Table Of ContentDimer Mott Insulator in an Oxide Heterostructure
Ru Chen,1 SungBin Lee,2 and Leon Balents3
1Department of Physics, University of California, Santa Barbara, Santa Barbara, CA, 93106
2Department of Physics and Centre for Quantum Materials,
University of Toronto, Toronto, Ontario M5S 1A7, Canada
3Kavli Institute of Theoretical Physics, University of California, Santa Barbara, Santa Barbara, CA, 93106
(Dated: January 21, 2013)
We study the problem of designing an artificial Mott insulator in a correlated oxide heterostruc-
ture. We consider the extreme limit of quantum confinement based on ionic discontinuity doping,
andarguethatauniquedimer Mott insulatorcanbeachievedforthecaseofasingleSrOlayerina
GdTiO matrix. InthedimerMottinsulator,electronsarelocalizednottoindividualatomsbutto
3
bonding orbitals on molecular dimers formed across a bilayer of two TiO planes, and is analogous
3 2
totheMottinsulatingstateofHubbardladders,studiedinthe1990s. Weverifytheexistenceofthe
1
dimerMottinsulatorthroughbothabinitioandmodelHamiltonianstudies,andfindforreasonable
0
2 valuesofHubbardU thatitisstableandferromagneticwithaclearbonding/anti-bondingsplitting
of order 0.65eV, and a significant smaller Mott gap whose size depends upon U. The combined
n
effectsofpolardiscontinuity,strongstructuralrelaxationandelectroncorrelationsallcontributeto
a
the realization of this unique ground state.
J
7
1 Recently, the growth techniques from semiconductor totheusualsingleatomlocalization,andiscrucialtothe
physics, such as Molecular Beam Epitaxy (MBE), have success of our scheme. We combine ab initio and model
]
l beenincreasingappliedtotransitionmetalandrareearth calculations to establish the existence and nature of the
e
materials to create correlated heterostructures.1 The re- DMI state theoretically.
-
r sulting atomic layer control promises the ability to de- The starting point for our work is the polar/ionic dis-
t
s signorbital,spin,andchargestates,createnewemergent continuity, which induces a large net charge, typically
.
t phenomena,andstudyfundamentalphysicsofcorrelated half an electron per planar unit cell, at a polar to non-
a
m quantum states in unprecedented new ways. A first step polar interface. The polar discontinuity has been iden-
in this direction would be the creation of the simplest tified as a possible mechanism of doping in many oxide
d- and most dramatic manifestion of electron-electron in- interface studies,1,3 but has only recently been quanti-
n teractions: the formation of a Mott insulators, a system tatively verified systematically4. In MBE grown het-
o whichwouldbeametalaccordingtobandtheory, butin erostructures of GdTiO and SrTiO , a carrier density
3 3
c which instead electrons localize because their motion is of n = 3.5×1014cm−2 (=1/2 e− per planar unit cell)
[ 2d
jammed by their mutual short-range Coulomb repulsion. for each GdTiO3/SrTiO3 interface has been systemati-
1 Mott insulators occur only when the electron density cally observed by Hall coefficient measurements4. These
v is commensurate with the underlying lattice, and typi- electrons fall into the empty d states of the SrTiO3, and
2
cally an (odd) integer number of electrons per atom is consequently high 3d carrier density can be achieved by
2
2 required. A charge density of one electron per atom is confinement in narrow quantum wells of SrTiO3 embed-
4 enormous, reaching of order n2d ≈ 7×1014cm−2 for a ded in thicker GdTiO3, with n3d=n2d/w, where w is the
. typical perovskite structure even if these electrons are well width. Recent transport experiments5 showed that
1
0 confined to a single atomic layer. This provides a chal- such wells with a width of a few SrTiO3 unit cells are
3 lengeforheterostructures,asthisn2d isalreadyanorder indeed strongly correlated metals with ultra-high carrier
1 of magnitude larger than can be achieved in the high- density n2d = 7×1014cm−2 arising from two interfaces,
: est density semiconductors, and even if it is created, corresponding to 1/2+1/2=1e− per planar unit cell.
