# Low temperature resistivity in nearly excitonic systems

Denis Jerome, Michel Rieux, Jacques Friedel

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Denis Jerome, Michel Rieux, Jacques Friedel. Low temperature resistivity in nearly excitonic systems. Philosophical Magazine, 1971. hal-04996337

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# Low-temperature Resistivity in nearly Excitonic Systems

By D. JEROME, M. RIEUX and J. FRIEDEL

Laboratoire de Physique des Solides†, 91 Orsay, France

[Received 18 January 1971]

## ABSTRACT

The effects of spin and charge fluctuations in semimetals close to an excitonic transition have been investigated. The calculation of the resistivity has been performed for a particular anisotropic two-band model and yields an exchange-enhanced contribution proportional to $T^2$ in the limit of zero temperature.

Low-temperature resistivity measurements are reported for ytterbium in the semimetallic range up to a pressure corresponding to the vanishing of the overlap.

The large and pressure-dependent $T^2$ contribution which is found is in fair agreement with the theory.

It is therefore suggested that manifestations of nearly excitonic systems are observed in Yb under pressure at low temperature.

## § 1. INTRODUCTION

THE problem of the occurrence of spin density fluctuations in the neighbourhood of a band antiferromagnetic instability has recently been investigated by Moriya (1970) for the case of a single band with two different extrema. In addition, the occurrence of various interesting features related to spin instabilities has been predicted in semimetals in the limit of a small overlap between the conduction band and the valence band.

It has been suggested by many authors that for semimetals with a low density of carriers a transition to a new distorted phase is likely to occur in the domain of the temperature-energy gap plane surrounding $G=T=0$ (Kohn 1968, Halperin and Rice 1968 a). This new phase, which is known as an ' excitonic phase ', is characterized by a static spin density oscillation coupled to a charge density oscillation. Jerome (1970) has performed a calculation of the dynamic spin susceptibility for the normal semimetallic phase in the neighbourhood of the instability. He has shown that in this limit spin fluctuations arise according to a dispersion law $\omega \sim q^2$ valid at small value of $q$ and $\omega$.

It has been known for some time that spin fluctuations exist in strongly paramagnetic metals (Schrieffer 1968) such as Pd and Pd-Ni alloys (Mills
and Lederer 1966 and Schindler and Rice 1967). Moreover, spin fluctuations contribute to the low-temperature electrical resistivity as a $T^2$ term.

Following the same lines we consider here the influence of an incipient excitonic transition on the low-temperature electrical resistivity.

We take into account in this paper a simplified picture of a two-band semimetal, namely, a single non-degenerate conduction band and a single non-degenerate valence band. In our model the current-carrying carriers are scattered by self-generated spin fluctuations. This latter fact is in contrast with strongly paramagnetic materials where s-electrons are scattered by the fluctuations among the spins of the d-electrons, while these are not supposed to contribute to the current flow.

In § 2 we have used the Boltzmann equation formalism together with the $T$ matrices for electron-hole scattering to derive an enhanced $T^2$ contribution to the low-temperature electrical resistivity. In § 3 we have extended the calculation to the charge fluctuations coupled to spin fluctuations. In § 4 we have considered the case of Yb which is known to be favourable for the observation of the excitonic instability. However, we have, as yet, no direct proof for the existence of an excitonic phase.

Resistivity measurements have been performed under pressure at low temperature on Yb and reveal a large $T^2$ contribution to the low-temperature resistivity.

In § 5 we have made a numerical estimate of the contribution of spin and charge fluctuations to the resistivity. The agreement is qualitatively satisfactory and seems to indicate that these fluctuations are responsible for the increasing $T^2$ term in the incipient excitonic system when the transition is approached from the semimetallic side.

