PHYSICAL REVIEW B, VOLUME 64, 064401

Spin versus lattice polaron:

Prediction for electron-doped CaMnO3

Yiing-Rei Chen and Philip B. Allen
Department of Physics and Astronomy, State University of New York, Stony Brook, New York 11794-3800
共Received 23 January 2001; revised manuscript received 23 April 2001; published 11 July 2001兲
CaMnO3 is a simple bipartite antiferromagnet 共AF兲 that can be continuously electron doped up to LaMnO3 .
Electrons enter the doubly degenerate E g subshell with spins aligned to the S⫽ 23 core of Mn4⫹ (T 3↑
2g ). We takethe Hubbard and Hund energies to be effectively infinite. Our model Hamiltonian has two E g orbitals per Mnatom, nearest-neighbor hopping, nearest neighbor exchange coupling of the S⫽ 23 cores, and electron-phonon
coupling of Mn orbitals to adjacent oxygen atoms. We solve this model for light doping. Electrons are confinedin local ferromagnetic 共FM兲 regions 共spin polarons兲 where there proceeds an interesting competition between
spin polarization 共spin polarons兲, which enlarges the polaron, and lattice polarization 共Jahn-Teller polarons兲,
which makes it smaller. A symmetric seven-atom ferromagnetic cluster (Mn27⫹
7 ) is the stable result, with a net
spin S⫽2 relative to the undoped AF. The distorted oxygen positions around the electron are predicted. The
possibility that two electrons will form a bipolaron has been considered. A fairly modest Coulomb repulsion
U c ⫽0.98 兩 t 兩 共where t⬇⫺0.75 eV兲 will destroy any simple bipolaron. Therefore we do not expect phaseseparation to occur. The model predicts a critical doping x⯝0.045 where the polaronic insulator becomes
unstable relative to a FM metal.
DOI: 10.1103/PhysRevB.64.064401

PACS number共s兲: 75.30.Vn, 75.50.Ee, 71.38.Ht

I. INTRODUCTION

CaMnO3 is a bipartite (G-type兲 antiferromagnetic 共AF兲
insulator1 with a Néel temperature T N ⫽125 K and almost
perfect cubic perovskite crystal structure. There is not a large
amount of literature on this material. It deserves attention asa particularly simple case of an AF insulator that can beelectron doped. The (T,x) phase diagram2,3 of theCa1⫺x Lax MnO3 series has attracted attention because of the
fascinating interplay of spin order, orbital order, and metallic
versus insulating transport. ‘‘Colossal magnetoresistance’’
occurring at concentration x⬇0.65 and T⬇250 K is themost dramatic manifestation.4 For small x, magnetization andconductivity measurements5,6 suggest local ferromagnetic
共FM兲 regions or ‘‘spin polarons’’ in the range 0.02⬍x⬍0.06. In this paper we use a model for pure CaMnO3 , thex⫽0 end member, and predict its behavior under light doping xⰆ1. We keep T equal to 0 and neglect lattice zero-point
energy, but allow both spin and lattice coordinates to relax.
Our main question is, what is the ground state of an excess electron in CaMnO3 ? The answer to this question will
also apply to lightly doped CaMnO3 provided the doped stateis homogeneous. Our model has four parameters. 共1兲 The
bandwidth parameter t governs the effective Mn E g electron
hopping between Mn3⫹ and Mn4⫹ through the intervening
oxygen. The spins of the two Mn ions must be parallel as in
the usual ‘‘double-exchange’’ model.7,8 共2兲 The magnetic exchange parameter J couples spins of first-neighbor Mn ions,due to virtual hopping of T 2g electrons. 共3兲 The electronphonon coupling constant g describes interactions between
Mn E g orbitals and the six nearest oxygen atoms. 共4兲 Oxygen
displacements u are opposed by the restoring force ⫺Ku,
where ␻ ⫽ 冑K/M is an Einstein frequency assigned to oxygen vibrations along the bonds. There are two important dimensionless parameters. Spin polarons9 are controlled by theratio ␤ ⫽t/JS 2 . Jahn-Teller 共JT兲 lattice polarons10 are controlled by the parameter ⌫⫽g 2 /Kt.11 Balancing these com0163-1829/2001/64共6兲/064401共8兲/$20.00

peting effects, we find the most favorable local FM spinarrangement, lattice distortion, and electron wave function.
As doping increases, we predict a transition from polaronic
insulator to FM metal.
II. MODEL HAMILTONIAN

The Mn ion in CaMnO3 has a configuration 3d 3 , i.e.,
the three spin-aligned T 2g states (xy,yz,zx) are filled withelectrons, while the two spin-aligned E g states 关 ␺ 2 ⫽(x 2⫺y 2 )/ 冑3, ␺ 3 ⫽3z 2 ⫺r 2 兴 are empty and lie above by the
crystal-field splitting. The empty opposite-spin T 2g states are
split to even higher energy by the Hund term J H . Light
electron doping puts carriers into the doubly degenerate E glevel. Hopping of the (dd ␲ )-type occurs from T 2g to T 2g
and of the (dd ␴ )-type from E g to E g , but no E g to T 2g
hopping matrix element exists because of the simple cubic
structure. Virtual T 2g hopping12 共at the cost of Hubbard energy U) gains delocalization energy if adjacent spins are
antiparallel. This gives an S⫽ 23 antiferromagnetic Heisenberg Hamiltonian with exchange coupling J⫽2(dd ␲ ) 2 /U,13
U being large compared with (dd ␲ ),14 and agrees with theexperimentally observed magnetic structure of pure
CaMnO3 . The ground state has ↑ spins on the A sublatticeជ •lជ)⫽1, where ជl labels the Mn sites兴 and ↓ spins关when exp(iQជ •lជ)⫽⫺1] with Qជon the B sublattice 关when exp(iQ⫽( ␲ , ␲ , ␲ ).
Ignoring for now the electron-phonon terms, the Hamiltonian for an excess electron is H⫽Ht ⫹HJ . The first term
contains hopping of Mn E g electrons to nearest neighbors,

