HAL Id: jpa-00247385
https://hal.archives-ouvertes.fr/jpa-00247385
Submitted on 1 Jan 1997
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Calculated Local Electronic Densities of States Induced by a point Defect in a CuO2 Plane
J.-P. Jardin, T. Hocquet, P. Germain, J. Labbé
To cite this version:
J.-P. Jardin, T. Hocquet, P. Germain, J. Labbé. Calculated Local Electronic Densities of States Induced by a point Defect in a CuO2 Plane. Journal de Physique I, EDP Sciences, 1997, 7 (9), pp.1113-1122. �10.1051/jp1:1997112�. �jpa-00247385�
J. Phys. I FYance 7 (1997) 1113-1122 SEPTEMBERI997, PAGE1113
Calculated Local Electronic Densities of States Induced by a Point Defect in a CUO~ Plane
J.-P. Jardin (~), T. Hocquet (~,*), P. Germain (~) and J. Labbd (~)
(~) Laboratoire PMTM (CNRS), Universit4 Paris Nord, 93430 Villetaneuse, France
(~) Laboratoire de Physique de la Matibre Condens4e de I'#cole Normale Supdrieure,
24 rue Lhomond, 75231 Paris Cedex 05, France
(Received 20 November 1996, revised 13 May1997, accepted 2 June 1997)
PACS.61.72.Bb Theories and models of crystal defects
PACS.71.20.-b Electron density of states and band structure of crystalline solids PACS.74.72.Dn La-based cuprates
Abstract. The effect of a point defect on the electronic density of states in
a Cu02 plane
of the La2-~Sr~Cu04 compound is calculated by a Green function method applied to a tight- binding itinerant electronic model. Different possible physical cases are discussed with respect to the shift of the atomic energy level of the impurity and to the modification of its effective transfer integral with its nearest neighbors. We show that a non vanishing magnetic moment
can exist at the impurity site, even when this one is non magnetic by itself.
1. Introduction
Experimental studies show that copper substitution by impurities with different electronic
structures strongly decreases the critical temperature Tc of high Tc superconducting
Lai.85Sro_15Cu04 Ill. It seems to be experimentally established that this strong reduction is associated to the formation of magnetic moments localized on the impurities or on their
nearest neighboring copper sites [2, 3]. These local magnetic moments would reduce Tc by a
pair breaking mechanism [4]. This same phenomenon of reduction of Tc by a pair breaking
mechanism occurs as well with impurities which are not magnetic by themselves, as Zn, Ga or Al (but which induce local magnetic moments), as with magnetic impurities as Fe, Co or Ni.
On the contrary, dilute oxygen vacancies do not seem able to appreciably reduce the value of Tci although these compounds are never perfectly stoichiometric as concerning the oxygen content, the value of Tc remains large. As a matter of fact, magnetic moments localized on
oxygen vacancies do not seem to exist.
In this paper, we calculate, in a tight-binding model for itinerant electrons in a Cu02 plane,
the effect of a point defect on the local electronic density of states at the defect site and
at its nearest neighboring copper sites. Our purpose is to show that, in this model, even
the impurities which are not magnetic by themselves, as Zn, Ga or Al, are able to strongly
increase these local electronic density of states close to the Fermi level, and thus to induce local (*) Author for correspondence (e-mail: hocquetfiphysique.ens.fr)
© Les #ditions de Physique 1997
a
Fig. 1. The copper lattice of the Cu02 plane, with lattice parameter a. A subsituted impurity is
at the site 0, and its four neighboring copper atoms at the sites I
= 1 to 4. t and t' are the effective transfer integrals between neighboring sites.
magnetic moments in a Cu02 plane. Indeed, the important fact is not to know if the impurity is magnetic by itself, but if it remains, or becomes magnetic when substituted to copper in a
Cu02 plane [5, 6]. In this paper, we discuss the conditions for that.