## § 2 CALCULATION OF THE SPIN FLUCTUATION CONTRIBUTION TO THE RESISTIVITY

We shall work with a very simple geometry consisting of a maximum in the valence band at $k=0$ and two equivalent minima in the conduction band at $k = \pm \mathbf{Q}$. The corresponding energies are $\epsilon_a(\mathbf{k})$ and $\epsilon_b(\mathbf{k} + \mathbf{Q})$ given by the formula

$$ \epsilon_a(\mathbf{k}) = \frac{1}{2m} \{k_F^2 - k^2\}, \quad \dots \quad \dots $$

$$ \epsilon_b(\mathbf{k} + \mathbf{Q}) = \frac{1}{2m} \{ (k_x + Q_x)^2 + (k_y + Q_y)^2 + \gamma (k_z + Q_z)^2 - \gamma^{1/3} k_F^3 \}. \quad (1) $$

The notation in eqn. (1) is that of Jerome (1970) and provides electrical neutrality between both bands. The coefficient $\gamma$ measures the anisotropy of the b-band with respect to the a-band. The effective mass for the b-band in the $z$ direction $m/\gamma$ ($0 < \gamma < 1$) is larger than the effective mass in the $x$ and $y$ directions.
We take the model Hamiltonian (Zittartz 1967):

$$ \mathcal{H} = \sum_{\mathbf{k},\sigma} \epsilon_a(\mathbf{k})a_{\mathbf{k},\sigma}^{\dagger}a_{\mathbf{k},\sigma} + \epsilon_b(\mathbf{k} + \mathbf{Q})b_{\mathbf{k}+\mathbf{Q},\sigma}^{\dagger}b_{\mathbf{k}+\mathbf{Q},\sigma} + \frac{1}{2}\sum_{\mathbf{q}\neq0} \frac{V(\mathbf{q})}{N}\rho(\mathbf{q})\rho(-\mathbf{q}), \quad \dots \quad (2) $$

where $N$ is the single band density of carriers $k_3^3/3\pi^2$ and $\rho(\mathbf{q})$ the partial density operator given by

$$ \rho(\mathbf{q}) = \sum_{\mathbf{k},\sigma} a_{\mathbf{k}\cdot\mathbf{q},\sigma}^{\dagger}a_{\mathbf{k},\sigma} + b_{\mathbf{k}\perp\mathbf{q}\cdot\mathbf{q},\sigma}^{\dagger}b_{\mathbf{k}\perp\mathbf{q}\cdot\mathbf{q},\sigma}. \quad \dots \quad (3) $$

In the effective interaction $V(\mathbf{q})$ we keep only the exchange Coulomb terms which are the most important terms (this is the dominant term approximation of Halperin and Rice (1968 b). Moreover, we make the assumption that the interaction depends only on the total momentum of the electron-hole pair.

The solution of the Bethe-Salpeter equation for the electron-hole $T$ matrix is required in the following calculation. Brinkman and Engelsberg (1968) have solved this equation in the problem of ferromagnetic spin fluctuations with the previous approximation for the effective interaction $V(\mathbf{q})$. The scattering is visualized on fig. 1a. In the initial state we have one hole in band (a) with momentum $k_2,\sigma_2$ and one electron in band (b) with momentum $k_3,\sigma_3$. In the final state we have one hole in band (a) with momentum $k_4,\sigma_4$ and one electron in band (b) with momentum $k_1,\sigma_1$.

<figure><img src="image_2.png" /><figcaption>Fig. 1</figcaption></figure>

Electron-hole scattering without exchange of carriers between bands (a) and with exchange of carriers (b).

The contribution of electron-hole scattering to the electrical resistance is then calculated by the standard variational method Ziman (1960):

$$ \Delta\rho = \frac{1}{8k_B T} \frac{\int d\mathbf{k}_1 d\mathbf{k}_2 d\mathbf{k}_3 d\mathbf{k}_4 \Gamma(\mathbf{k}_2\mathbf{k}_3, \mathbf{k}_4\mathbf{k}_1) \{\phi_{\mathbf{k}_2} + \phi_{\mathbf{k}_3} - \phi_{\mathbf{k}_1} - \phi_{\mathbf{k}_4}\}^2}{\left| \int e^{i\mathbf{k}\cdot\mathbf{\phi}} \frac{\partial f^0(\mathbf{k})}{\partial\epsilon(\mathbf{k})} d\mathbf{k} \right|^2} \quad (4) $$
The deviation of the Fermi function from its equilibrium value

$$f^0(\mathbf{k}) = [\exp\{\epsilon(\mathbf{k})/k_B T\} - 1]^{-1}$$

is written as $-\phi_k \partial f^0(\mathbf{k}) / \partial \epsilon(\mathbf{k})$.

