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# Description of nearly antiferromagnetic systems

Denis Jerome

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Denis Jerome. Description of nearly antiferromagnetic systems. Solid State Communications, 1970. hal-05000919

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Distributed under a Creative Commons Attribution 4.0 International License
DESCRIPTION OF NEARLY ANTIFERROMAGNETIC SYSTEMS

D. Jerome

Laboratoire de Physique des Solides*, Faculté des Sciences, 91 Orsay, France.

(Received 11 August 1970 by P.G. de Gennes)

This note attempts to show that spin fluctuations exist in a nearly antiferromagnetic system at $T = 0$°K. The calculation of the spin susceptibility has been performed for a particular anisotropic two-band model and is an extension of Moriya's formulation. A spin density fluctuations mode $\omega \sim q^2$ becomes heavily damped in the vicinity of the antiferromagnetic transition. It is suggested that spin and charge density fluctuations contribute a large $T^2$ term in the low temperature resistivity of a nearly excitonic system.

WE SHALL study in this note the behavior of a semimetal at $T = 0$°K in the limit of vanishingly small overlaps. We look only at the spin instability.

We must keep in mind that in the low temperature region spins and charges are presumably strongly coupled as it will be mentioned in the course of this paper.

To be able to make tractable calculations, we take a particular anisotropic band model: an almost full (a) band with a spherical Fermi surface and an almost empty (b) band with an ellipsoidal Fermi surface.¹ We call $\vec{Q}$ the vector joining the minimum of band (b) to the maximum of band (a).

The dynamics of the system is governed by the following model Hamiltonian.²

$$ \mathcal{H} = \sum_{\vec{k}, \sigma} \{ \epsilon^a(\vec{k}, \sigma) a_{\vec{k}, \sigma}^{\dagger} a_{\vec{k}, \sigma} + \\ + \epsilon^b(\vec{k} + \vec{Q}, \sigma) b_{\vec{k}+\vec{Q}, \sigma}^{\dagger} b_{\vec{k}+\vec{Q}, \sigma} + \\ + \frac{1}{2} \sum_{\vec{q} \neq 0} \frac{V(\vec{q})}{N} \rho(\vec{q}) \rho(-\vec{q}) \} \quad (1) $$

where $a_{\vec{k}}^{\dagger}$ and $b_{\vec{k}+\vec{Q}}^{\dagger}$ are the electron creation operators in bands (a) and (b), $V(\vec{q})$ is an effective Coulomb interaction and $N$ is the single band density of carriers.

The Hartree correction to the energies is included in $\epsilon^a$ and $\epsilon^b$ with the condition $\vec{q} \neq 0$ in equation (1).

The partial electron density operator is given by the equation:

$$ \rho(\vec{q}) = \sum_{\vec{k}, \sigma} a_{\vec{k}+\vec{q}, \sigma}^{\dagger} a_{\vec{k}, \sigma} + b_{\vec{k}+\vec{q}, \sigma}^{\dagger} b_{\vec{k}, \sigma} \quad (2) $$

We now study the motion of an electron-hole pair of momentum $\vec{q} + \vec{Q}$ and opposite spins under the influence of the Hamiltonian (1) plus an external field of wave vector $\vec{q}$ and frequency $\omega$ characterized by

$$ \mathcal{H}_{\text{ext}} = (\lambda_{\vec{q}+\vec{Q}} \psi(\vec{q}, \omega) e^{+i\omega t} + cc) \eta_{\vec{q}}^{t} \quad (3) $$

where $\lambda_{\vec{q}+\vec{Q}}$ is the operator

$$ \sum_{\vec{p}} \lambda_{\vec{p}+\vec{q}} = \sum_{\vec{p}} b_{\vec{p}+\vec{q}+\vec{Q}}^{\dagger} a_{\vec{p}+\vec{q}} $$

The commutator of $\lambda_{\vec{p}+\vec{q}}$ with (1) and (3) leads to the following equation within the RPA approximation:

<footnote>*Laboratoire associé au C.N.R.S.</footnote>
$$ egin{aligned} & \left[ -\omega + i\eta + \epsilon_{\uparrow}^{b}(\vec{p} + \vec{q} + \vec{Q}) - \epsilon_{\downarrow}^{a}(\vec{p}) - \sum_{\vec{k}} \frac{V(\vec{p} - \vec{k})}{N} n_{\uparrow}^{b}(\vec{k} + \vec{q} + \vec{Q}) + \sum_{\vec{k}} \frac{V(\vec{p} - \vec{k})}{N} n_{\downarrow}^{a}(\vec{k}) \right] \lambda \vec{p} \vec{q} \\ &= - \left[ n^{b}(\vec{p} + \vec{q} + \vec{Q}) - n_{\downarrow}^{a}(\vec{p}) \right] \left[ \sum_{\vec{k}} \frac{V(\vec{p} - \vec{k})}{N} \lambda_{\vec{k}} \vec{q} + \psi \right] \end{aligned} \quad (4) $$