The local Green functions are calculated in Section 2. The existence of bound states is discussed in Section 3. The modification of the local densities of states induced by a point defect is calculated, and discussed according to the values of the parameters in Section 4. We show the possibility of the occurrence of a magnetic moment at the impurity site in Section 5.
2. Solution of the Hamiltonian for a Point Defect
To calculate the effect of copper substitution by impurities or of oxygen vacancies on the electronic structure of a Cu02 plane in the metallic paramagnetic state (Fig. I), we start from the simple model in which the tight-binding energy of an electron with wave vector k(k~,ky)
in the perfect Cu02 plane with lattice parameter a is
E(k)
= Eat 2t(cos k~a + cos kya), (1)
in which Eat is the effective Hartree-Fock energy of an electron occupying the d~2-y2 orbital of
a copper site, and -t (t > 0) is the indirect hopping integral between two nearest neighboring
copper atoms through the p~ or py orbital of the intercalated oxygen atom [7, 8].
In our model, copper substitution is characterized by two distinct parameters. The first one is the modified atomic level E([
= Eat + sa on the impurity; this parameter will depend on spin a if a local magnetic moment appears at the impurity site, as we will see in Section 5. The second parameter is the modified value t' = t + for the hopping term between the impurity
and its four nearest neighboring copper atoms. The case of an oxygen vacancy will be described by the strong reduction of the hopping integral between the two related copper atoms.
Taking Eat as the origin of the energies, we use the following Hamiltonian for a single defect in a perfect Cu02 plane:
11 ~ llo + V, with
~lo = -t ~ ~ a[~a~, j2)
a ~#j
N°9 LOCAL D-O-S- IN A Cu02 PLANE 1115
where I and j denotes nearest neighboring copper sites of the lattice. In the case of copper substitution,
4
V
= Vi
=
~j E«a(«ao« d~(a(«aw + a)~ao«) (3)
a ~=l
where the substituted impurity is at the site 0, its four neighboring copper atoms being at the sites I
= 1 to 4. In the case of an oxygen vacancy between two nearest neighboring copper
atoms at sites 0 and I,
V = V2 " -&(a(~ai« + a)~ao«). (4)
Local densities of states n[(E) will be calculated from the local Green functions G$(E+),
where G[(E+)
= (I[G~(E+)[j) and G(z)
= I/(z it).
CALCULATION OF THE LOCAL GREEN FUNCTIONS. We start &om the Dyson equation
Glz)
= glz) + glz)VGlz), 15)
where g(z)
= II (z 7lo). In the case of a copper substitution, one gets from equations (3, 5)
4
G& = ga + gios«G& ~lgioGs + G&gi~), 16a)
1=1
4
Gig = gut + guusaGli £lguuGs + Gligu~), (fib)
i=I
where I is anyone of the lattice sites, including the impurity site I
= 0. In these equations, the summations on the four nearest neighbors sites I
= I to 4 are easily performed by using the identities
zG(z)
= I + iiG(z), 29(Z) ~ l + ~lo9(Z), leading to
4 1
~ (7a)
£ Gj
= ~ [l + (sa z)Gooj,
~=i
~
~j4 ~a ~a ~ ~a j~b)
it ~, 0t'
1=1
Lgoi
= (ii zgoo), 17C)
~i
Lgii4
= -)got 17d)
1=1
Then, straightforward calculations give the local Green functions at the impurity site:
~~0(Z)
=
g00(Z)
~~ ~~~~ ~~~ + ~«1900jz) j~~
with r
= t'It, and also at all the other sites:
Gl(z)
" 9a(z) +
r2
j$-)~~~~goo(z)~"~~~~°~~~~
~~ ~ ~~ ~~~
As E(k)
= E(-k), we have got(z)
= gio(z). The periodicity of ~o leads to ga(z)
= goo(z). In the special case where I
= I to 4 ii.e. one of the four nearest neighbors of the impurity), it follows from (7c) that
~°~ ~"
t
~~ ~~°°~' and thus we have
Gsiz)
= 9001z) + &11 z9001z))~r~ iji-/)lliigooiz) ii°)
In the case of an oxygen vacancy, one gets from equations (4, 5)
Goo " goo dgooGio dgoiGoo, (lla)
Gio " gio &gioGio &giiGoo, II16)
and taking into account equation (7c), we find:
g~o(Z)
' ~~~~
001~~ - ~~~~~~ ii + iii - z9°°i~~~l ~ ~~~~°~~~
In the absence of the intercalated oxygen atom, the transfer integral
between the two elated
opper sites reduces to the rect
one, which is very small, as
it is equivalent to
ake & = Then,ssion
12)
ecomes
~°°~~~ " ~~~~~~ " 13 + lz -
4t)goolz~()) )
for the Green function on the two
disconnected copper atoms.