In eqn. (4) Γ is the scattering rate and $\phi_k$ the usual trial function proportional to $\mathbf{k}$. $\mathbf{u}$ where $\mathbf{u}$ is a unit vector along the direction of the applied electric field.

The momenta of the system must satisfy the following conservation law

$$\mathbf{k}_3 - \mathbf{k}_2 = \mathbf{k}_1 - \mathbf{k}_4 + n\mathbf{g}, \quad \dots \quad (5)$$

where $\mathbf{g}$ is a reciprocal lattice vector and $n$ is an integer. The cases $n=0$ and $n \neq 0$ correspond to the normal exchange processes and Umklapp processes respectively. We therefore write

$$\left. \begin{array}{ll} \mathbf{k}_2 = \mathbf{p} - \frac{1}{2}\mathbf{q}, & \mathbf{k}_4 = \mathbf{p}' - \frac{1}{2}\mathbf{q}, \\ \mathbf{k}_3 = \mathbf{p} + \frac{1}{2}\mathbf{q} + \frac{1}{2}n\mathbf{g}, & \mathbf{k}_1 = \mathbf{p}' + \frac{1}{2}\mathbf{q} - \frac{1}{2}n\mathbf{g}. \end{array} \right\} \quad \dots \quad (6)$$

As far as spins are concerned, there are two possibilities. The triplet pairing corresponds to $\sigma_2 = -\sigma_3 = \sigma_4$ and $\sigma_3 = \sigma_1$, whereas the singlet pairing corresponds to $\sigma_2 = \sigma_3 = \sigma_4 = \sigma_1$. In this section we concentrate on the triplet pairing and leave the discussion of the singlet pairing for § 3.

Since the momenta of the particles in bands (a) and (b) are given by eqn. (6), the vector $\mathbf{Q}$ joining the minimum of the conduction band to the maximum of the valence band must fulfil the relation

$$\mathbf{Q} = \frac{1}{2} n \mathbf{g} \quad \dots \quad (7)$$

in the limit of small Fermi surfaces. We therefore notice that the vectors joining the valence band maximum to each of the conduction band minima are equal to plus or minus half a reciprocal lattice vector. It is thus justified to take the simple description of the band structure in terms of one non-degenerate valence band and one non-degenerate conduction band, fig. 2. For simplification we pursue the calculation with $n=1$.

After some algebra on the Bethe-Salpeter equation the $T$ matrix for triplet pairing is expressed (Brinkman and Engelsberg 1968) in terms of a zero range effective interaction

$$T^1(\mathbf{q} + \mathbf{Q}, \omega) = \frac{V/N}{1 - \frac{V}{N} \chi_0(\mathbf{q} + \mathbf{Q}, \omega)} \quad \dots \quad (8)$$

In the above equation $\chi_0$ is the transverse susceptibility for non-interacting electrons and holes (Jerome 1970).
<figure><img src="image_3.png" /><figcaption>Fig. 2</figcaption></figure>

Schematic model for the band structure.

The matrix given by (8) has a pole at $\omega = \mathbf{q} = 0$ when the Stoner condition

$$1 - \frac{V}{N} \chi_0(\mathbf{Q}, 0) = 0 \quad \dots \quad (9)$$

is fulfilled.

The spin density fluctuation modes should lead to a large contribution to the electrical resistance in the vicinity of the pole.