64 064401-1

4⫹

Ht ⫽t

关 L 共 ជl , ជl ⫾ẑ 兲 c †3 共 ជl ⫾ẑ 兲 c 3 共 ជl 兲兺l,⫾⫹rotations to x̂,ŷ directions兴 ,

␪1 ␪2
␪1 ␪2
L 共 1,2兲 ⫽cos cos ⫹sin sin e ⫺i( ␾ 1 ⫺ ␾ 2 ) .
2
2
2
2

共1兲
共2兲

©2001 The American Physical Society

YIING-REI CHEN AND PHILIP B. ALLEN

PHYSICAL REVIEW B 64 064401

HJ ⫽兺 JSជ 共 ជl 兲 •Sជ 共 ជl ⬘ 兲 .

具 l,l ⬘ 典

共5兲

Mean-field theory gives the value JS 2 ⫽3.23 meV for theexchange coupling 关we use a quantum treatment for spin 23
and the measured Néel temperature T N⫽125 K 共Ref. 19兲兴.
Corrections to mean-field theory for the Heisenberg
antiferromagnet20–22 increase JS 2 to 4.74 meV. This gives
t/JS 2 ⫽158, with a probable uncertainty of 10%. The phononparts of the Hamiltonian are given in Sec. V.
III. UNIFORM SOLUTIONS

FIG. 1. Schematic spin structures for the antiferromagnetic G
共AFG兲, ferromagnetic 共FM兲, and antiferromagnetic C 共AFC兲 structures and interpolating canted structures.

Here c 3 ( ជl ) destroys an electron state ␺ 3 (rជ ⫺ ជl ) on the Mn
atom at ជl , ជl ⫾ẑ labels the Mn neighbors above and below theone at site ជl , and t is the (dd ␴ ) integral from the SlaterKoster two-center theory.15 We use a value t⫽⫺0.75 eV
obtained from fitting the band structure of CaMnO3 共Refs. 16
and 17兲 共see Sec. III兲. The factor L(1,2) comes from locallyrotating the axis of spin quantization for the ith E g electron
into the direction ( ␪ i , ␾ i ) of the i th S⫽ 23 core spin, treating
the angles ( ␪ i , ␾ i ) as classical parameters and discardingfrom the Hilbert space the state with spin opposite to the corespin 关i.e., assuming J H →⬁ 共Refs. 17 and 18兲. In this paper,
we usually take the spins to be perfectly ordered at T⫽0,that is, ␪ ⫽ ␪ 1 ⫺ ␪ 2 equals 0 or ␲ , corresponding to L(1,2)
equal to 1 or 0 for FM or AF oriented neighbors. We will
also consider uniformly canted states where a relative angle␪ ⫽ ␲ ⫺ ␪ 0 occurs 共with ␾ 1 ⫺ ␾ 2 ⫽0), so that L takes thevalue sin(␪0/2). These spin orientations are shown in
Fig. 1.
The rotation of ␺ 3 to the x̂ axis is (⫺ ␺ 3 ⫹ 冑3 ␺ 2 )/2⫽3x 2 ⫺r 2 , and to the ŷ axis it is (⫺ ␺ 3 ⫺ 冑3 ␺ 2 )/2⫽3y 2⫺r 2 . Using these, we rewrite Ht in the usual orthonormal
basis ( ␺ 2 , ␺ 3 ),
Ht ⫽t

冉

c 2 共 ជl ⫾ ␦ˆ 兲冊

兺 共 c †2共 ជl 兲 c †3共 ជl 兲兲 T␦ c 共 ជl ⫾ ␦ˆ 兲 ,l, ␦ ⫽x,y,z
3

共3兲

where it is understood that the hopping only operates between parallel spin Mn atoms. The hopping matrices areTx ⫽冉 冊 冉 冊
3
4

⫺

⫺

冑3
4

冑3
4

1
4

Tz ⫽

,

Ty ⫽

3
4

冑3
4

冉 冊
0

0

0

1

.

冑3
4

1
4

,

共4兲

The HJ term is the AF nearest-neighbor T 2g exchangeFirst consider the hypothetical case of a uniform FM spinorder. Then a doped-in electron could hop without paying aHund penalty, and extended Bloch states would form withwave functions⌿ k ⫽d 2

1

1

ជ ជ
ជ ជ
e ik • l ␺ 2 共 ជl 兲 ⫹d 3
e ik • l ␺ 3 共 ជl 兲 , 共6兲
兺
兺
冑N l
冑N l

where d 2 ,d 3 are coefficients. Diagonalizing Eq. 共3兲, the resulting energy eigenvalue, taking into consideration that t is
negative, is
E 共 kជ 兲⫽⫺cos k x ⫺cos k y ⫺cos k z兩t兩⫿ 共 cos2 k x ⫹cos2 k y ⫹cos2 k z ⫺cos k x cos k y⫺cos k y cos k z ⫺cos k z cos k x 兲 1/2.

