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Crystallization of electrons confined to a lattice.

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Ann. Physik 6 (1997) 178-186
Annalen
der Physik
0 Johann Ambrosius Barth 1997
Crystallization of electrons confined to a lattice
P. Fulde
Max-Planck-Institut fur Physik komplexer Systeme, Bayreuther Str. 40, D-01187 Dresden, Germany
Received 30 October 1996, accepted 20 December 1996
Dedicated to Bruno Liithi on the occasion of his 65th birthday
Abstract. A necessary condition for electron crystallization is the dominance of the electronic Coulomb repulsion as compared with the kinetic energy. We show that 4f systems are good candidates
for electron or hole crystallization to occur since the small radius of the 4f’ shell leads to small hybridizations and hence kinetic energy of the ,f’ electrons. Crystallization may take place in lattices
with several equivalent 4f sites per unit cell when the electron or hole number is less than the number of sites. We give evidence that charge ordering in Yb4As3 is an example of the mechanism considered here. We also compare it with those considered by Wigner, Verwey, Mott and Hubbard.
Keywords: Electron crystallization; Charge ordering; Yb4As3
1 Introduction
Electron crystallization was first considered by Wigner [l, 21. In a pioneering paper
he considered the low-density limit of an electron gas with the positive charge uniformly spread through the system (jellium model). In particular, he investigated the
ground state and its energy. At high densities the ground state has a homogeneous
charge distribution since due to Pauli’s principle the kinetic energy of the electrons is
more important than their mutual Coulomb repulsion. But when the density is sufficiently low it is the interaction energy which is dominating the kinetic energy. In that
case the ground state must be one in which the electrons stay as far apart from each
other as possible. As Wigner pointed out that is the case when they form a lattice.
The kinetic energy reduces to the zero-point motion of the electrons around their
equilibrium positions. Clearly, that energy must be less than the potential energy due
to the electron-electron repulsions in order that localization can occur. The exchange
interaction is of little importance for a Wigner crystal, the reason being that it depends exponentially on the interelectron distance which is sizeable. A good review of
the different energy contributions to a Wigner lattice is found in Ref. [3]. In accordance with the above, the low-temperature specific heat C( T ) is found to be proportional to T 3 . Possible realizations of Wigner lattices are semiconducting inversion
layers [4] or rare-earth pnictides with low carrier concentrations [5] (see also Ref.
P. Fulde, Crystallization of electrons confined to a lattice
179
[3]). Wigner’s scenario for electron crystallization did not remain the only one. Verwey developed a model to explain the formation of the charge-ordered low temperature phase of Fe304 (magnetite) [6, 71. Mott [8] and Hubbard [9, 101 suggested two
other physical processes which lead to electron localization (metal-insulator transition). Mott predicted for a chain of atoms with one electron per atom a transition
from delocalized to localized electrons when the interatomic distance becomes sufficiently large. Hubbard considered a lattice with an on-site Coulomb repulsion U and
found electron localization at half filling when U exceeds a critical value U,.
All four cases have in common that electron crystallize because their Coulomb repulsion is more important than their kinetic energy. However, they differ in the physical details and therefore lead to quite different experimental results. While the
Wigner lattice forms out of a structureless, i.e., homogeneous electron system, the
other three cases deal with electrons confined to a lattice. The Verwey transition has
been originally interpreted as an order-disorder transition in which the kinetic energy
of the electrons plays no role. It is driven by the interplay between the intersite Coulomb repulsions and the entropy term in the free energy. As mentioned before, in the
Mott-Hubbard case it is the on-site Coulomb repulsion of electrons which may lead
to their localization, provided it exceeds their kinetic energy.
The aim of the present communication is to point out that electron (or rather hole)
crystallization in Yb4As3 carries features of all four cases discussed above. It also
differs sufficiently from each of them in its physical details, that it seems justified to
consider it as a case of its own. The main idea is the following:
Consider a lattice structure with vo sites per unit cell. Assume that we deal with a
system having m electrons (or holes) per unit cell. At high temperatures this system
will be metallic, since the electrons will move from site to site. Furthermore, assume
that the hopping matrix elements between different sites are small as it will be the
case when we deal, e.g., with 4f or 5f orbitals. Then we expect that at low temperatures due to the mutual Coulomb repulsion between the particles an electron (hole)
crystallization will take place even though the electron (hole) concentration m/Q is
high (Q is the volume of the unit cell). Therefore, in distinction to a Wigner lattice
but similar as in the Venvey and Mott-Hubbard cases the distances between the localized electrons may be small and the (indirect) exchange will generally be important.
