative compressibility

         κ−1
A/C = 2       + A/Cgeom
        (en)2
nterface (10, 11). Simple modifi- capacitance in the frequency range between 1 Hz                   For device 2, the interface underneath the




                                                                                                                                                                    Downloaded from w
   structures described could yield a to 2 MHz with ~20-mrad resolution in the phase gate was not conductive at Vg = 0, even though
 enhancement greater than 100%. measurement of the impedance. We were able to the regions away from the YBCOPenetration field measurement. (A) Sketch of the
                                                                                                                          Fig. 3. circular pads
                                         Measured negative compressibility
 onic correlations (12–14) may be vary the DC voltage on the top gate and track were conductive (the two-terminal resistanceWith the interface grounded, we applie
                                                                                                                          surement setup. be-
  eer yet Au pad with 350 mm was deposited to serve as with gate voltage (16).
  ate, larger capacitances (15).                the capacitance change tion C increased rapidlytween thef.Nb ohmic contacts is of order 500 ohms E from the back gate to the inte
                                                                                                   at low Measured atexternal electric field
                                                                                                                                                  0
  ated capacitor devices on it was foundWe measured the capacitance between a top C at 4.2 K). greater than and samples during electric field that penetrates the grou
  nse top gate because LAO/                          that at 300 K, f = 5 Hz, the peak of was 10% In testing these detected the at
 ructuresgold caused less leakage than did YBCO (16). atthat of Chd. This room-temperature different coolings from room temper-
  een        through in situ growth of gate and the interface             frequencies f ranging least four behavior pro-
                                                                                                                          interface layer. The penetration field Ep is determine
m,(YBCO) or Au films on thecurrent to the 8 to 2000 Hz while varying a top evidence ruling4.2 Kanomaliesmeasuring noticed that from the device top gate. (B) The p
    to Although a leakage sur- from gate appeared vides additional gate ature to out for each sample, we the current
AO. As below –0.6 V 1A, we then −1 resistance dropswasof the dielectricsample theSTO as the origin varied slightly with ther-
                     κ
  r to shown in Fig. [the gate-to-channel Vg. Device 1
                                                voltage                  fabricated on a function of depletion voltage of
                                                                                                                          Iy divided by the measurement frequency for device 1 at T
            A/C = 2      + A/C
            to the megohm range near with 12 unit(fig. S3)LAO. On the top surface Li, C. Richter, S. Paetel, T. Kopp, J. of
                                                 depletion cells of geomthe capacitanceLu of mal cycling. However, the depletion voltage Mannhart, R.C. Ashoori pen
                                                                                          enhancement.                    at three excitation frequencies. At V near zero, the
                    (en)
  hysics, Massachusetts Institute of Technology
 nce-02139, USA. 2the room-temperature C2 Vg film, a circular YBCO top gatecapacitor C with charge12–unit cellproportional to f and is constant overga broad range of –0
   MA       (16)], Center for Electronic Cor- the LAO curves are
                                                    –                        Charging a Science 332,with e does not LAO was always neg-
                                                                                         with a devices
                                                                                                           825 (2011)
  gnetism, University of Augsburg, of the YBCO-gated 200 mm was patterned. Figure 2A in voltage dV = e/Cgeom.
   the similar to those Augsburg diameter of devices simply require a change ative, whereas that for devices with 10–unit cell
                                                displays the capacitance C versus V curves of LAO layers was positive. This difference sug- V, Iy /f displays a frequency depe
                                                                                                                          V. For Vg  –0.18
  nce shown in Fig. 2, A and B. As Vg decreased from Because ofg the finite dn/dm, an extra voltage
                                                device 1 at 4.2 K. Similar curves are shown in gests that the YBCO top gate tends to depleteeffects of current leakage through
                                                                                                                          probably caused by
high 0 should decreased slowly. However, near deple- (dm/dn)/eA is required, where m is the chemical
  spondence V, C be addressed. E-mail:
 u                                              Fig. 2B for device 2 with an LAO thickness of the interface underneath the gate: The larger the
  dd
 ick-
ple layout A                                                     B B                                C                           A                                            B
nce bridge
           A                                                                                                               balancing point
  fer-                Device 1         12 u.c. LAO
  tch of the ohmiccircular top gate with d = 200 µm
mall.
                      contact
                                                                                                       Ve µ          C         Cs             Vb
                200 nm Nb
 ce layout.
   the
  f LAO (10
mall                f
                                   top gate T = 4.2 K      Chd
                                                                                  T = 4.2 K
                                100 nm YBCO
 ls thick) is
 d at top
  o the                             LAO film
 Fig. STO
  nated                                                                                                     pre-amplifier
                                                                                                                 Chd
kHz gates
O top                                    f
                                  1 mm STO