共7兲

At cos kx⫽cos ky⫽cos kz⫽1, the energy is minimum, E(kជ⫽0)⫽⫺3 兩 t 兩 , the state being doubly degenerate. Although
the actual spin arrangement is not FM, the result 兩 3t 兩 sets a
useful scale for the maximal energy gain from electron delocalization. Also, the dispersion relation 共7兲 can be fitted to a
FM CaMnO3 majority-spin band structure calculated indensity-functional theory,16 determining the hopping energyt⫽⫺0.75 eV.
For light doping, the magnetic energy Eq. 共5兲 of antiferromagnetic order (⫺zJNS 2 /2 for classical spins; N is thenumber of Mn ions兲 is much larger than the hopping energy.
As discussed by de Gennes,23 the best uniform solution 共fora one-band model兲 is a compromise where antiferromagnetic
spins cant uniformly toward FM solutions. Start from theNéel AF structure and let all spins tilt towards the x̂ direction
with angle ␪ /2, as shown in Fig. 1. This costs magnetic energy ⌬E J ⫽zNJS 2 (1⫺cos ␪)/2. A doped-in electron can now
delocalize, reducing its energy by ⫺3 兩 t 兩 sin(␪/2). For small
doping x⫽n/NⰆ1, the total hopping energy is ⌬E t ⫽⫺3n 兩 t 兩 sin(␪/2). Therefore when sin(␪/2)⫽x 兩 t 兩 /4JS 2 , the total energy ⌬E⫽⌬E J ⫹⌬E t is minimum, ⫺3Nx 2 t 2 /8JS 2 .
From this estimate, the system cants all the way to FM when
the optimal ␪ equals ␲ , which occurs at x⫽4JS 2 / 兩 t 兩⯝0.025.
Because the doubly degenerate E g electrons hop anisotropically, a better canted solution exists. Let all spins withជ •łជ)⫽1 关 K
ជ is (0,␲ , ␲ )] tilt towards ⫹x̂ and all spins
exp(iK

064401-2

SPIN VERSUS LATTICE POLARON:

PREDICTION . . .

PHYSICAL REVIEW B 64 064401
TABLE I. Ground-state energy of symmetric clusters, with only
spin polaron effects included, i.e., from exact diagonalization of
Ht ⫹HJ in the 2M dimensional subspace, where M is the number of
Mn sites in the clusters and each Mn atom has two relevant E gorbitals. The 25-site cluster gives the lowest ground-state energy.
The last column uses t⫽⫺0.75 eV and JS 2 ⫽4.74 meV.

FIG. 2. Energy per electron of various doped states. Electron
concentration x is measured in units of 1/␤ ⬅JS 2 / 兩 t 兩 . Bold lines
denote the most favorable uniform states. Nonuniform 共polaron兲
states 共discussed in Sec. VI, using ⌫⫽0.25兲 are denoted by dotdashed lines for the various values of ␤ indicated. Among different
extended states, AFC lies lowest from ␤ x⫽2⬃4, and FM takes
over above ␤ x⫽4 (x⯝0.025). For light doping, the inhomogeneous polaron states are stable. As x increases, uniform states takeover. The transition from polaron-doped AFG to FM occurs at x
⯝0.045 共using ⌫⫽0.25, t⫽0.75 eV, JS 2 ⫽4.74 meV, ␤ ⯝158).
Notice that the value of ␤ determines the critical doping x c of thephase transition and even the transition type, as small ␤ results in
an AFG to AFC transition, rather than to FM. But for small ␤ thepredicted x c can be large enough to exceed the light-doping approximation used here.

ជ •łជ)⫽⫺1 tilt towards ⫺x̂ with an angle ␪ /2, aswith an exp(iKshown in Fig. 1. The magnetic energy cost is reduced to⌬E J ⫽zNJS 2 (1⫺cos ␪)/6, but the delocalizaton energy lowering is also reduced to ⌬E t ⫽⫺2n 兩 t 兩 sin(␪/2). This type of
canting terminates in C-type antiferromagnetism when x⫽2JS 2 /t, as shown in Fig. 1. Figure 2 summarizes our results for the energies of uniform solutions vs x.
IV. PURE SPIN-POLARON EFFECT

Experimentally, CaMnO3 is insulating at a small doping x.
This rules out uniform solutions with electrons doped intodelocalized states. The simplest picture is that each doped
electron is localized on one Mn site, creating a local Mn3⫹
with spin S⫽2 in an AF background, for a spin excess of 21 .
This does not agree with the measured saturated
magnetization,5 which has been interpreted in terms of local
spin flipping 共net excess spin of 2兲 in the region 0.02⬍x⬍0.08. Local spin flipping leads to local FM regions 共called
spin polarons兲 around doping centers, allowing the doped
electron to gain delocalization energy. In this section we discuss candidate localized ground states of a single E g electron, using the same Hamiltonian H⫽Ht ⫹HJ .
First we make a continuum 共effective-mass兲 approximation in the spirit of Nagaev.9 Inside the local FM region,

