\documentstyle[psfig]{article}

\title{ Influence of strong electron correlations on the spin susceptibility in
High-T$_c$ cuprates}
\author{{\it \ I. Eremin} \\
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia\\
{\em E-mail: Ilya.Eremin@ksu.ru}
}
\date{{\small Received November 22, 1997\\ Accepted
December 8, 1997\\ }}

\begin{document}

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\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 1}, {\sc 2 (1997)}\hfill}
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%{\Large \bf Vol. 1, \ \  No. 2, \ \ 1997}
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\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 1  \\  {\it No.} 2, 1997}
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\begin{center}
{\bf Abstract}
\end{center}

A new formula for the dynamic spin susceptibility has been analyzed with
taking into account the strong electron correlations. The correlations
sufficiently modify the Stoner-like factor and as a consequence change the
phase diagram of the instability. It is remarkable to note that the
instability area moves to the low doping. The possible approximates
(lorentzian and gaussian) of the real part of the spin susceptibility have
been considered. The behavior of the correlation length with doping and
temperature in both cases is discussed.


\begin{description}
  \item[PACS]: 74.25.Jb; 74.72.Bk 

  \item[Keywords]: High-T$_c$ superconductivity, strongly correlated electronic 
system, spin susceptibility 
\end{description}

\bigskip
Selected for publication by Programme Committee of youth 
scientific school "Actual problems of a magnetic resonance 
and its applications: Magnetic resonance 
in high $T_c$ materials", Kazan, November 20-22, 1997



\newpage

\section{Introduction.}

It is well known that strong electron correlation effects play the crucial
role in High-T$_c$ cuprates \cite{anderson, pines}. However, up to
now there is no clear understanding of how these effects will change the
behavior of the electronic system. For example, the $2\Delta _0/k_BT_c$
ratio gives an important information about the nature of the pairing
mechanism. But as long as we cannot take into account the strong electron
correlations we are not be able to say anything about the correct value of
this ratio. The special interest concerns the consideration of the influence
of the strong electron correlation effects on the expression for the dynamic
spin susceptibility because the susceptibility itself is directly measured
by the different experimental technics such as nuclear magnetic resonance
(NMR), electron spin resonance (ESR) and inelastic neutron scattering (INS).

Here, the new formula for the dynamic spin susceptibility has been analyzed
with taking into account the strong electron correlations. The correlations
sufficiently modify the Stoner-like factor and as a consequence change the
phase diagram of the instability. It is remarkable to note that the
instability area moves to the low doping. Then the possible approximates
(lorentzian and gaussian) of the real part of the spin susceptibility have
been examined. The behavior of the correlation length with doping and
temperature in both cases is discussed. The expression for the uniform spin
susceptibility below T$_c$ has been calculated. I compare the calculated
curve to the experimental Cu(2) Knight shift and extract the ratio $2\Delta
_0/k_BT_c$.

\section{Model Hamiltonian.}

In the calculation of the spin susceptibility I start from the two bands
model proposed by authors \cite{plakida}, \cite{eremin}: 
\begin{eqnarray}
H & = & \sum \varepsilon _d\Psi _i^{\sigma ,\sigma }+\sum E_{pd}\Psi
_i^{pd,pd}+\sum t_{ij}^{(11)}\Psi _i^{pd,\sigma }\Psi _j^{\sigma ,pd}+ 
							\nonumber\\
& & +\sum t_{ij}^{(22)}\Psi _i^{\sigma ,0}\Psi _j^{0,\sigma }+ 
+\sum t_{ij}^{(12)}(-1)^{\frac 12-\sigma }\left[ \Psi _i^{\sigma ,0}\Psi
_j^{\overline{\sigma },pd}+\Psi _i^{pd,\sigma }\Psi _j^{0,\overline{\sigma }
}\right]+ \nonumber\\
& & +\sum J_{ij}\left[ (\vec {S_i}\vec {S_j})-\frac{n_in_j}4\right]
\label{first}
\end{eqnarray}

Here, $\Psi _i^{\sigma ,0}$ and $\Psi _i^{pd,\sigma }$ are the Hubbard like
quasiparticle creation operators, for the lower and upper bands,
respectively. $\vec S$ are copper spin operators, $t_{ij}^{(11)}$ and $%
t_{ij}^{(22)}$ are hopping integrals between sites and $t_{ij}^{(12)}$ is a
hybridization parameter. $\varepsilon _d$ and $E_{pd}$ are the site energies
of the copper holes and copper-oxygen singlets. The last term in the
Hamiltonian (\ref{first}) is the superexchange interaction between the
nearest copper spins.