The perfect crystal reen
jK I() for (E(
> 4t, (14a)
~
~
~~~~
~~)~)~K (() ~K
(fij
for (E( < 4t, (14b)
~ ~
where sgn(E) is the sign of E, and K(u) is the complete elliptic integral of the first kind:
K(lL)
#
~~~ ~~
~ (15)
3. Condition for Existence of Bound States
Bound states exist only when the Green function have real poles z
= Ep outside of the band, I.e. with [Ep[ > 4t. Thus, in the case of copper substitution, it follows from equations (8) or (9) that the energies Ep of the bound states must be solutions of the equation
~
j~ l~ ~
~~~P jilt)
~ ep (r2 1)ep + i~'
N°9 LOCAL D-O-S- IN A Cu02 PLANE 1117
2
++
~ ~
'++
, '
,
~i~~/
~ ~
-2
Fig. 2. Comprehensive view of the various solutions of the local densities of states, with bound states, versus the reduced modified atomic level i~
= s«/4t and the transfer integral ratio r
= t'/t.
with ep = Ep /4t, r
= t'It and
i~ = s«/4t. We find that, according to the values of parameters
r and i~, there are 0, or 2 bound states:
. for r < I and
[i~[ < I r~, there exists no bound state (region I in Fig. 2);
. for [il[ > ii r~[, there exists only one bound state, the energy Ep of which has the same
sign as il (region II for il > 0 and III for il < 0);
. for r > I and [il[ < r~ I, there exists two bound states, one with Ep > 4t and the other
one with Ep < -4t (region IV).
The weights uJi of a bound state Ep on the different lattice sites t are equal to the residues of the poles z
= Ep of the Green function Ga(z). For instance, from equation (8), one gets
~~
~~~~~ ll(~ ~ ~~~j~~~)~ + l T~
~~~~
~
where E(u) is the complete elliptic integral of the second kind:
r/2
E(u)
=
fi d9. (18)
And, from equation (10), one gets, for I
= I to 4,
~
l~ ep)~~ j~g~
1 ~~ o.
It is easy to verify from equation (13) that, in the case of an oxygen vacancy, a bound state never exists.
4. Modified Local Density of States
The local density of states N/(E) on a site I, for E lying inside the continuous part of the
spectruri ([E[
< 4t), can be obtained from the well known formula
N/(E) = Im G[g(E+). (20)
N~(e)
1,> 0.724
e
,< =0
q = o.5
q = i
a) ~
Nj(e)
e
"'=
q = o.5
~~'~ ~~~~
q = I
b) ~
Fig. 3. Local densities of states on the impurity (a) and on the nearest neighbors 16), calculated for r
= t'It
= 1 and il = s~ /4t
= 0, -0.5 and -1.
This calculation involves the knowledge of the perfect crystal local Green function
goolE+)
= RIE) i~nlE), 121)
where the real part R(E) and the unmodified density of states n(E) per atom are given by equation (14b). In the case of a copper substitution by impurities, equations (8, 20) give the
modified local density of states at the impurity site:
~«j~~ r~nlE)
° lr~ ilr~ I)E + E«iRlE)l~ + ~~ilr~ I)E + E«l~n~lE) j~~~
The local modified density of states NT(E) on the nearest neighboring copper sites I
= I to 4
is also easily calculated from the equation (10), and it can be used for numerical calculations.