Clustersize M

Number of
spins flipped

1
7
25
51
57
63

0
1
6
13
14
19

Energy gain fromhopping ( 兩 t 兩 )Ground-stateenergy 共eV兲00⫺1.242⫺1.406⫺1.097⫺0.989⫺0.869⫺ 冑3⫽⫺1.732⫺2.330⫺2.449⫺2.380⫺2.600electrons hop like free electrons with inverse band mass
1/m * ⫽(a/ប) 2 兩 t 兩 given by Eq. 共7兲, with a the perovskite lattice constant a⯝3.73 Å . This electron sits in a sphericalwell with infinite walls at radius R, whose depth is ⫺3 兩 t 兩 .
G-type antiferromagnetism resumes for r⬎R. The groundstate energy of the electron in this polarized region is then冉

⌬E t ⫽ 兩 t 兩 ⫺3⫹

␲ 2a 2
2R 2

冊

,

共8兲

where the second term is the zero-point energy in the well.
The magnetic energy cost from spin canting is⌬E J ⫽zJS 2 共 4 ␲ R 3 /3a 3 兲 ,共9兲where the last factor is the number of Mn atoms in the cluster. Optimizing ⌬E⫽⌬E t ⫹⌬E J over R, the optimal size
cluster has a number of Mn atoms inside the sphere equal to

冉冊

冉 冊

R 3 4
␲兩t兩
4
␲
⫽ ␲
3
a
3
4zJS 2

3/5

⯝26.

共10兲

This is close to the optimum 25-site symmetric cluster that
we will find by exact diagonalization and show in Table I.
However, we will also find a smaller asymmetric cluster oflower energy.
It is interesting that continuum theory gives a lower energy if the region r⬍R is canted rather than fully FM. The A
sublattice is fixed for all r, but the B sublattice is tilted toward the A sublattice by an angle ␪ for r⬍R. Then the optimum tilting angle is 180° 共FM兲 for ␤ ⫽t/JS 2 ⬎ ␤ c⫽(7 ␲ 2 /18) 5/24z/ ␲ ⯝220, while for smaller ␤ , the optimal
tilting angle is sin(␪/2)⫽ ␤ / ␤ c . Our estimated value is ␤⯝158, which gives ␤ / ␤ c ⯝0.72. So the spins inside radius R
are approximately 90° apart, and the optimum cluster size
increases to 31.
Now we repeat the calculation using the true discrete
Hamiltonian. First consider flipping only one spin. In thisway, the spin-flipped Mn 共the central site兲, along with its sixnearest neighbors, form a seven-site cluster with all sevenspins parallel. The cluster is invariant under transformationsof the point group O h . If the central Mn spin is kept un-

064401-3

YIING-REI CHEN AND PHILIP B. ALLEN

PHYSICAL REVIEW B 64 064401
TABLE II. Ground-state energy of asymmetric clusters, with
only spin-polaron effects included. The as3-4 cluster gives an evenlower ground-state energy than the 25-site cluster shown in Table I.
Clustersize M

Number of
spins flipped

Energy gain fromhopping ( 兩 t 兩 )Ground-stateenergy 共eV兲

12 共as2-1兲
13 共as2-2兲
16 共as3-2兲
17 共as3-1兲
17 共as3-3兲
17 共as3-4兲

2
2
3
3
3
3

⫺1.936⫺2⫺2.015⫺1.936⫺1.984⫺2.145⫺1.338⫺1.386⫺1.341⫺1.281⫺1.317⫺1.438

V. LATTICE POLARON EFFECT

FIG. 3. Different asymmetric clusters: as2-1 and as2 have two
spins flipped; the four types of as3 have three spins flipped.

flipped, but instead the spins of those six nearest neighbors
are flipped, a 25-site O h -symmetric cluster is formed. Similar
steps can be taken to obtain larger and larger symmetric clusters. We will not look at candidate states with canted spins
except for one special case to be mentioned later.
Each spin flip costs magnetic energy 6⫻2JS 2 . Since
spins are parallel inside the cluster, the electron can hop
among the E g orbitals of all the spin-aligned Mn ions, with acorresponding energy lowering from delocalization. Table I
shows numerical values of ground-state energy found by exact diagonalization of 2M ⫻2M Hamiltonian matrices forsymmetric clusters of M Mn atoms, ranging in size from M⫽1 to 63. The ground states are all doubly degenerate (Egrepresentation兲 because of the O h symmetry of the cluster.
Asymmetric clusters are also possible. If we flip one spin
at site ជl and another one at site ជl ⫹x̂⫹ŷ, a 12-site cluster isformed. Starting from this 12-site cluster, there are four different ways to create a larger cluster with one more flipped
spin, as shown in Fig. 3. Other possibilities are less closely
packed, such as the 13-site cluster shown in the figure, andother 3-spin-flipped cases not shown here. The ground-stateenergy of these examples are calculated, as shown in Table
II. For our chosen values of t and JS 2 , the most favorable
spin polaron is an asymmetric 17-site cluster with threeflipped spins.
There are many possible variations with inhomogeneously
canted spins, of which we considered only one, a 25-sitecluster with the seven inner atoms canted rather than ferromagnetically aligned with the outer 18 atoms. This interpolates between the seven- and the 25-atom cluster. Without
canting, the seven- and 25-atom cluster become equal in energy for the value 兩 t 兩 /JS 2 ⯝100, smaller than our preferredvalue of 158. It turned out that in the range 94⬍ 兩 t 兩 /JS 2⬍104, the locally canted state was lower in energy than either the seven- or the 25-atom pure ferromagnetic cluster.