                                                                                                                          from www.sciencemag.org on May 12, 2011
                                                                                                                                      lock-in
 er a pat-
    and                                                                                                                             amplifier
 ethe layer,
    LAO
 wercontacts are deposited close to the corners of the wafer. (B) tion voltage Ve. In the other arm, another ac voltage Vs with the same fre-                  f 5
 BCO/LAO/STO sample with leads attached. The wafer is square quency is applied to a standard capacitor Cs. The signal at the balancing point
  e is                                                                                                                                                         f 7
owsof 5 mm. The diameter of the YBCO circular top gates varies is measured with a pre-amplifier and a lock-in amplifier. During the mea-
 gth                                                                                                                                                           f 8
nd 500 mm. (C) Setup sketch of the capacitance bridge. In one surement, with the phase and the amplitude of Ve held stable, Vs is varied both
  cell
 dge, C stands for the sample capacitor, excited by an ac excita- in phase and in amplitude to null the signals at the balancing point.
 ow-
 t of
  fre-                          www.sciencemag.org SCIENCE VOL 332 13 MAY 2011                                                                   825
 V≤
 ken
   f=
   Vg.
  nce                                                                                                                   Fig. 4. The inverse of compressibility dm/dn determined
 rve.      C 4.0 Device 3            12 u.c. LAO
                                                                    D                                                   field measurements on (A) device 1 and (B) device 2. Th
gnal                circular top gate with d = 350 µm                                                                   at the interface is determined by integrating the C vers
).
               3.5                                        Chd              enhanced capacitance                                         negative compressibility
                        T = 300 K
                                                                                                                        lowest frequency achieved. (C) The density n dependence
   we
             3.0
Summary and conclusions

‣ 2D electron liquid at LaAlO3/SrTiO3 interfaces
   as compared to semiconductor interfaces, it is not just a quantum well confining free electrons;
   it is the potentials of the Ti ions in 1–3 uc from the interface which confine the charge carriers


‣ Correlations are sufficiently strong
   that they influence the spectrum as measured by STS                       I, U ≥ 8t
   that a ferromagnetic state is formed, possibly not homogeneously


‣ Magnetism  superconductivity coexist at LaAlO3/SrTiO3 interfaces
   possibly ferromagnetic puddles in the superconducting background
   role of oxygen vacancies which introduce a strong ferromagnetic coupling?
   finite momentum d-wave pairing?


‣ Negative electronic compressibility at low density
   strong increase of the capacitance of the LaAlO3/SrTiO3 interface

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Charge, spin and orbitals in oxides