v
the electron density per atom will be greatly reduced by To maximize the 3d electron density and approach
i
X the electrons tendency to spread out in the third dimen- the Mott limit, we take this approach to its logical end
r sion. In this paper, we show that these difficulties can and consider the case of a single SrO layer embedded in
a
beovercomebyjudiciousdesignofaDimerMottInsula- GdTiO . However, even in this case of ultimate confine-
3
tor (DMI), a state envisioned decades ago in the context ment, we do not achieve a 3d density of 1 e− per atom.
of one-dimensional Hubbard ladders,2 and created here Thisisbecausethedopedelectronsgosymmetricallyinto
in two dimensions at a single monolayer of SrO embed- the two interfaces TiO layers on either side of the SrO
2
ded in a GdTiO matrix. In the DMI, the requisite high plane. While this situation appears unfavorable for the
3
charge density is achieved by combining ionic disconti- formation of a Mott insulator, all is not lost. On funda-
nuity doping, quantum confinement, and the formation mentalgrounds,theconditionfortheformationofaMott
of electronic dimers. The dimers are bonding orbitals on state depends only on the charge per unit cell defined by
electron pairs, to which electrons are Mott localized in- the translational symmetry of the system. Here the two
steadoftoindividualatoms. Dimerformationhalvesthe interface planes form a bilayer, with translational sym-
charge density needed to reach the Mott state, relative metry only within the plane, and there is indeed a unit
2
Ti (a) (SrTiO3)1(GdTiO3)3
a GSrd º e ( ) 115600 ʇ (SrGTdiOT3iO)1(3G bduTlkiO3)5
O ngl ʇ ʇ Ê
b y Ti a 140
z O- 160 (b)
x Ti-
z=-1 z=0 z=1 z=2 ʇ
150
ʇ Ê
140
FIG. 1: (Color online) One unit cell of the relaxed structure 0 1 2 3 4
ofthesuperlattice(SrTiO ) (GdTiO ) . TiO octahedraare O z-coordinate
3 1 3 3 6
drawn in blue, to emphasize tilts.. The superlattice repeats
periodicallyalongthezdirection. ThebilayerconsistsofTiO
2
planesatz=0andz=1. Hereaandbdenotethetwodifferent FIG. 2: (Color online) Ti-O-Ti bond angles along (a) verti-
Ti sublattices. cal direction and (b) in-plane direction with respect to the
O (in the Ti-O-Ti bond) z positions. O along vertical direc-
tion resides in the RO (R=Sr, Gd) plane, which is why the
coordinate appears to be half-integer.
charge per planar unit cell. This indicates that a Mott
stateispossibleinprinciple. Torealizeit,wemustsome-
how induce the two Ti atoms in a unit cell to act as a √ √
single ‘superatom’. 2a× 2a×c unit cell to allow for the possibility of
octahedral tilts, where a is set to be the value of the ex-
Itisinstructivetoviewthebilayeronitsside,withthe
perimental SrTiO lattice constant, 3.905˚A. The struc-
Ti atoms projected into an x-z plane (here we use stan- 3
turaloptimizationisdonebothontheatomiccoordinates
dard cubic coordinates for the perovskite structure, and
and c/a ratio, within the GGA + U approximation. We
z is the growth direction). The Ti-O-Ti network then
focus on U =U-J=4eV on the Ti d orbitals, an accept-
forms a ladder with Ti-O-Ti bonds between the two in- eff
ablevalueforthetitanates8. Inaddition, wefurtheradd
terfaces making the rungs of the ladder, and intra-plane
U =U-J=8.5eVontheGdf orbitalssincetheenergyof
bondsprojectingtotheladderslegs. InaHubbardmodel eff
the occupied Gd f bands lie much lower than the Fermi
description, symmetry dictates that the hopping ampli-
energyinpractice. Thisvaluewillnotaffecttherelevant
tude for electrons along the legs, t, and along the rungs,
electronic properties.
