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)
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
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
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 ?
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)
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)
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