The difference of the suggested model in \cite{plakida}, \cite{eremin} from
the standard Hubbard Hamiltonian connects to the following fact. A hole
doped into the plane forms a Zhang-Rice singlet and does not go into the
upper copper Hubbard band \cite{rice}. The standard Hubbard Hamiltonian with
a large Coulomb repulsion is replaced by a Hubbard-like Hamiltonian with the
upper singlet band and small Coulomb repulsion. The influence of the upper
Hubbard bands is taken by superexchange interaction.

\section{Dynamic Spin Susceptibility.}

The dynamic spin susceptibility formula was deduced by the Green function
method: 
\begin{equation}
\chi (q,\omega )=-\frac 1N\left\langle \left\langle S_q^\sigma |S_q^{%
\overline{\sigma }}\right\rangle \right\rangle  
\label{gr}
\end{equation}
where $S_q^\sigma $ is a Fourier component of the copper spin. Using the
standard Hubbard I approximation scheme in the case of $E_{pd}-2\varepsilon
_d>>t^{(12)}$ the general expression for the dynamic spin susceptibility has
been got and details are described elsewhere \cite{ilya1}. In high-T$_c$
compounds the copper band is completely filled. Upon doping the carriers go
to the singlet band and $E_{k1}$ is filling up. In this case the expression
for the static spin susceptibility in the singlet band can be written as
follows: 

\begin{equation}
\chi ^{\prime }(q)=\frac{\chi _0^{11}(q)}{J_q\chi _0^{11}(q)+\chi _1^{11}(q)}
\label{real}
\end{equation}

where 

\begin{equation}
\chi _0^{11}(q) =\frac 1N\sum \frac{n_{k+q}^{(1)}-n_k^{(1)}}{
P_{pd}t_k^{(11)}-P_{pd}t_{k+q}^{(11)}}  
\label{hi} 
\end{equation}

and

\begin{equation}
\chi _1^{11}(q) =\frac 1N\sum \frac{
n_k^{(1)}t_k^{(11)}-n_{k+q}^{(1)}t_{k+q}^{(11)}}{
P_{pd}t_k^{(11)}-P_{pd}t_{k+q}^{(11)}}  
\label{hi1}
\end{equation}

Here, $J_q=J_0(\cos q_x+\cos q_y)$ and $P_{pd}=(1+\delta )/2$ is a
thermodynamic average of the anticommutator and $\delta $ is a doping level
per two copper sites, $n_k^{(1)}=<\Psi ^{pd,\sigma }\Psi ^{\sigma ,pd}>_k$
is a number of the quasiparticles of the copper-oxygen singlet band and 
\begin{eqnarray}
t_k^{(11)} & = & 2t_1^{(11)}\left( \cos (k_x)+\cos (k_y)\right) +4t_2^{(11)}\cos
(k_x)\cos (k_y)+ \nonumber\\
& & +2t_3^{(11)}\left( \cos (2k_x)+\cos (2k_y)\right)  
\label{tk}
\end{eqnarray}
with $t_1^{(11)}$, $t_2^{(11)}$, and $t_3^{(11)}$ referring to hopping to
the first, second, and third Cu neighbors, respectively

The obtained formula (\ref{real}) looks quite different comparing to the
ordinary RPA expression due to the strong electron correlations. In
particular, those effects can be clear seen on the phase diagram of the
instability determined by the condition 
\begin{equation}
J_q\chi _0^{11}(q)+\chi _1^{11}(q)=0  
\label{instability}
\end{equation}