  • 1. Aspects of the eory of Oxide Interfaces Thilo Kopp Center for Electronic Correlations and Magnetism Universität Augsburg International 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 coworkers Natalia Pavlenko Lviv, Ukraine Jochen Mannhart MPI for Solid State Research, Stuttgart George Sawatzky UBC Vancouver Peter Hirschfeld University of Florida, Gainesville Evgeny Tsymbal University of Nebraska, Lincoln Florian Loder EKM, Universität Augsburg Arno Kampf EKM, Universität Augsburg Cyril 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 Oxides parallel 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 Oxides parallel 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 SrTiO3 MIT @ 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 2p DOS (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 2 DOS (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 2 DOS (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 2 DOS (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 2 DOS (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 0 2 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 interfaces dipolar distortion of LaO plane + displacement of AlO2 plane versus polar catastrophe 1.5 eV
  • 23. LDA+U for LaAlO3/SrTiO3 interfaces dipolar distortion of LaO + displacement of AlO2 plane plane 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 eV U values from T. Bandyopadhyay, D. D. Sarma, Phys. Rev. B 39, 3517 (1989)
  • 26. Tunneling spectra compared to LDA+U DOS Tunneling spectra compared to LDA+U DOS Experiment: NDC Theory: 3d-DOS interface Ti atom
  • 27. The shape of the 2-DEG quantum well III–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 . the 6805-2 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 systems ons result in conducting 2DEGs at the inserted ns 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-laser O3 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 that th 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 critically in “Metallic and insulating oxide interfaces controlled by electronic correlations” r- Fig. 1. (A) Schematic he representation of on a3 TiO on the rare-earth ion, and that this dependence a SrTiO / 2-terminated SrTiO3 substrate, followed tron a- arises from strong electronic correlations.RO/SrTiO hetero- 1-ML by deposition of a SrTiO3 overlayer of varying Typ 3 of We grew epitaxial SrTiO3 heterostructures thickness (20). A thick SrTiO3 overlayer approx- structure. The atomic struc- a 1 he containing a symmetric TiO2/RO/TiO2near the interface is a single RO monolayer embedded in an ture interface imates SrT 8) (Fig. 1A), resulting in RTiO3-like structure at+1 valent SrTiO3 matrix. Thicknesses of inserted enlarged. The infinite forc O3 RO layer donates elec- the interface. Using pulsed-laser deposition, 1-monolayer (ML)–thick RO and 1-unit-cell (uc)– of a in 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-ML a– LaO and 10-uc SrTiO3 lay- focu als ers in sequence on a TiO2- or, terminated SrTiO3 substrate. elec te (C) AFM image of a 10-uc nd the a. SrTiO3/1-ML LaO/SrTiO3 the al heterostructure showing an 10− vi- atomically smooth surface. A. opti il: to f inse and con RUARY 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%5 w. 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 structural cts 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 ordered nd tial variations. The work presented here is impor- e- tant in elucidating correlation effects in systems nd metal ULa,Y = 8 external with atomic-scale perturbations (32) andeV al- perturbation-induced changes in oxidespurious mixing with to avoid 2DEG sys- ty tems (8, 15–17). The abilityTi-states and grow to design nd heterostructures with atomic-scale variations, and n. the demonstrated strong dependence of correlated es 2DEGs on these variations, open1 eV J = new directions mi for oxide 2DEG heterostructures. independent of material rather te or References and Notes ML UTiH.= 4 eV al., Science 305, 646 (2004). with optical band gap 1. Yamada et in agreement ce 2. A. Ohtomo, D. A. Muller, J. L. for bulk YTiO3 (exp. + theor.) Grazul, H. Y. Hwang, or ML 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 at LaAlO3/SrTiO3
  • 34. Tutorial: Ferromagnetism electron gas: exchange hole triplet pair-correlation functiong↑↓ (r) singlet pair-correlation function g↑↑ (r) g↑↓ (r) 1 2 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) 1 2 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) 1 2 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) 1 2 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: Ferromagnetism F. 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: Ferromagnetism F. 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 interfaces A. 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 no sample 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 where imensions 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 phase ected using a wire bonder. One of using 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 SC align perpendicular or parallel to In Fig. 3͑b͒ we show the temperature dependence of the