So far we have considered only intraband mechanism excluding especially terms involving exchange of carriers between both bands. The only graphs representing intraband scattering give a major contribution in the derivation of the excitonic instability. We neglect terms represented by graphs such as fig. 1 (b).

The main problem is the evaluation of the numerator of eqn. (4). The scattering rate $\Gamma$ is the product of the $T^1$ matrix (8) and of Fermi occupation numbers for incoming and outgoing particles. In the numerator of eqn. (4) we average over all directions of $\mathbf{u}$ and transform integration over $\mathbf{k}$-space into integration over energy and Fermi surface. Taking into account the momentum and energy conservation law we end up with the expression

$$ \mathcal{N} = \frac{1}{8k_B T'} \frac{2\pi}{\hbar} \frac{g^2}{3} \left(\frac{1}{8\pi^3}\right)^4 \times \frac{|T'^1|^2 \delta\{\epsilon(\mathbf{k}_3) + \epsilon(\mathbf{k}_2) - \epsilon(\mathbf{k}_1) - \epsilon(\mathbf{k}_4)\}}{v_F^4 \hbar^4 [\exp\{\epsilon(\mathbf{k}_1)/k_B T'\} + 1][\exp\{-\epsilon(\mathbf{k}_4)/k_B T'\} + 1]} \times \frac{\delta(\mathbf{k}_1 + \mathbf{k}_2 - \mathbf{k}_3 - \mathbf{k}_4 - \mathbf{g}) \, dS \, d\epsilon}{[\exp\{\epsilon(\mathbf{k}_2)/k_B T'\} + 1][\exp\{-\epsilon(\mathbf{k}_3)/k_B T'\} + 1]} \quad (10) $$

We now write

$$\omega = \epsilon(\mathbf{k}_3) + \epsilon(\mathbf{k}_2),$$

$$\omega' = \epsilon(\mathbf{k}_1) + \epsilon(\mathbf{k}_4),$$
and perform the straightforward integration over energies. We set $x = \omega/k_B T$ in the result

$$ \mathcal{N} = \frac{(k_B T)^2}{8} \frac{2\pi}{\hbar} \left(\frac{1}{8\pi^3}\right)^4 \frac{g^2}{3} \frac{1}{v_F^4 \hbar^4} \int dS \int_{-T_F/T}^{+T_F/T} dx \frac{x^2}{(e^x - 1)(1 - e^{-x})} |T^1|^2. \quad (11) $$

As long as the **q** and **Q** vectors satisfy the relation

$$ 0 < q < 2k_F \ll Q \quad \dots \quad (12) $$

the surface integration over the four Fermi surfaces in eqn. (11) is easy to perform and leads to the result

$$ \mathcal{N} = \frac{(k_B T)^2}{8} \frac{2\pi}{\hbar} \left(\frac{1}{8\pi^3}\right)^4 \frac{1}{V_F^4 \hbar^4} \frac{g^2}{3Q^2} 16\pi^3 k_F^5 \int_{-T_F/T}^{+T_F/T} dx \frac{x^2 G(x)}{(e^x - 1)(1 - e^{-x})} \quad (13) $$

where $G(x)$ is the integral over $q$:

$$ G(x) = \int_{0}^{2k_{F}} |T^{1}(\mathbf{q} + \mathbf{Q}, xk_{B}T)|^{2} q \, dq. \quad \dots \quad (14) $$

The integral (13) is tractable if one makes a few simplifications. Since the factor $x^2/(e^x - 1)(1 - e^{-x})$ decreases rapidly as $x$ increases from zero it can be assimilated to a $\delta$ function.

In the limit $T \ll T_F$, the integral (13) becomes

$$ \int_{-T_F/T}^{T_F/T} \frac{dx \, x^2 G(x)}{(e^x - 1)(1 - e^{-x})} \approx 2 \int_0^\infty \frac{x^2 G(x)}{(e^x - 1)(1 - e^{-x})} \approx 2G(0)J_2(\infty), \quad (15) $$

where $J_2(\infty) = \frac{2}{3}\pi^2$ is a Bloch-Gruneisen integral.