In Fig.1 we present the result of our calculation versus doping level. As
one can see, the instability region displaces to the low doping regime on
the contrary to the $t-J$ model prediction where at the low doping level
there is not any instability \cite{rosner}. This fact has cast some doubts
on the application of the $t-J$ model for the real cuprates. In our case the
situation is differed because at low doping there is a large area of the
instability. In this connection we could propose the following scenario for
the high-T$_c$ cuprates. At low doping, the $t_1^{(11)}/J_0$ ratio is big
enough and there is an instability with incommensurate wave vector in the
system. Upon doping the $t_1^{(11)}/J_0$ ratio has reached the value (at
half-filled band) that satisfies the instability with commensurate wave
vector. And in the overdoped regime the $t_1^{(11)}/J_0$ is too small for
getting any instability at all. We have to mention that our result will
coincide with $t-J$ model phase diagram \cite{rosner} in the case of the
exchanging of the doping level $\delta $ on $1-\delta $ and putting zero
hopping integrals $t_2^{(11)}$ and $t_3^{(11)}$.

\begin{figure}[tbp]
\centerline{\psfig{figure=mrsjf1.eps,clip=}}%
%\end{center}
\caption{The calculated phase diagram for the copper - oxygen singlet
band. The phase boundary line means the appearance of the instability with
any wave vector. At low doping the phase boundary corresponds to the wave
vector $Q=(\pi ,\pi )$ albeit in the overdoped regime it displaces to the
incommensurate wave vectors. The perfect nesting does not show up at
half-filling because of the non-zero parameter $t_3^{(11)}$. }
\end{figure}

Now, we turn our consideration to the possible approximations for the static
spin susceptibility expression. Assuming two different formulas we then look
for the temperature and doping dependencies of the correlation length and
the amplitude in both cases. First, we consider the lorentzian approximation
formula: 

\begin{equation}
\chi _L^{\prime }(q)=\frac{A\xi ^2}{1+\xi ^2(q-Q)^2}
\label{loren}
\end{equation}

where $\xi $ is a correlation length, $A$ is an amplitude and $Q=(\pi ,\pi )$
. This approximation formula is a currently central topic now in the Nearly
Antiferromagnetic Fermi Liquid (NAFL) phenomenological theory for high-T$_c$
cuprates \cite{pines}. After fitting we have found that upon doping and
temperature the amplitude practically does not change and $\xi $ slightly
decreases with increasing temperature and doping from $1.3$ till $1.8$
constants of $a$. The use of the second approximation: 
\begin{equation}
\chi _G^{\prime }(q)=4\pi A\xi ^2\exp \left\{ -\xi ^2(q-Q)^2\right\}
\label{gauss}
\end{equation}
provides another temperature and doping dependencies of the correlation
length and amplitude. The amplitude in this case strongly depends on doping
and increases with decreasing temperature. Therefore, the correlation length
is less sensitive to variations of $\delta $. These results coincides well
with experimental results \cite{bobroff} for $^{17}O$ NMR probe the
susceptibility, where assuming a gaussian approximation formula the
correlation length was found is nearly T independent albeit the amplitude is
T dependent. In this connection we have to remark that the results of
approximation are differed depending on the choosing of approximation.

In Fig.2 we present the result of the calculation of the static spin
susceptibility for the half-filled band together with approximations As one
can see, among both approximations the lorentzian is preferable.

\begin{figure}[tbp]
\centerline{\psfig{figure=mrsjf2.eps,clip=}}%
%\end{center}
\caption{The static spin susceptibility along the diagonal of the
Brillouine zone. Black curve - calculated results for the expression
(\protect\ref{real}) with $\delta =0.33$, $T=100K$,
$J/t_1^{(11)}=0.2$, dotted curve -
the result of the approximation by lorentzian with $\xi =1.35a$ and $A=4.9$,
dashed - the result of the approximation by gaussian with $\xi =0.92a$ and
$A=0.75$. }
\end{figure}

\section{Uniform Spin Susceptibility below T$_c$.}

In the external magnetic field along z-axis the Hamiltonian can be written: 
\begin{equation}
H=H_0-g\beta H_z\frac 12\sum \left( \Psi _i^{\uparrow \uparrow }-\Psi
_j^{\downarrow \downarrow }\right)
\label{field}
\end{equation}
The anticommutator in this case has a view: 
\begin{equation}
P^{\uparrow ,\downarrow }=P\pm <s_z>
\label{comm}
\end{equation}
where $<s_z>$-thermodynamical expectation value of the copper spin.