The degenerate ground state, found in the previous section for symmetrical spin polarons, is Jahn-Teller 共JT兲unstable.24 We now add to our Hamiltonian lattice distortions, controlled by the electron-phonon interaction. The
only lattice degrees of freedom included are oxygen motions
along the bonds to the nearest two Mn ions. If an oxygenmoves along this bond a distance u, it gets closer to one Mn
atom and farther from another. This will raise by gu 共for thecloser Mn兲 or lower by gu 共for the farther Mn兲 the energy of
any occupied Mn E g state of the type that points toward theoxygen 共that is, the 3z 2 ⫺r 2 state if oxygen motion in the ẑ
direction is considered.兲 We use adiabatic approximation
共oxygen mass →⬁) and treat the oxygen distortions as classical parameters. Each Mn ion is surrounded by six oxygens
whose distortion amplitudes 关 u ␦ ( ជl ⫾ ␦ˆ /2), ␦ˆ ⫽x̂,ŷ,ẑ 兴 formbasis vectors for a representation of O h , namely, a 1g 丣 e g
丣 t 1u . The vector representation t 1u contains the distortionsu z ( ជl ⫹ẑ/2)⫹u z ( ជl ⫺ẑ/2), etc. The remaining three degrees of
freedom 关 Q z ⫽u z ( ជl ⫹ẑ/2)⫺u z ( ជl ⫺ẑ/2),etc.兴 form basis vectors of a 1g 关 Q 1 ⫽(Q x ⫹Q y ⫹Q z )/ 冑6 兴 and e g 关 Q 2 ⫽(Q x⫺Q y )/2,Q 3 ⫽(2Q z ⫺Q x ⫺Q y )/ 冑12],
in
Van
Vleck
notation.25 The oxygen distortions are limited by a harmonic
restoring force, e.g., ⫺Ku z ( ជl ⫾ẑ/2).
The lattice elastic energy and electron-phonon interaction
terms of the Hamiltonian are26
HL⫽

K
2

冊 冉 冊 冉 冊册
兺冋 冉
l

1 2
1 2
1 2
u x ជl ⫹ x̂ ⫹u y ជl ⫹ ŷ ⫹u z ជl ⫹ ẑ ,
2
2
2
共11兲

Hep⫽⫺

4g

冑3 兺l

关 c z† 共 ជl 兲 c z 共 ជl 兲 Q z 共 ជl 兲

⫹rotations to x̂, ŷ directions兴 .

共12兲

Hep can be split into two parts, JT and ‘‘breathing:’’
Hep⫽HJT⫹Hbr ,where, in Van Vleck notation,

064401-4

共13兲

SPIN VERSUS LATTICE POLARON:

HJT⫽2g兺l关 c †2 共 ជl 兲 c †3 共 ជl 兲兴

Hbr⫽⫺2 冑2g

冉PREDICTION . . .

Q 3 共 ជl 兲

Q 2 共 ជl 兲

Q 2 共 ជl 兲

⫺Q 3 共 ជl 兲

PHYSICAL REVIEW B 64 064401

冊冉 冊
c 2 共 ជl 兲

c 3 共 ជl 兲

VI. THE SEVEN-SITE CLUSTER: SPIN AND LATTICEPOLARON,共14兲

兺l Q 1共 ជl 兲共 c †2共 ជl 兲 c 2共 ជl 兲 ⫹c †3共 ជl 兲 c 3共 ជl 兲兲 .
共15兲The vector distortions (t 1u ) do not couple to E g electronstates and therefore do not appear.
A simple case shows how HJT splits energy degeneracy.
Suppose a doped electron is localized at a single Mn site,with no hopping or spin flipping considered. The two E gstates at that Mn are the only degrees of freedom for theelectron and are originally degenerate. When the six surrounding oxygen distortions are considered, the degeneracy
is lifted in HJT :