t ,areunequal. Correlatedelectronladderswerestudied
⊥
intensively in the 1990s,2 and in particular it was shown We now discuss the key features of the relaxed struc-
when t /t is sufficiently large (approximately t /t > 1 ture, which becomes independent of n for n ≥ 3, and is
⊥ ⊥
for the one-dimensional Hubbard ladder), electrons form shown in Fig. 1. First, relaxation is significant at the
an unconventional Mott state of bonding orbitals on the two interface layers, but decays quickly into the GdTiO3
rungs of the ladder - the one dimensional analog of the region. Second, the Ti-O-Ti bond angles are highly di-
DMI. Qualitatively, we anticipate the same physics ap- rection dependent near the interface, as shown in Fig. 2.
pliestotheHubbardbilayers,providedt /tissufficiently The “vertical” Ti-O-Ti bond connecting the two inter-
⊥
large. faces is slightly distorted, with a 160◦ angle. The next
Hopping parameters in transition metal oxides are vertical Ti-O-Ti bond away from the interface is already
largely controlled by the metal-oxygen-metal bond an- highly distorted, with only a 3◦ angular difference from
gle, due to the directionality of d and p orbitals, and that of bulk GdTiO3, and more distant bonds are nearly
are generally largest when the bond angle is closest to indistinguishable from bulk. On the other hand, the in-
180◦. Intuitively, we expect that the interlayer Ti-O- planebondsaredistortedbutdifferentfromthebulkeven
Ti bonds are the most SrTiO -like, while those within in the interfacial layers, with bond angles of about 153◦.
3
the TiO planes conform more closely to those of the Third, the Ti-O bond length varies by only 6% between
2
GdTiO . Since SrTiO is nearly perfectly cubic, while the longest and shortest bonds in the entire superlattice,
3 3
GdTiO is one of the most highly distorted titanates, and is less significant compared to the dramatic bond
3
this appears quite favorable. We checked this intuition angle variations.
with ab initio density functional theory (DFT) calcula- Is the bond angle difference, together with other mi-
tions for periodic superlattices of the single SrO layer, nor structural relaxation effects sufficient to promote
(SrTiO ) (GdTiO ) , with n=3,5. Calculations were a DMI? We first check the electronic structure within
3 1 3 n
performedintheWien2k6implementationandthegener- the GGA+U approximation using the relaxed structure.
alized gradient approximation7 (GGA). An RKmax pa- Searching for possible magnetic structures, we obtained
rameter 7.0 was chosen with RMTs of 1.91 a.u., 1.69 the lowest energy for a state with ferromagnetic align-
a.u., 2.29 a.u. and 2.27 a.u. for Ti, O, Sr and Gd, ment of Ti spins within each TiO plane, with interfa-
2
respectively. The DFT calculation is carried out in a cial Ti spins antiparallel to those in the GdTiO region
3
3
(((bbb)))
up up Direction Direction
(a) (c)
0.8 (im; jn) [1, 1, 0] [-1, 1, 0] (im; jn) [0, 0, 1]
(1ayz; 1byz) 0 -0.35 (0ayz; 1ayz) -0.6
0.4
(1ayz; 1bxz) 0.14 0.14 (0ayz; 1axz) 0
0. (1axz; 1byz) -0.12 -0.12 (0axz; 1ayz) 0
eV) 0.4 TTii xxyz (1axz; 1bxz) -0.35 0 (0axz; 1axz) -0.63
es/ Ti yz (1axy; 1bxy) -0.34 -0.34 (0axy; 1axy) 0
DOS (stat 00..88 (b) douwpn (d) douwpn TABLE I: Hopping parameters for two TiO2 interface layers
from fits to the (SrTiO ) (GdTiO ) superlattice in unit of
0.4 3 1 3 3
eV. The index, for example 1ayz, stands for sublattice a Ti
0. at z =1 with basis yz state. All the major hoppings (larger
than 10% of the largest hopping magnitude) are kept here.