Using the equation: 
\begin{equation}
\sum <\Psi ^{pd,\uparrow }\Psi ^{\uparrow ,pd}>_k=\sum <\Psi ^{pd,\downarrow
}\Psi ^{\downarrow ,pd}>_k
\label{start}
\end{equation}
in the fast fluctuating regime \cite{texas} we have deduced the expression
for the uniform spin susceptibility \cite{ilya2}: 
\begin{equation}
\chi (\delta ,\theta )=\frac{(1+\delta )^2\chi _{pl}(\delta ,\theta )}{%
4\delta +Z(\delta ,\theta )}
\label{suscep}
\end{equation}
where $\chi _{pl}(\delta ,\theta )$ is an ordinary Pauli-Lindhard
susceptibility below T$_c$ for the ordinary Fermi liquid, $Z(\delta ,\theta
) $ is a contribution due to strong electron correlation effects: 
\begin{eqnarray}
Z(\delta ,\theta )=-\frac{(1+\delta )^2}4\sum F_k\left\{ \frac{
E_{1k}-E_k^{11}}{E_{1k}}\frac{\partial f(E_{1k})}{\partial E_{1k}}+
\right.    \nonumber\\
+ \frac{E_{1k}+E_k^{11}}{E_{1k}}\frac{\partial f(-E_{1k})}{
\partial (-E_{1k})}
\left. \right\}
\label{z}
\end{eqnarray}
where 
\begin{equation}
F_k=2\left( t_k-\frac{t_k<S_iS_j>}{P^2}\right) -\frac{4J_0}{(1+\delta )^2}
\label{Fk}
\end{equation}
and 
\begin{equation}
E_{1k,2k}=\pm \left[ (E_k^{11})^2+\mid \Delta _k\mid ^2\right] ^{\frac 12}
\label{dispersion}
\end{equation}
where $\Delta _k=\Delta _0\left( \cos k_x-\cos k_y\right) $ is a
superconducting gap function, 
\begin{equation}
E_k^{11}=t_k\left( P+\frac{<S_iS_j>}P\right) +\sum_{k_1} \frac{
2J(k_1-k)}P\left\langle \Psi ^{\uparrow pd}\Psi ^{pd\uparrow }\right\rangle
_{k_1}-\mu
\label{rothd}
\end{equation}
$<S_iS_j>$ is a spin correlation function for the copper neighbors, $\mu $
is a chemical potential

In the limit case of the zero gap the formulae (\ref{suscep}) agrees with
the expression for the spin susceptibility in the normal phase \cite{texas},
\cite{jain}. The interested reader I refer to the paper \cite{ilya2}.

On the Fig. 3 ESR experimental results for Gd:YBa$_2$Cu$_3$O$_7$ \cite
{Janossy} are compared to our numerical calculations. The chemical potential
was chosen on 10 meV below Van-Hove singularity ( we employ the ''hole''
picture ) in according to the experimental observation \cite{gof1}. The
results fit well the experimental data and yields $2\Delta _0/k_BT_c=4.87$.

\begin{figure}[tbp]
\centerline{\psfig{figure=mrsjf3.eps,clip=}}%
%\end{center}
\caption{Temperature dependence of the normalized Knight shift for
Cu(2) in the plane (magnetic field $\perp $ $c$ axis): experiment, black
squares (taken from \protect\cite{Janossy}); theory, straight line.}
\end{figure}

\section{Conclusion.}

In this paper, the new expression for the dynamic spin susceptibility for
the singlet copper-oxygen band has been analyzed.

i. After calculating the phase diagram we have found that the instability
region moves to the low doping regime comparing to pure Fermi-liquid regime.

ii. The behavior of the correlation length and amplitude of the possible
approximations of static spin susceptibility is differed. For the gaussian
the correlation length practically does not depend on doping but amplitude
changes significantly. For the lorentzian the situation is upside down -
correlation length slightly decrease with increasing doping albeit amplitude
is not altered. A comparison between both approximations leads to conclusion
that lorentzian is more favorable for further calculations.

I am acknowledge ISSEP ''Graduate Students'' program (Grant No. a97-1981)
for the financial support.

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\end{document}