Electron crystallization takes place here on a sublattice of the solid. Also, the lowtemperature specific heat need not to be proportional to T3, but can vary, e.g., like
C(T) T , depending on the crystal structure. In the next section we specify the
model Hamiltonian for the system and formulate the theory and, in Section 3, we discuss the application to Yb4As3. Section 4 contains the conclusions.
N
2 Formulation of the theory
We describe the electrons in a lattice by the model Hamiltonian
The operators C f ( C j 0 ) create (annihilate) electrons with spin CJ in atomic orbitals i.
The tij and Vqke are the matrix elements for the kinetic energy plus external potential
180
Ann. Physik 6 (1997)
and for the interaction energy, respectively. Within the spirit of the semiempirical calculational schemes we keep only the Coulomb integrals V;jkk = Jij # 0. This corresponds to a Complete Neglect of Differential Overlap (CNDO) approximation. In an
improved treatment one includes also the on-site exchange matrix element
Vjkjk = Vjklci = Kik where z and k refer to orbitals on the same site. To good approximation one may use for the Jij an interpolation due to Ohno [ll] and Klopman [12]
I
Here Rij denotes the distance between the atoms on which the orbitals i and j are
centered. Both, Rij and Jij are expressed in atomic units. For a system with one orbital per site we set Jii = U / E H where EH = 27.21 eV is the Hartree energy. For the
kinetic-energy part of tjj one may choose the form
where S, is the overlap matrix.
Consider a lattice with vo equivalent sites per unit cell. Furthermore, let us assume
one orbital per site and m electrons per unit cell. Electron crystallization will take
place when the expectation value of the Hamiltonian
is lower for a state with localized, i.e., crystallized electrons than for one with delocalized electrons. This is the case when the loss of kinetic energy due to electron
crystallization is overcompensated by the reduction in the electronic Coulomb repulsion.
In order to estimate the changes in ( H ) when electrons localize we introduce the
bond-order matrix
Pij = C ( C i ’ , C j , ) .
(5)
a
For independent electrons Eq. (4) reduces to the form
where we have kept the on-site interactions in the original form. In order to include
screening in the metallic phase one must multiply Jij (see Eq. ( 2 ) )by a screening factor exp[-Rij/Ro] where Ro is the Debye screening length. We drop the first term on
the right hand side, because tij = to and the trace of the bond order matrix gives the
P. Fulde, Crystallization of electrons confined to a lattice
181
total electron number, i.e., it contributes a constant only. Forf-electron systems Jii is
large and therefore double occupancies of orbitals are forbidden. Therefore we drop
the second term too but keep in mind that an orbital is either empty or singly occupied, but never doubly occupied. The effective hopping term - $ JijPjiPu results from
(.inj) - (ni)(nj)and ensures that, e.g., in the case of a H2 molecule the ionic configurations are properly taken into account. When the correlations are strong, like inf
electron systems, we expect that term to be less important and therefore shall neglect
it. This amounts to neglecting intersite density correlations, and requires further investigations. They may result in short-range crystalline correlations similarly as one
may have short-range antiferromagnetic correlations in a spin liquid or spatial correlations in a Wigner liquid [7]. For the present purposes we shall use the following
simplified form of Eq. (6)
subject to the subsidiary condition that double occupancies of orbitals are forbidden.
The resulting modifications of Pq as compared with the uncorrelated case can be estimated by adopting Gutzwiller's approximation [13]. The latter was developed for the
case of equivalent lattice sites, i.e., for a uniform charge distribution. Let us denote
by no = m/vo 5 1/2 the average site occupancy. According to Gutzwiller, the effect
on Py of excluding double occupancies is given by
where PI;")denotes the matrix element in the absence of correlations.
When electron crystallization takes place the charge distribution is no longer uniform. In that case it is more difficult to estimate the effects of correlations on Pi,. We
consider the following estimate reasonable. First, we have to modify the Gutzwiller
factor in Eq. (8) for the crystalline phase where different sites i have different average occupational numbers ni. We do this by setting
For ni = nj = no this expression reduces to the previous one.
In addition to this modification a second one is required. It must account for the
reduction in kinetic energy, or more precisely of (c&cjo), when the average electron
numbers on sites i a n d j are different. A spin-density wave state with, e.g., a density
of spin-up electrons differing on two sublattices may serve as a guide here. We suggest the following correction factor
Therefore we find that
182
Ann. Physik 6 (1997)
+ +
and it is n1 ... nvo= m.
From the energy expectation value (1) one can determine the energy of the ground
state and the condition for electron crystallization.
As an example we consider the case of a cubic structure with vo = 4 and m = 1.
We divide the lattice into sublattices A and B where sublattice B contains three times
as many sites as A. Sites of sublattice A have three nearest neighbor sites which all
belong to B. Those of sublattice B have one nearest neighbor belonging to A and
two belonging to B. In the uniform case the site occupancy is no = 1/4 while in the
nonuniform case the one of a site belonging to A is nA = 1 - 6 and that of B sites is
nB = 6/3.