  • 43. Magnetism at LaAlO3/SrTiO3 interfaces J.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 0 z 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 -0.05 z 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 7 z 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 µB z 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 vacancies Ti O Ti O Ti O O Ti 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 0.5 enhanced DOS ρ(EF ) 0 enlarged 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 0 substantial amount of the excess charge -0.5 with O-vacancy transferred to t2g spin-up orbitals -1 1 3dxz DOS Ti 3dxz,yz dominant 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 index strong magnetic moment in the interfacial 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 electrons on more than two Ti sites 0.2 with O vacancy identify: 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 vacancies scenario: 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 scenario scenario 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 j HK = −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 scenario scenario 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?
  • 62. Superconductivity at 200 mK in 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 /cm2 A. 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. Superconductivity Order 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 pairing Superconductivity in the presence of ferromagnetism Finite momentum pairing P. 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 Interaction Spin singlet component of a nearest neighbor interaction: Fourier transform to momentum space: with and Do ground state solutions exist with ∆(k,q) ≠ 0 and ∆(k,-q) ≠ 0 for a specific q ? Yes !
  • 67. Energy at T = 0 Here: q = (q, 0) along x-direction and mean 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 Density Charge 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 interface Effective model Mechanism for superconductivity ?
  • 70. Theoretical design of the interface L1 dpd electric field L2 a a
  • 71. Theoretical design of the interface L2: metallic layer accumulation of charge at interface through field doping kinetic energy: ntot = n0 + n(Ez ) 2D band: bandwidth 8t interaction 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 layer two-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 interface interaction 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,σ interaction 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 pd second order perturbation theory for zero field: 2 V exciton Veff |zero field = −2 pd Δ pd V pd positive: attractive interaction Possibility 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 solution 1. 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 2 3. 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 of carrier 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 pairing Extension: nonlocal effective interaction (through 2 microscopic processes) attractive in d-wave channel
  • 80. d-wave pairing estimates 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 regime remember 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 compressibility Compressibility 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 −Q How 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−1 A/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 = aB T. K, J.Mannhart, J. Appl. Phys. 106, 064504 (2009)
  • 87. Negative compressibility κ−1 A/C = 2 + A/Cgeom (en)2
  • 88. nterface (10, 11). Simple modifi- capacitance in the frequency range between 1 Hz For device 2, the interface underneath the Downloaded from w structures described could yield a to 2 MHz with ~20-mrad resolution in the phase gate was not conductive at Vg = 0, even though enhancement greater than 100%. measurement of the impedance. We were able to the regions away from the YBCOPenetration field measurement. (A) Sketch of the Fig. 3. circular pads Measured negative compressibility onic correlations (12–14) may be vary the DC voltage on the top gate and track were conductive (the two-terminal resistanceWith the interface grounded, we applie surement setup. be- eer yet Au pad with 350 mm was deposited to serve as with gate voltage (16). ate, larger capacitances (15). the capacitance change tion C increased rapidlytween thef.Nb ohmic contacts is of order 500 ohms E from the back gate to the inte at low Measured atexternal electric field 0 ated capacitor devices on it was foundWe measured the capacitance between a top C at 4.2 K). greater than and samples during electric field that