0.4
0.8 down down
-2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 pure GGA calculation was carried out. The eigenvalues
Energy (eV) Energy (eV)
of the f states of Gd were shifted manually away from
the Fermi energy. We constructed 30 maximally local-
FIG. 3: (Color online) Layer-resolved electronic density of ized Wannier functions9,10 (MLWF) around the Fermi
states of Ti 3d states of (a) sublattice a and of (b) sublattice energy, with the 3 t2g orbitals forming the basis. The
b at z=1 (at the interface); Ti of (c) sublattice a and of (d) hopping parameters were then calculated by evaluating
sublattice b at z = 2 (center of GdTiO region). The Fermi the matrix element of the MLWF. Including these hop-
3
energy is set to zero. ping parameters, the tight-binding model of the bilayer
takes the form
(cid:88) (cid:88)
(energy difference compared to parallel to those in the H = tmnc† c (1)
tb ij imα jnα
GdTiO region is small). With this configuration, and
3 (cid:104)ij(cid:105)mn,α
U = 4eV, the interfacial and bulk density of states
eff
(DOS) is shown in Fig. 3. We observe a bulk gap of where i and j are nearest neighbor sites, m and n are
1.25eV,comparabletotheoreticalvaluesintheliterature, orbital indices and α is the spin index. The hopping
butremarkablyamuchreducedbutstillnon-zerogapat parametersforthetwointerfacialTilayersaretabulated
the interface of approximately 0.2eV. This is a signature in Table 1, which contains the full orbital dependence of
of the DMI state. Within GGA+U, the DMI persists for thehoppingterms. Themaximumintra-planeandinter-
U (cid:38)3.5eV,withbothbulkandinterfacialgapsreduced plane hopping matrix elements are t ∼ 0.35eV, t ∼
eff ⊥
for smaller U. To see the DMI more directly, we decom- 0.63eV, respectively, which gives strong support for the
posed the DOS by Ti site and the 3 t orbitals (defined DMI picture. The separation energy between bonding
2g
by pseudo-cubic axes). Within the interface plane, the and anti-bonding state is consistent with this magnitude
major DOS just below the Fermi level has no xy charac- inter-plane hopping.
ter,consistinginsteadofspindownpredominantlyxz/yz We supplement these hopping parameters with
orbitals on alternating a/b sublattices. Consequently, interactions,8,11 to form the effective Hamiltonian8,11
we identify this state as the occupied bonding orbital
of the DMI, with the xz/yz orbital degeneracy split by H =H +H
tb int
octahedral rotations. The anti-bonding state, centered (cid:88)(cid:104) (cid:88) (cid:88)
H = U n n +U(cid:48) n n
at around 0.2eV, is again mainly composed of yz or xz int im↑ im↓ im↑ in↓
orbitals. The separation (of subband centers) from the i m m(cid:54)=n
bonding state gives the bonding/anti-bonding splitting + 1(U(cid:48)−J) (cid:88) n n +J (cid:88) c† c c† c
of 0.65eV. The gap is much smaller than this splitting, 2 imα inα im↑ in↑ in↓ im↓
m(cid:54)=n,α m(cid:54)=n
however, due the width of the bonding and anti-bonding
(cid:88) (cid:105)
bands. Away from the interface region, the electronic +J(cid:48) c† c c† c ,
im↑ in↑ im↓ in↓
structure resembles that of bulk GdTiO . For the Ti
3 m(cid:54)=n
in the center of the GdTiO region, we observe domi- (2)
3
nant xy/yz and xy/xz states alternating between two
orthorhombic sublattices. Quantitative analysis shows whereU andU(cid:48) representon-siteintra-orbitalandinter-
theoccupationatxy,yz andxz statesare37%,16%and orbital Coulomb repulsion between up and down spin,
47% for Ti at sublattice a, similar to in bulk GdTiO . respectively, and J is the Hund coupling. We will not
3
TostudytheDMIstatemoreexplicitly,weconstructed restrict the condition to Slater-Kanamori interaction pa-
a model extended Hubbard Hamiltonian by extracting rameter U(cid:48) = U − 2J and J = J(cid:48), but rather simply
hopping parameters from the ab initio calculations. Us- assume J = J(cid:48) and explore the phase diagram by vary-
ing the optimized structure obtained within GGA+U, a ing all the other parameters.
4
4 enhancestheferromagneticinsulatingstate. Forourbest
guess at physically appropriate values, e.g. U = 4.5eV
CO+I andtheSlater-KanamoriU(cid:48) =U−2J,theferromagnetic
3 AFM+I
DMI obtains. The Hartree-Fock results for these values
) for the band gap and orbital ordering are very similar to
V
e FM+I those of the previously discussed GGA+U calculations.