Only the exchange scattering term occurs in the right hand side of equation (4) with the approximation (2) and the Hamiltonian (1). The mean value $\langle \lambda_{\vec{q}+\vec{Q}}^b \rangle$ in the normal ground state is given by the susceptibility.

$$ \langle \lambda_{\vec{q}+\vec{Q}} \rangle = \chi(\vec{q} + \vec{Q}, \omega) \psi(\vec{q}, \omega) \quad (5) $$

From now on, we make the assumption of a strong screening and take $V(\vec{q})$ as a constant.

The susceptibility (5) is obtained in terms of the non interacting electrons susceptibility

$$ \chi(\vec{q} + \vec{Q}, \omega) = \frac{\chi_0(\vec{q} + \vec{Q}, \omega)}{1 - \frac{V}{N} \chi_0(\vec{q} + \vec{Q}, \omega)} \quad (6) $$

with

$$ \chi_0(\vec{q} + \vec{Q}, \omega) = \chi'_0(\vec{q} + \vec{Q}, \omega) + i\chi''_0(\vec{q} + \vec{Q}, \omega) = -\sum_{\vec{p}} \frac{n_{\uparrow}^{b}(\vec{p} + \vec{q} + \vec{Q}) - n_{\downarrow}^{a}(\vec{p})}{E^b(\vec{p} + \vec{q} + \vec{Q}) - E^a(\vec{p}) - \omega + i\eta} \quad (7) $$

In equation (7), the energies $E^b$ and $E^a$ include the Hartree-Fock renormalisation. An instability of the normal ground state against a distorted state is obtained when

$$ 1 - \frac{V}{N} \chi'_{0}(\vec{q} + \vec{Q}, \omega) = 0 \quad (8) $$

is satisfied for $\omega = 0$. It will be shown shortly that the instability (8) first sets in at a wave-vector $\vec{q} = 0$. A non-zero value of the Fourier component of the magnetization proportional to $\langle \lambda_{\vec{q}} \rangle$ exists in the distorted phase even in the absence of the external field $\psi(\vec{q}, \omega)$. If $\vec{Q}$ is equal to half the length of a reciprocal lattice vector, an antiferromagnetic spin density wave has arisen at the instability. We notice that we have just obtained an excitonic instability

coming from the semimetallic side.²

If coupling to the phonon system is neglected the charge density wave instability occurs always after the antiferromagnetic instability in a perturbation treatment of Hamiltonian (1).³ We only consider the problem of the magnetic instability in this paper.

In order to make the computation of the susceptibility (6) we choose a particular geometry⁴

$$ egin{aligned} E^a(\vec{k}) &= \frac{1}{2m} (k_F^2 - k^2) \\ E^b(k + \vec{Q}) &= \frac{1}{2m} (k_x + Q_x)^2 + (k_y + Q_y)^2 + \gamma(k_z + Q_z)^2 - \gamma^{1/3} k_F^2 \end{aligned} \quad (9) $$

In equation (9) we forgot spin indices since we do the calculation in the paramagnetic phase. The density of carriers in one band is related to $k_F$ by the equation

$$ N = k_F^3 / 3\pi^2 \quad (10) $$

Extending Moriya's treatment¹ to the two-band model we shall expand $\chi'_0$ and $\chi''_0$ in the vicinity of $\vec{Q}$ according to

$$ \begin{aligned} \chi'_{0}(\vec{q} + \vec{Q}, \omega) &= \chi_{0}(\vec{Q}) (1 - a(\vec{Q}, \vec{q}))q^2 - b(\vec{Q}, \vec{q})\omega^2 + \dots \\ \chi''_{0}(\vec{q} + \vec{Q}, \omega) &= \chi_{0}(\vec{Q}) c(\vec{Q}, \vec{q})\omega \end{aligned} \quad (11) $$