冉

Q3

Q2

Q2

⫺Q 3

c3

␤†兲

冉

⫺Q

0

␣

0

Q

HJT⫽2g 共 c †2 c †3 兲
⫽2g 共 ␣
where

冉 冊
␣

␤

†

† ⫽

†

冉

␾
cos
2

␾
⫺sin
2

␾
sin
2

␾
cos
2

冊冉 冊
冊冉 冊
c2

␤

,

共16兲

共17兲

冊冉 冊
c †2

c †3

and (Q 2 ,Q 3 )⫽Q(sin ␾,cos ␾). The change in energy due to
HJT⫹Hbr⫹HL is ⌬E⫽⫺2 冑2gQ 1 ⫾2gQ⫹12(Q 21 ⫹Q 2 )/2.
The energy splitting ⫾2gQ comes only from the JT term.
The optimal distortions are Q 1 ⫽4 冑2g/K and Q⫽4g/K,
which give the maximum energy lowering of the groundstate ⌬E⫽⫺6g 2 /K.
The angle ␾ does not enter ⌬E. This continuous degeneracy can be lifted either by adding kinetic energy and quantization of lattice degrees of freedom 共dynamic JT effect兲 orelse by introducing higher-order anharmonic terms in
Hep . 27,28 However, later in the discussion of the seven-sitecluster, we shall show how this continuous degeneracy isnaturally removed by the increase of electronic and lattice
degrees of freedom.
It is interesting to consider what would happen if spins
were ferromagnetically ordered, so that magnetism would
not assist localization. Then polaron formation can only occur through lattice distortion and is prohibited when the delocalization energy per electron is larger than the polaronenergy, i.e., when ⌫⫽g 2 /K 兩 t 兩 is less than a critical value
close to 0.5. We believe that ⌫ is close to 0.25, so that in the
hypothetical ferromagnetic case, polarons would not form.
Antiferromagnetic confinement is needed before a lattice polaron effect occurs. LaMnO3 is different in this respect; its
cooperative Jahn-Teller ground state makes polaron formation easier.11

With the full Hamiltonian considered, the seven-site cluster turns out to be the most important one, as shown later inthis section. Only a single spin is flipped. Inhomogeneous
canting will not be favored for this state. By understanding
its ground state algebraically, we learn several interesting
aspects of the influence of Hep in the Hamiltonian. The dimensionless parameter ⌫⬅g 2 /Kt characterizing the strengthof electron-phonon coupling has a value near 0.25 for
LaMnO3 , and we assume that the value for for CaMnO3 issimilar, i.e., 0.20⬍⌫⬍0.30.30,29 We will measure energies inunits of 兩 t 兩 , using dimensionless lattice variables Q ⬘⬅ 冑K/ 兩 t 兩 Q. The prime in Q ⬘ is suppressed from here on.
To study the ground state of a single electron in the sevensite cluster, a 14-dimensional Hilbert space is used, consisting of atomic E g orbitals from each of the seven Mn atoms.
Later when we turn on Hep , we will find that for small ⌫,fewer than 14 basis functions are needed for the ground-statecalculation. The 14-dimensional space can be decomposed
into seven irreducible representations of point group O h ,namely, A 1g 丣 A 2g 丣 T 1u 丣 T 2u 丣 3E g . When lattice distortions are absent (⌫⫽0,Hep⫽HL⫽0), these functions diagonalize the 14-dimensional Ht . The three E g -type basis func⫹
0
0⫺⫺
tions ( ␺ ⫹2 , ␺ 3 ), ( ␺ 2 , ␺ 3 ), and ( ␺ 2 , ␺ 3 ) are degenerate
separately with eigenvalues ⫹ 冑3 兩 t 兩 , 0, and ⫺ 冑3 兩 t 兩 . Thesesix states, along with the A 1g state 共with eigenvalue 0兲, willbe the main states of interest when ⌫⫽0. All other states
stay absent as long as ⌫ is small.
When ⌫⫽0, Hep and HL are turned on. Amplitudes oflattice distortions appear linearly in the matrix representationof Hep and quadratically in that of HL . Since these lattice
distortions are of order 冑⌫ when ⌫Ⰶ1, Hep and HL can betreated as perturbations to Ht . In the following, all lattice
distortion modes will be introduced. Then second-order per⫺turbation of the original ground states ( ␺ ⫺2 , ␺ 3 ) can be expressed in terms of these modes and shows that most of thesemodes do not participate in the ground-state lattice distortion. This will in turn eliminate the need for considering
electron states of symmetries that couple to the absent modes
only. Finally, from the form of the perturbed ground-stateenergy, the pattern of its optimized lattice distortion will bederived and compared to the exact numerical result.
There are 36 oxygens adjacent to one or more Mn atoms
in the seven-atom cluster, so there are 36 distortion parameters in Hep . These can be organized into sets of basis vectors for irreducible representations of the point group O h ,namely, 3a 1g 丣 a 2g 丣 5t 1u 丣 3t 2u 丣 4e g . The a 1g -type modes
are denoted as (q i1 , where i⫽1,2,3), and the e g -type modes
are denoted as (q i2 ,q i3 ), where i⫽1,2,3,4. The matrix elements of Hep can be reexpressed in terms of these modes,
2/2.
and HL is simply 兺 modesq modes
Second-order perturbation theory shows that for small ⌫,many of these modes appear only quadratically with positivecoefficients and hence should be optimized to 0. For larger⌫, these coefficients start to turn negative, and the corresponding distortions start to develop. The critical values are⌫⫽0.443 for the appearance of the t 1u and q 43 distortion,