U′ ( 2 CO+M In summary, we have argued for the existence of a
Dimer Mott Insulator (DMI) for a single SrO layer em-
1 PM+M bedded in a thick GdTiO matrix, using both ab ini-
FM+M 3
tio and model calculations. The DMI state is unique
to the bilayer TiO structure created by a single SrO
2
0
layers; we have indeed verified that a metallic state is
0 1 2 3 4 5
U (eV) obtained in GGA+U for the case of two SrO layers em-
bedded in GdTiO (see supplemental material). Insu-
3
lating behavior in such structures experimentally should
FIG.4: (Coloronline)Hartree-FockphasediagramforHund’s therefore be attributed to the combined effects of disor-
coupingJ =0.6eV.Here,weabbreviatethephasesasfollows: der(e.g. SrTiO thicknessfluctuations)andinteractions.
3
PM+M = paramagnetic metal; CO+M= Weakly charge or- The DMI for a single SrO layer could be experimentally
deredmetal;CO+I=chargeorderedinsulator;FM+M=fer- probed by many experiments, including transport, op-
romagnetic metal; FM+I = ferromagnetic insulator; AFM+I tical measurements of the gap and bonding-antibonding
= antiferromagnetic insulator.
splitting, and angle-resolved photoemission. Observing
the magnetic structure is more difficult, but might be
possible with ferromagnetic resonance or optical dichro-
Fig.4showsthephasediagramasafunctionofU and
ism. This work suggests many directions for future the-
U(cid:48) obtained by the Hartree-Fock approximation, with
oretical and experimental research. We anticipate that
fixed Hund’s coupling J =0.6eV. When the Coulomb re-
the Mott state can be controlled and modified by vary-
pulsion is small, the ground state is just paramagnetic
ingcomposition, strainandthegrowthdirection. Itmay
metal. In the unphysical region where inter-orbital re-
be possible to create antiferromagnetic DMIs by varying
pulsion is dominant, U(cid:48) >U, electrons reside in a single
the rare earth ion; however, theory is needed to gauge
orbitalperTiandinfactchargeorder. Ofmostinterestis
whetherthisalsomaydestabilizethedimerformationit-
thelowerrightpartofthephasediagram,whereU >U(cid:48).
self. Choiceofsubstratealsoeffectsthestrainandgrowth
Inthisregion,ferromagnetismarisesforsufficientlylarge
directionofthetitanatefilms. WhethertheDMIpersists
U. In the weak inter-orbital repulsion limit, the elec-
when the GdTiO grows along the (110) direction is an
trons are distributed in all three t orbitals. Under this 3
2g important question for future study. More speculatively,
condition,thesystemisalwaysmetallicorsemi-metallic,
we might contemplate the possibility of superconductiv-
since those electrons in xy orbitals are non-bonding be-
ityinducedbydopingtheDMI,byanalogytothesuper-
tweenlayersandhencemetallic. ForsufficientlylargeU(cid:48),
conductivity predicted theoretically and observed exper-
however, the inter-orbital repulsion eventually disfavors
imentally in ladder systems.
and empties the xy states, in favor of the xz/yz orbitals
which have lower energy through inter-layer hopping. In We thank Jim Allen, Susanne Stemmer, Dan Ouel-
this way the bonding state becomes fully occupied and lette, Pouya Moetakef and Chuck-Hou Yee for helpful
a (ferromagnetic) DMI is achieved. Further increase of discussions. We acknowledge support from the Center
U(cid:48) enhances the gap and eventually prefers an antifer- for Scientific Computing at UCSB: NSF CNS-0960316.
romagnetic DMI, since Hund’s coupling becomes ineffec- This research is supported by DARPA grant No.W911-
tive if there is strictly one electron per site, and super- NF-12-1-0574(L.B.andR.C.)andtheMRSECProgram
exchange becomes dominant. For completeness, we also of the National Science Foundation, Award No. DMR
have tested J=0,1eV, which shows the Hund’s coupling 1121053, NSERC, CIFAR (SB.L.).
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