$\chi'_0$ and $\chi''_0$ are given from equation (7) by

$$ \begin{aligned} \chi'_{0}(\vec{q} + \vec{Q}, \omega) &= -P \sum_{\vec{p}} \frac{n^{b}(\vec{p} + \vec{q} + \vec{Q}) - n^{a}(\vec{p})}{E^{b}(\vec{p} + \vec{q} + \vec{Q}) - E^{a}(\vec{p}) - \omega} \\ \chi''_{0}(\vec{q} + \vec{Q}, \omega) &= \pi \sum_{\vec{p}} n^{b}(\vec{p} + \vec{q} + \vec{Q}) - n^{a}(\vec{p}) \\ \delta E^{b}(\vec{p} + \vec{q} + \vec{Q}) - E^{a}(\vec{p}) - \omega &= 0 \end{aligned} \quad (12) $$

The integral $\chi'_0(\vec{Q}, 0)$ would be logarithmically divergent at the Fermi surface for the isotropic case ($\gamma = 1$). Following a treatment similar to Zittartz's formulation⁴ we shall keep in the evaluation of $\chi'_0(\vec{Q}, 0)$ only the leading term proportional to $\text{Log}|1 - \gamma|$ in the limit $1 - \gamma \ll 1$. One obtains thus
$$ \chi'_{0}(\vec{Q}, 0) \approx - \frac{mk_F}{\pi^2} \log|1 - \gamma| \quad (13) $$

With equation (8) and (13), for a given shape $\gamma$ and a given interaction $V$, the instability occurs at a critical overlap $G_0$ such as (Fig. 1).

$$ G_0 \sim 3V \log|1 - \gamma| \quad (14) $$

It is seen from (14) that in the case of perfect nesting of the two surfaces the normal state is unstable against an arbitrarily small interaction.⁵ A careful look at the integrand in $\chi'_0(\vec{q} + \vec{Q}, \omega)$ shows that $b(\vec{Q}, 0)$ is finite and positive as well as a $(\vec{Q}, 0)$.

The integral $\chi''_0(\vec{q} + \vec{Q}, \omega)$ is reduced to a surface integral over the surface $S(\omega)$ defined by the condition $E^b(\vec{k} + \vec{q} + \vec{Q}) - E^a(\vec{k}) = \omega$ and $E^b(\vec{k} + \vec{q} + \vec{Q}) > 0$, $E^a(\vec{k}) < 0$.

$$ \chi''_0(q + Q, \omega) = \frac{1}{4\pi^2} \int_{S(\omega)} \frac{dS}{|\nabla[E^b(\vec{k} + \vec{q} + \vec{Q}) - E^a(\vec{k})]|} \quad (15) $$

For an easy calculation we must take $\vec{q}$ along the $k_z$ axis.

At small values of $\omega$, equation (15) is reduced to two contour integrals along the lines $L_1$ and $L_2$, Fig. 2, and becomes

$$ \chi''_0(\vec{q} + \vec{Q}, \omega) = -\frac{\omega m^2}{4\pi^2} \int_{L_1 + L_2} \frac{dk}{k_{\perp} |k_z(1-\gamma) - \gamma q|} \quad (16) $$

in cylindrical coordinates.

The expansion of $k_z$ in equation (16) up to the second order in $q$ gives the result

$$ \frac{\chi''_0(\vec{q} + \vec{Q}, \omega)}{\chi_0(\vec{Q})} = \frac{-\pi\sqrt{3}m\omega}{k_F^2(1-\gamma)\log|1-\gamma|} \left[ 1 - \frac{\gamma}{2(1-\gamma)} \frac{1}{(1-\gamma^{1/3})} \left( \frac{q}{k_F} \right)^2 \right] \quad (17) $$

It is important to notice that $\chi''_0(\vec{q} + \vec{Q}, \omega)$ does not diverge at $\vec{q} = 0$.

$Im\chi(\vec{q} + \vec{Q}, \omega)$ is obtained from (6) and (17) by the equation

$$ \frac{\mathrm{Im}\chi(\vec{q} + \vec{Q}, \omega)}{\chi_0(\vec{Q})} = \frac{c\omega}{(K_0^2 + aq^2 + b\omega^2)^2 + c^2\omega^2} \quad (18) $$

where $K_0^2$ is the inverse of the Stoner enhancement factor. At a given $\omega$, $Im\chi(\vec{q} + \vec{Q}, \omega)$ shows a maximum for a value $q$ satisfying the law

$$ \omega = K_0^2 \left( \frac{1}{\alpha} + \frac{\beta}{\alpha} q^2 \right) \quad (19) $$

where $\alpha$ and $\beta$ depend on the shape of the Fermi surfaces

$$ \begin{aligned} \alpha &= \frac{\pi\sqrt{3}m}{k_F^2(1-\gamma)\log|1-\gamma|}, \\ \beta &= \frac{\gamma}{2(1-\gamma)} \frac{1}{1-\gamma^{1/3}} \frac{1}{k_F^2}, \end{aligned} \quad (20) $$