We consider first the case of a uniform charge distribution. For simplicity, only
nearest-neighbor hopping matrix elements ty = t are taken into account. We find for
the ground-state energy per unit cell
(0)
z is the number of nearest neighbors, i.e., z = 3 in the present case and PO = Pij
where i , j are nearest-neighbor sites. The notation J A ~
means that, without loss of
generality, the index i of Jv refers to a site of sublattice A.
In the crystallized phase the energy for a fixed value of 0 5 6 5 3/4 is
The first two terms represent the kinetic energy while the last three terms are due to
the Coulomb repulsion. The Gutzwiller factor y y plays an unimportant role here because only nearest-neighbor hopping terms tv are considered. It is important though,
when hopping matrix elements between different A sites are included and it is crucial
when the low-energy excitations of the system are considered.
In the limit of small values of 6 we find
Here we have set
P. Fulde, Crystallization of electrons confined to a lattice
183
jeA
The minimum of E(6) is obtained for 6 = a:/(4a$) and is given by
We note that if the kinetic-energy contribution is much smaller than the Coulomb reJ,q. The
pulsion, then the condition E < Eo is equivalent to (1/3) C . J . > Z j f A
is fulfilled.
nearest neighbors of an A site are the B sites and therefore tiT:c$dition
When the kinetic energy is of comparable size as the Coulomb repulsion one must
minimize Eq. (14) and check the condition E < EO from case to case.
3 Application to Yb4As3
We suggest that the present theory applies to YbdAs3 [14-191 where charge ordering
has been observed [20]. Yb4As3 has an anti-thorium phosphite structure. The Yb
ions are situated on four families of interpenetrating chains. These chains are
oriented along the diagonals of a cube. The Yb-Yb distance within a chain is 3.80 1$
and larger than the one for Yb ions belonging to different chains which is 3.40 A for
nearest neighbors. By counting valence electrons one finds that one out of four Yb
ions has a valency of 3+ and therefore a hole in the 4f shell while the remaining
ions have a valency of 2+ and a filled 4f shell. At temperatures above 300 K the
system is metallic with a carrier number of 1/4 per Yb ion. At T, = 300 K a structural phase transition takes place which is accompained by a volume conserving, trigonal distortion [21]. Accordingly, one family of chains is shortened while the other
three are elongated. With increasing distortion an ordering of the Yb3+ ions takes
place. Since their ionic radius is smaller than that of the Yb2+ ions they go into the
short chains. Finally, at low enough tem eratures T , predominantly chains in one direction, e.g., [111] are occupied by VbP+ ions. If the charge ordering were perfect
the system would be insulating since we would be dealing with half-filled Hubbard
chains (with each site occupied by a Yb3+ ion we have one hole in the 4f shell per
site). However, the system remains a semimetal with one charge carrier per 103Yb
sites indicating incomplete charge ordering. We interpret this as self-doping.
A theory describing the physics of that phase transition has been suggested which
is based on a band Jahn-Teller effect [22]. Below T, the 4-fold degenerate quasi onedimensional 4f bands split into one plus three bands. A similar theory was developed
by Labb6 and Friedel [23] for the martensitic phase transition in some of the A 15
compounds (e.g., V3Si,Nb3Sn). But in distinction to the latter the strong electron
correlations of the 4f holes play an important role here [22]. The theory can explain
the observed low-temperature specific heat of Yb4As3 which varies like C = yT
with a large Sommerfeld coefficient y. The theory presented here provides additional
justification for such an approach.
184
Ann. Physik 6 (1997)
According to the present theory we may also view the ordering of the Yb3+ ions
as a crystallization of holes. Due to the small radius of the atomic 4f shell the effective f bandwidth is of order 0.2 eV [24]. The difference in Coulomb repulsion of the
4f holes for the uniform distribution and the case of complete charge ordering has
been calculated by von Schnering by performing the Ewald summations [25].Not accounting for screening the result is AE N 1.8 eV per formula unit. Compared with
the transition temperature T, this energy difference is very large and suggests that
even in the high temperature phase short-range intersite correlations are strong. In
any case, these estimates show that hole crystallization may indeed take place as described above. Within this picture the lattice distortion is caused by the hole crystallization because the smaller Yb3+ ions make the chains shrink they are occupying. A
volume conserving, i.e., trigonal distortion costs the least elastic energy. The interpretation given here explains in a natural way the one-dimensional character of the low
T structure. It is the one which minimizes the repulsion of holes when the lattice degrees of freedom are included and therefore forms independently of the relative size
of the interchain and intrachain hopping matrix elements. This provides justification
for the concept of degenerate, effectively one-dimensional bands used in the band
Jahn-Teller description, even when the interchain hopping matrix elements are appreciable. Also, from Eq. (17) it is evident that the crystallization is not complete so that
the system appears to be self-doped (6 # 0). This would explain why the system is a
semimetal. For a description of the low-energy excitations of Yb4As3 one must also
include hopping matrix elements between sites in the same chain and also keep finite
the Coulomb repulsions U of two holes on the same Yb site. This results in an effective spin-spin coupling between Yb3+ ions in the short chains and leads to the t - J
model. The observed low-temperature specific heat with the large y coefficient was
explained this way [22]. A more quantitative treatment of Yb4As3 in the frame of
the present theory is beyond the scope of this paper and is deferred to a future investigation.