064401-5

YIING-REI CHEN AND PHILIP B. ALLEN

PHYSICAL REVIEW B 64 064401⌫⫽1.30 for t 2u , and ⌫⫽0.819 for a 2g and q 42 . It will thus
happen that as ⌫ increases, the symmetry of the ground-stateelectron wave function and lattice distortion pattern is gradually lowered by the successive appearance of these distortions.
We therefore ignore the above modes for the actual range⌫⬇0.25⫾0.05. This eliminates the presence of electronstates of A 2g , T 1u , and T 2u symmetry. The remaining modes
are three sets of (q i1 ,q i2 ,q i3 ), with i⫽1,2,3, of the inner,intermediate, and outer oxygen layers, respectively. Degen⫺erate first-order perturbation theory for ( ␺ ⫺2 , ␺ 3 ) shows thatthe ground-state energy of Ht splits into
E (1)⫽⫺ 冑3⫺ 冑⌫ 共 冑2Q 1 ⫿ 冑Q 22 ⫹Q 23 兲 ,
兩t兩

共18兲

1
1
1
Q 1 ⬅ q 11⫹ q 21⫹ q 31 ,
3
6
12
1
1
1
Q 2 ⬅ q 12⫹ q 22⫹ q 31 ,3
6
12

FIG. 4. One of the possible ground-state wave functions in theseven-site cluster. The corresponding lattice distortion pattern has
q 1 and q 3 components.

1
1
1
Q 3 ⬅ q 13⫹ q 23⫹ q 33 .
3
6
12
Thus at first order, there is still a continuous degeneracy, inthe sense that the energy JT splitting depends only on Q
⬅ 冑Q 22 ⫹Q 23 , not on ␾ ⬅tan⫺1 (Q 3 /Q 2 ).
To include second-order perturbations, for simplicity, we
treat analytically only the distortion modes from the inner
oxygen layer (q 11 ,q 12 ,q 13). The number of related elec⫹
tronic states is now reduced to five, namely, (A 1g , ␺ ⫹2 ,␺3 )⫺⫺and ( ␺ 2 , ␺ 3 ). In this five-dimensional subspace, the Schrödinger equation to be solved can be expressed as follows 共notincluding HL , which is always proportional to the identity
matrix兲:

冉

II
H II
t ⫹Hep⫺E

h ep

T
h ep

I
H It ⫹Hep
⫺E

冊冉 冊
⌿ II
⌿I

⫽0,

共19兲

⫺
⫹
⫹
II
where ⌿ I ⬅( ␺ ⫺2 , ␺ 3 ) and ⌿ ⬅(A 1g , ␺ 2 , ␺ 3 ). We obtain
I
T
II
⫺h ep
(H II
an effective Hamiltonian Heff⫽H It ⫹Hept ⫹Hep⫺E) ⫺1 h ep for ⌿ I . Since E should be very close to ⫺ 冑3 兩 t 兩
for small perturbation, we take E as ⫺ 冑3 兩 t 兩 ⫻1(3⫻3) andIIin the denominator. The Heff obtained this way
ignore Hep⫺
shows that the degeneracy of ( ␺ ⫺2 , ␺ 3 ) is now lifted to become 共including HL)

冉

冊

4 冑3
9
E (2)
⫺
⫽⫺ 冑3⫺ 冑2A⫺
共 A 2 ⫹ ␳ 2 兲 ⫾ 冑⑀ ,
兩t兩
3
2⌫
4
32
8 冑6
A ␳ 2⫹ ␳ 2
⑀ ⬅ ␳ 4⫹ A 2␳ 2⫹
3
3
3
4⫹ ␳ 3 共 4 冑2A⫹ 冑3 兲 cos共 3 ␪ 兲 ,3
where we introduce the notation

共20兲

A⬅

B⬅

冑⌫
3

冑⌫
3

q 11 ,

q 12⬅ ␳ sin ␪ ,␳ ⬅ 冑B 2 ⫹C 2 ,

C⬅

冑⌫
3

q 13⬅ ␳ cos ␪ .

We see that cos 3␪ should be ⫹1 to minimize the groundstate energy. Hence the degeneracy of ground states is notcontinuous anymore and has become threefold. This featureagrees with the numerical result. The ground-state electronic
wave function of ␪ ⫽0 is shown in Fig. 4. A rotation thatbrings ẑ to x̂ (ŷ) will generate the wave functions of ␪
⫽2 ␲ /3 ( ␪ ⫽4 ␲ /3).
VII. NUMERICAL RESULTS

Now we consider the full Hamiltonian H⫽Ht ⫹HJ⫹Hep⫹HL for all clusters examined in Sec. IV. FM spinalignment facilitates hopping and causes energy loweringfrom delocalization, therefore encouraging the polaron to
grow. However, as ⌫ increases, the JT splitting will become
the dominating influence on the ground-state energy. Greater
localization enhances the JT energy lowering, which increases localization of the electron. On the other hand, HJ
and HL serve as penalties for spin misalignment and lattice
distortions. Our numerical studies find the optimal resolutionof the competition between these effects. The seven-site cluster becomes favored from ⌫⬎0.18, as shown in Fig. 5. andFig. 6. For all clusters considered, when ⌫ becomes large
enough, the size of the ground-state wave function shrinks to
become that of the seven-site polaron, as shown in Fig. 7.
Enlarging the size of the FM cluster to enhance delocalization energy is then disfavored by the strong lattice polaron

064401-6

SPIN VERSUS LATTICE POLARON:

PREDICTION . . .