N°9 LOCAL D-O-S- IN A Cu02 PLANE 1119
No(e)
e
r= 1.3
1.6
a) r
Nj(e)
e
'"* r=1.3
'~ l.6
b) r
Fig. 4. Local densities of states on the impurity (a) and on the nearest neighbors (b), calculated for e~
= 0 and r = t'It = 1, 1.3 and 1.6.
It appears that, in our model with a single point defect, the local density of states N/(E)
can be modified on the defect site or its near neighbors only, but remains almost unmodified
on the sites far from the defect.
For r
= I and il ~ 0, the occurrence of a bound state, which lies very close to one of the
band edges as far as [il[ is not too large ([Ep[ 4t $ 4t/10 when [il[ $ 0.75), is associated to a
strong reduction o>f the local density of states in the middle of the band on the impurity site;
this reduction is large enough to make disappear the Van Hove singularity (Fig. 3a) [10]. At
the nearest neighbors, the singularity remains, but is reduced (Fig. 3b).
For il = 0 and r ~ l, the local densities of states N((E) and Nf(E) remain symmetrical
with respect to the middle E
= 0 of the band. For r > I, the occurrence of two symmetrical
bound states, which> in this case also remain very close to the band edges for r not too large ((Ep[ 4t £ 4t/10 when r £ 2), is associated to a reduction of the local density of states in the middle of the band (Fig. 4). But this reduction is much larger at the impurity site (Fig. 4a)
N~(e)
e
r = 0.6
0.3
a) I r
N~ie)
e
r = 0.3
b) r
Fig. 5. Local densities of states on the impurity (a) and on the nearest neighbors (b), c~lculated
for e~
= 0 and r = t'/t
= 1, 0.6 and 0.3.
than at the nearest neighboring copper sites (Fig. 4b). For
r < I, the absence of any bound state coincides with a large increase of the local density of states in the middle of the band at the impurity site (Fig. 5).
For r < I and il ~ 0, with [il[ < I r~ (region I in Fig. 2), the'local density of states
N((E) at the impurity site becomes highly unsymmetrical, with a large increase which occurs for energies which have the same sign as il. This increase is large in -an energy range which lies the closer to the center E
= 0 of the band as il is the smaller, and which is the narrower
as r is the smaller. For instance, when [il[ 10.I and r
= 0.5, one gets a relatively sharp peak (Fig. 6a), which can be considered as a virtual bound state.
In the case of an oxygen vacancy, although there exist no bound state, the local density
of state on the two copper sites between which the oxygen atom is missi.ng is simply a little
smeared out, and the singularity disappears.
N°9 LOCAL D-O-S- IN A Cu02 PLANE 1121
E~/4t
~
~
~
~~
X 4 CQ
~
~~ 2
o
~
b
~)/s
i
2 ~ j~
Ky
o
i o.5 o o.5
e = E/4t
»
Fig. 6. Local densities of states on the impurity calculated for eat
= -0.8t, Uo
= U = 2t, x = 0.15
and r
= t'It
= 0.5. With these values of the parameters, the two solutions are: (a) mo
= 0, no = 1.23,
we show the total density of states 2No(E) on the impurity; (b) mo
= o.70, no
= 1. ii (the more stable
one), we show the two densities of states Nj(E) and N)(E)
on the impurity.
5. Formation of a Magnetic Moment at the Impurity Site
A large increase of the local electronic density of states at the impurity site occurs when the effective transfer integral t' between the substituted impurity and its nearest neighbors is
smaller than its corresponding value t in the perfect lattice under the condition that the shift in its atomic energy level is smaller than 4t[1- it'It)~]. Thus we can expect the formation of a
local magnetic moment at the impurity site when these conditions are fulfilled. Of course, this
implies that this large increase in the local density of states occurs in the immediate vicinity
of the Fermi level.