4 Conclusions
We have shown that electron or hole crystallization can take place in lattices where
the number of electrons (or holes) is less than the number of equivalent sites. A prerequisite is that the kinetic energy of the electrons is sufficiently small as compared
to their mutual Coulomb repulsion when placed on different sites. Natural candidates
are rare-earth systems because of the small kinetic energy of the 4f electrons. We
have given arguments that Yb4As3 meets those requirements and that the charge ordering in that system can be viewed as a crystallization of holes.
Finally, we want to discuss similarities and differences of hole crystallization in
Yb4As3 as compared with the crystallization processes suggested by Wigner [1, 21,
Verwey [6], Mott [8] and Hubbard [lo, 111.
The approach of Wigner [ l ] is based on a homogeneous electron system and requires very low electron densities in order to yield crystallization [l]. As a consequence, the electrons are rather far apart in the crystallized phase and the low-energy
excitations result from vibrations of the electrons around their equilibrium positions.
The low-temperature specific heat is therefore C = aT3. In the present approach like
in the ones of Verwey, Mott and Hubbard the electrons are confined to a lattice and
their density may be rather large when they crystallize. For example, in Yb4As3 the
P. Fulde, Crystallization of electrons confined to a lattice
185
kinetic energy of the 4f holes is very small due to the small overlap of the atomic
4f function with the ones of neighboring ions. Due to this special feature the exchange coupling is appreciable in the crystallized phase and influences strongly the
low-energy excitations. In particular, due to the special crystal structure of YbdAs3 it
leads to a C = yT behaviour at low T.
Charge ordering in magnetite appears to have some common features with the one
in Yb4As3. A more specific comparison is presently not possib1e;mainly because the
low temperature phase of Fe304 is still a subject of controversy. The transition has
been originally interpreted as an order-disorder one [6), but the ordered structure suggested by Verwey has turned out to be in contradiction with neutron diffraction experiments. More recently, attempts have been made to include the kinetic energy of
the electrons (see, e.g., Ref. [7]). In order to split the degeneracies of ordered configurations one must include electron repulsions beyond nearest neighbours. To our
knowledge there is no theory available which does that and includes the kinetic energy of the electrons.
When compared with the Mott [8] and Hubbard [9, 101 scenarios for electron crystallization the differences of the present approach are also obvious. In their case it is
the on-site Coulomb repulsion of electrons which leads to localization for systems
with one electron per site. Mott considered a decrease of the kinetic energy by increasing the interatomic distance between hydrogen atoms. Hubbard investigated the
excitation spectrum of the Hamiltonian named after him for a half-filled band and
showed that a gap opens up when the on-site repulsion exceeds the kinetic energy.
Again, in the present case it is the Coulomb interaction between electrons on different sites which leads to crystallization when the electron number is less than the site
number. Thereby it is important not to reduce the model to one with nearest-neighbor
repulsions only. That would lead to a number of generate configurations for the
ground state and miss important physics. Note also that, in distinction to the MottHubbard scenario, Yb4As3 remains a semimetal in the low temperature phase. We
have shown that the zero-point fluctuations of the 4f holes prevent complete charge
ordering and we speculate that this is the origin of self doping. In that context work
on low-dimensional systems using an extension of the Hubbard Hamiltonian should
be mentioned (see, e.g., Refs. [6, 26, 271). The excitation spectrum is generally gapless.
It remains a task for the future to find other systems to which the theory might be
applicable. There is no reason why the crystallized phase could not have long-range
magnetic order. Also, it is worthwhile to reconsider the A15 compounds from the
present point of view.
J
I would like to thank B. Schmidt and Dr. P. Thalmeier for long-standing fruitful cooperations on
strongly correlated electron systems, in particular on Yb4As3. I also want to thank Prof. Dr. H.G.
von Schnering for discussions concerning chemical aspects of electron crystallization.
References
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Ann. Physik 6 (1997)
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