The dispersion law (19) represents a spin fluctuations mode heavily damped in the vicinity of the AF transition. This mode $\omega \sim q^2$ contrasts with the phonon-like mode for the paramagnons in nearly ferromagnetic metals.⁶,⁷ We have been looking, so far, only at properties of the AF instability at $T = 0°K$. But in the low temperature region ($T \sim T_{cm}$) in Fig. 1 density fluctuations will no longer be the only fluctuations present in the system since they are strongly coupled to charge density fluctuations. These two kinds of fluctuations may influence the low temperature transport properties. Due to 'Umlapp' scattering of carriers against spin and charge fluctuations an extra $T^2$ contribution should occur in the resistivity.

A relevant system for studying the incipient AF excitonic phase is Yb under pressure. Yb is a semimetal at $P = 0$ with a small density of carriers, $N = 2 \times 10^{-2}/\text{atom.}^8$ Galvanomagnetic measurements under pressure have shown that the overlap $G$ between the valence band and the conduction band vanishes around 11 kbar.

The Fermi surface of Yb is made of several pockets of electrons and holes,⁹,¹¹ located close to symmetry points on the Brillouin zone.

A large $T^2$ contribution to the low temperature resistivity has been found in Yb.⁸ The magnitude of this contribution increases noticeably in the
<figure><img src="image_3.png" /><figcaption>FIG. 1. Phase diagram for the AF excitonic instability. The gap $G_0$ corresponds to the AF instability. A static spin density wave exists in the shaded area.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 2. Diagram of the two Fermi surfaces displaced by $\vec{q}$ along $k_x$ axis, showing the intersections $L_1$ and $L_2$ of equation 16.</figcaption></figure>
vicinity of the metal-insulator transition.¹² This
may be attributed to the existence of spin and
charge fluctuations. More results on the $T^2$
term will be the subject of a forthcoming paper.

*Acknowledgements* – I wish to thank Professor A. Blandin, Drs. T.M. Rice, P. Lederer and M. Héritier for very helpful discussions. I am particularly grateful to Professor J. Friedel for his continuous encouragements and a critical reading of the manuscript.

REFERENCES

1. The single-band model has been looked at by MORIYA T., *Phys. Rev. Lett.* 24, 1433 (1970).

2. For the most recent reviews on the problem of nearly zero gap semimetals, see: KOHN W., in *Many Body Physics*, Les Houches, Summer School, Gordon and Breach, (1968) and HALPERIN B.I. and RICE T.M., *Rev. Mod. Physics*, 40, 755 (1968).

3. HALPERIN B.I. and RICE T.M., *Solid State Physics*, Vol. 21, Academic Press, New York, (1968).

4. ZITTARTZ J., *Phys. Rev.* 162, 752 (1967).

5. FEEDERS P.A. and MARTIN P., *Phys. Rev.*, 143, 245 (1966).

6. DONIACH S. and ENGELSBERG S., *Phys. Rev. Lett.* 17, 750 (1966).

7. SCHRIEFFER J.R., *J. appl. Phys.*, 39, 642 (1968).

8. JÉROME D., RIEUX M. and ACHARD J.C., in *Proc. of Int. Conf. Solids under Pressure*, Grenoble, C.N.R.S., (1970) to be published.

9. TANUMA S., ISHIZAWA Y., NAGASAWA H. and SUGAWARA T., *Phys. Lett.*, 25A, 669 (1967).

10. TANUMA S., DATARS W.R., DOI H. and DUNSWORTH A., *Solid State Commun.* 8, 1107 (1970).

11. JOHANSEN G. and MACKINTOSH A.R., *Solid State Commun.*, 8, 121 (1970).

12. JULLIEN R. and JÉROME D., to be published in *J. Phys. Chem. Solids*, (1970).

Nous avons démontré dans cette communication l'existence de fluctuations de spin au voisinage d'une transition antiferromagnétique à T = 0°K. La susceptibilité de spin a été calculée pour un modèle à deux bandes anistropes en s'inspirant des remarques de Moriya. Nous avons obtenu un mode de fluctuations de densité de spin ω ~ q² fortement amorti au voisinage de la transition antiferromagnétique. Les fluctuations de spin peuvent contribuer à un terme en T² dans la résistivité de basse température d'un système au voisinage d'une transition excitonique.