PHYSICAL REVIEW B 64 064401

FIG. 5. Numerical results for ground-state energy (E gs ) of symmetric clusters, obtained by diagonalizing Ht ⫹HJ ⫹Hep⫹HL in the
2M dimensional subspace.

effect. It is also clear that for extremely large ⌫, the one-sitelattice polaron will be the only form of electron state thatexists.
Our numerical calculation predicts the lattice distortion
pattern of the seven-site ground state. For ⌫⫽0.25, t⫽0.75 eV, and K⫽27.2 eV/Å 2 共obtained from Raman
scattering in LaMnO3 ), the ground state shown in Fig. 4 has
oxygen displacement parameters q 11⯝1.1, q 13⯝0.78, andq 12⫽0, which gives an outward displacement of 0.15 Å inthe ẑ direction and smaller outward displacements of
0.036 Å in both x̂ and ŷ directions for the six oxygens
surrounding the central Mn. Displacements of oxygens in the

FIG. 6. Numerical results for ground-state energy (E gs ) of
asymmetric clusters, obtained by diagonalizing Ht ⫹HJ ⫹Hep⫹HL
in the 2M dimensional subspace. The symmetric seven-site cluster
is also shown here as it becomes the optimal solution when ⌫exceeds 0.2.

FIG. 7. Numerical results for the ground-state radius of symmetric clusters. The radius is measured in units of lattice constant a 共theMn-Mn separation distance兲 and is defined to be 关 兺 i (rជ i⫺ 具 rជ 典 ) 2 ␺ i2 兴 1/2, where the index i runs over all Mn sites inside thecluster, and 具 rជ 典 ⬅ 兺 i rជ i ␺ i2 .

intermediate and outer layers are at least 10 times smaller. As
predicted perturbatively, t 1u , t 2u , a 2g , q 42 , and q 43 distortions are absent.
VIII. TWO-ELECTRON PROBLEM

When two electrons are too close to be considered isolated, it is interesting to see if they form a bipolaron. Without
introducing additional parameters in our Hamiltonian, welook at the two-electron problem for the extreme cases where
the on-site Coulomb repulsion, or the Hubbard U, is 0 or ⬁.
Here again we consider the FM clusters mentioned earlier
and look at ⌫⫽0.25 only.
For U⫽0, the ground-state energy of a bipolaron is given
when the single-electron ground state and first excited state
are both occupied. With the lattice distortion re-optimized
numerically, we find that the seven-site cluster has the lowest
bipolaron ground-state energy, which is 0.74兩 t 兩 lower than
two separate polarons. However, when U→⬁ 共with nonearest-neighbor repulsion considered兲 in the same sevensite cluster 共with lattice distortion again reoptimized兲 the bipolaron ground state is lifted up to become 0.86兩 t 兩 higher
than two separated polarons. We then varied U to find thecritical value above which a bipolaron does not occur. Considering only the seven-site cluster, we obtain U c ⬇0.98兩 t 兩 , avery small value, about 0.74 eV.
For a more careful investigation, different shapes of clusters should be considered, as well as different spin orientations inside the clusters. Since bipolarons have a larger size
than polarons, the variety of possible shapes and spin textures is therefore also larger. A continuum calculation similar
to that in Sec. IV shows that, without electron-electron interaction, the optimal size of a FM bipolaron contains 50 Mn
atoms. For a realistic theory, on-site and nearest-neighbor

064401-7

YIING-REI CHEN AND PHILIP B. ALLEN

PHYSICAL REVIEW B 64 064401

Coulomb repulsions should both be included. Our results
suggest that bipolarons are likely to be unstable relative topolarons, and phase separation is not likely to happen.
IX. DISCUSSION

When doping x is nonzero, one should ask whether localized polaron solutions will distribute homogeneously or will
attract, causing phase separation.31 We have not addressed
this issue, which requires a more complicated calculation
with additional Coulomb parameters in the Hamiltonian. Experiment is consistent with a concentration interval up to x
⫽0.08 where polarons are homogeneously distributed.5 Ourmodel shows that by this concentration, randomly distributed
seven-site polarons will overlap significantly. However, ourmodel also shows that at concentrations when x⬎0.045, polarons should be unstable relative to an undistorted groundstate with FM spin order and metallic conduction by the
doped electrons. This effect is not seen in experiments. Apparently alternate ground states, possibly involving organization of polaronic distortions, occur and enable the system to

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remain nonmetallic. Without additional physics 共such as defects兲, our model cannot account for the observation5 that La
concentrations with x less than 0.02 yield less of an excess
moment than expected from seven-site polarons.
Our model describes the competing spin- and lattice polaron effects. We believe that it contains the main featuresneeded to describe the system. A test would be measurementof the oxygen displacements, which our model predicts. The
model omits nonadiabatic phonon effects, spin quantization,
temperature, and polaron-polaron interactions.32 For higher
doping levels or T⬎0, these may have a larger influence andpresent challenges that could be worth pursuing if experimental guidance improves.

ACKNOWLEDGMENTS

We thank J. J. Neumeier for suggesting this investigation,
J. J. Neumeier and J. L. Cohn for comments on the manuscript, and V. Perebeinos and A. Abanov for discussions. This
work was supported by NSF Grant No. DMR-0089492.

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There is a factor of 2 change in the normalization of the displacement variables Q⬀u a ⫺u b in the present paper, relative to theconvention used in Ref. 11. For this reason, even though thecoupling constant g is exactly the same, the Hamiltonian Eqs.
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