penetrates the grou nse top gate because LAO/ that at 300 K, f = 5 Hz, the peak of was 10% In testing these detected the at ructuresgold caused less leakage than did YBCO (16). atthat of Chd. This room-temperature different coolings from room temper- een through in situ growth of gate and the interface frequencies f ranging least four behavior pro- interface layer. The penetration field Ep is determine m,(YBCO) or Au films on thecurrent to the 8 to 2000 Hz while varying a top evidence ruling4.2 Kanomaliesmeasuring noticed that from the device top gate. (B) The p to Although a leakage sur- from gate appeared vides additional gate ature to out for each sample, we the current AO. As below –0.6 V 1A, we then −1 resistance dropswasof the dielectricsample theSTO as the origin varied slightly with ther- κ r to shown in Fig. [the gate-to-channel Vg. Device 1 voltage fabricated on a function of depletion voltage of Iy divided by the measurement frequency for device 1 at T A/C = 2 + A/C to the megohm range near with 12 unit(fig. S3)LAO. On the top surface Li, C. Richter, S. Paetel, T. Kopp, J. of depletion cells of geomthe capacitanceLu of mal cycling. However, the depletion voltage Mannhart, R.C. Ashoori pen enhancement. at three excitation frequencies. At V near zero, the (en) hysics, Massachusetts Institute of Technology nce-02139, USA. 2the room-temperature C2 Vg film, a circular YBCO top gatecapacitor C with charge12–unit cellproportional to f and is constant overga broad range of –0 MA (16)], Center for Electronic Cor- the LAO curves are – Charging a Science 332,with e does not LAO was always neg- with a devices 825 (2011) gnetism, University of Augsburg, of the YBCO-gated 200 mm was patterned. Figure 2A in voltage dV = e/Cgeom. the similar to those Augsburg diameter of devices simply require a change ative, whereas that for devices with 10–unit cell displays the capacitance C versus V curves of LAO layers was positive. This difference sug- V, Iy /f displays a frequency depe V. For Vg –0.18 nce shown in Fig. 2, A and B. As Vg decreased from Because ofg the finite dn/dm, an extra voltage device 1 at 4.2 K. Similar curves are shown in gests that the YBCO top gate tends to depleteeffects of current leakage through probably caused by high 0 should decreased slowly. However, near deple- (dm/dn)/eA is required, where m is the chemical spondence V, C be addressed. E-mail: u Fig. 2B for device 2 with an LAO thickness of the interface underneath the gate: The larger the dd ick- ple layout A B B C A B nce bridge A balancing point fer- Device 1 12 u.c. LAO tch of the ohmiccircular top gate with d = 200 µm mall. contact Ve µ C Cs Vb 200 nm Nb ce layout. the f LAO (10 mall f top gate T = 4.2 K Chd T = 4.2 K 100 nm YBCO ls thick) is d at top o the LAO film Fig. STO nated pre-amplifier Chd kHz gates O top f 1 mm STO from www.sciencemag.org on May 12, 2011 lock-in er a pat- and amplifier ethe layer, LAO wercontacts are deposited close to the corners of the wafer. (B) tion voltage Ve. In the other arm, another ac voltage Vs with the same fre- f 5 BCO/LAO/STO sample with leads attached. The wafer is square quency is applied to a standard capacitor Cs. The signal at the balancing point e is f 7 owsof 5 mm. The diameter of the YBCO circular top gates varies is measured with a pre-amplifier and a lock-in amplifier. During the mea- gth f 8 nd 500 mm. (C) Setup sketch of the capacitance bridge. In one surement, with the phase and the amplitude of Ve held stable, Vs is varied both cell dge, C stands for the sample capacitor, excited by an ac excita- in phase and in amplitude to null the signals at the balancing point. ow- t of fre- www.sciencemag.org SCIENCE VOL 332 13 MAY 2011 825 V≤ ken f= Vg. nce Fig. 4. The inverse of compressibility dm/dn determined rve. C 4.0 Device 3 12 u.c. LAO D field measurements on (A) device 1 and (B) device 2. Th gnal circular top gate with d = 350 µm at the interface is determined by integrating the C vers ). 3.5 Chd enhanced capacitance negative compressibility T = 300 K lowest frequency achieved. (C) The density n dependence we 3.0
  • 89. Summary and conclusions ‣ 2D electron liquid at LaAlO3/SrTiO3 interfaces as compared to semiconductor interfaces, it is not just a quantum well confining free electrons; it is the potentials of the Ti ions in 1–3 uc from the interface which confine the charge carriers ‣ Correlations are sufficiently strong that they influence the spectrum as measured by STS I, U ≥ 8t that a ferromagnetic state is formed, possibly not homogeneously ‣ Magnetism superconductivity coexist at LaAlO3/SrTiO3 interfaces possibly ferromagnetic puddles in the superconducting background role of oxygen vacancies which introduce a strong ferromagnetic coupling? finite momentum d-wave pairing? ‣ Negative electronic compressibility at low density strong increase of the capacitance of the LaAlO3/SrTiO3 interface