We wish to discuss the stability of the state of the system where, the host crystal being in its normal non magnetic phase, a non vanishing local magnetic moment mo exists only at the
impurity site. As
a result, the shift in the atomic energy level at the impurity site depends on
the spin a according to the formula
s« = sat + (no amo)Uo )nU, (23)
where sat is the shift in the bare particle energy, the two other terms giving the variation of the mean Coulomb energy. In this formula, n and U are the values of the electronic population
and of the Coulomb repulsion parameter at a host lattice site, whereas no and Uo are their values at the impurity site. In this calculation, we will assume that n has the same value I x
at all the host lattice sites, where z is the doping ratio in La2-zSr~Cu04. The values of no and mo have to be self consistently calculated from their expressions
a
= /~~iNliE) + NliE)idE,
-4t (24)
mo = /~~iNliE) NliE)idE,
-4t
where N((E) depends on no and mo, as its results from the equation (22). The chemical
potential EF is referred to the center of the band, and is related to the doping ratio x by
I x
= 2
/
~ n(E)dE. (25)As an example, we present here a numerical solution which shows the possibility of the existence of a non vanishing local magnetic moment mo at the impurity site even when Uo is not larger
than U. We choose U
= 2t and x
= 0.15, so that there are no magnetic moments at the host lattice sites in our model Ill]. Then, with Uo " U, sat
" -0.8t, and r
= t'It
= 0.5, we find
mo " 0.70 and no
" I-II- Of course, with the same values of the parameters, there exists
another solution with mo
" 0, and in that case one finds no
" 1.23; but, by calculating their
energies, we have verified that this second solution is less stable than the previous one. Figure 6 shows the calculated local electronic densities of states at the impurity site, associated with these two solutions.
The solution presented here is oversimplified, in particular because it does not take into
account electronic charge transfers between the impurity and its neighbors. This raises the
problem of the screening of the impurity potential by the electron gas. A correct calculation of this screening would have to be made self-consistently, by taking into account the modifications of the electronic populations not only at the impurity site, as we did, but also at the surrounding
copper sites, and thus of the atomic energy levels on these latters. Also the discussion of the
occurrence of magnetic moments on the neighbors would need the previously mentioned self-
consistent treatment.
References
Ill Xiao G., Cieplak M-Z-, Xiao J-Q- and Chien C-L-, Phys. Rev. B 42 (1990) 8752.
[2] Mahajan A-V-, Alloul H., Collin G. and Marucco J-F-, Phys. Rev. Lett. 72 (1994) 3100.
[3] Zagoulaev S., Monod P. and Jdgoudez J., Phys. Rev. B 52 (1995) 10474.
[4] Arai J. and Shimizu H., Physica C 161 (1989) 475.
[5] Tarascon J-M-, Greene L-H-, Bardoux P., Mckinnon W.R., Hull G-W-, Orlando T-P-, Delin K-A-, Foner S. and NcNiff Jr. E-J-, Phys. Rev. B 36 (1987) 8393.
[6] Nakano T., Oda M., Manabe C., Momono N., Miura Y. and Ido M., Phys. Rev B 49
(1994) 16 000.
[7] J. Labbd J. and J. Bok J., Europhys. Lett. 3 (1987) 1225.
[8] Hocquet T., Jardin J-P-, Germain P. and Labbd J., Phys. Rev. B 52 (1995) 10 330.
[9] Economou E-N-, Green's functions in quantum physics, Solid-State Sciences (Springer- Verlag, 1983).
[10] Rigorously, according to the formula (22), N((E) vanishes for E = 0 when s~
= 4til ~ 0;
but the energy range in which N((E) vanishes is so small that it does not appear at the scale of the band width. Thus, this vanishing of N((E) at E
= 0 has no physical meaning.
ill] Hocquet T., Jardin J-P-, Germain P. and Labb6 J., J. Phys. I France 4 (1994) 423.