The $T^1$ matrix is related to the susceptibility $\chi_0$ by eqn. (8).

According to Jerome (1970),

$$ \chi_0 = \chi_0' + i\chi_0'' $$

and

$$ \left. \begin{aligned} \chi_0'(\mathbf{q} + \mathbf{Q}, \omega) &= \chi_0(\mathbf{Q})(1 - a\mathbf{q}^2 - b\omega^2), \\ \chi_0''(\mathbf{q} + \mathbf{Q}, \omega) &= \chi_0(\mathbf{Q})\omega c_0(1 - \alpha\mathbf{q}^2), \end{aligned} \right\} \quad \dots \quad (16) $$

with

$$ c_0 = \frac{-\pi\sqrt{3m}}{(1-\gamma)k_F^2 \log|1-\gamma|} \quad \text{and} \quad \alpha = \frac{\gamma}{2(1-\gamma)(1-\gamma^{1/3})k_F^2}. $$

We also make the assumption that $a \sim 1/12k_F^2$ (the value for a spherical Fermi surface) and that $b \sim 1/k_F^4$ from dimensional arguments.
The calculation of $G(x)$ is performed in the Appendix. We derive from eqn. (A 5) its value at $x=0$, $T \ll T_F$ and $\kappa_0^2 \neq 0$:

$$G(0) = \left(\frac{V}{N}\right)^2 \frac{2k_F^2}{\kappa_0^2(\kappa_0^2 + 4ak_F^2)}, \quad \dots \quad (17)$$

where $\kappa_0^2 = 1 - (V/N)\chi_0'(\mathbf{Q},0)$ is the inverse Stoner enhancement factor. The integration of the denominator of eqn. (4) is easy for a spherical Fermi surface and gives (Ziman 1960)

$$\mathcal{D} = \frac{1}{9\pi^4} \frac{e^2}{\hbar^2} k_F^6. \quad \dots \quad (18)$$

Putting together eqns. (13), (15), (17) and (18), the contribution of spin fluctuations to the electrical resistance becomes

$$\Delta\rho_{\text{sp}} = \frac{1}{128\pi^2} \frac{m^3}{e^2\hbar^2} v_F \left(\frac{V}{N}\right)^2 \frac{1}{\kappa_0^2(\kappa_0^2 + 4ak_F^2)} \left(\frac{k_B T}{\epsilon_F}\right)^2 \dots (19)$$

As a function of the density of carriers (19) becomes

$$\Delta\rho_{\text{sp}} = \frac{1}{96\pi^4} \frac{k_B^2 m^4}{e^2\hbar^5 N} \left(\frac{V}{\hbar N}\right)^2 \frac{1}{\kappa_0^2(\kappa_0^2 + 4ak_F^2)} T^2. \quad \dots (20)$$

## § 3. EFFECT OF CHARGE FLUCTUATIONS

So far we have discussed only the occurrence of a spin distortion in the excitonic state. We should, however, point out that a distortion characterized by charge density oscillations can also arise at the excitonic instability. With the dominant-term approximation Rice and Halperin (1968 b) have shown that, discarding the spin-orbit coupling, the spin and the charge density oscillations occur at a common value of the overlap $-|G_0|$ at $T=0$°K, figure 3. The $T^0$ matrix for singlet pairing

<figure><img src="image_4.png" /><figcaption>Fig. 3</figcaption></figure>

Schematic diagram for the excitonic phase.
can be derived from the Bethe-Salpeter equation according to Brinkman and Engelsberg (1968). One gets

$$ T^0(\mathbf{q} + \mathbf{Q}, \omega) = \left(\frac{V}{N}\right)^2 \frac{\chi_0}{\left(1 + \frac{V}{N}\chi_0\right)\left(1 - \frac{V}{N}\chi_0\right)} \quad \dots \quad (21) $$

It is clear from (8) and (21) that $T^1$ and $T^0$ possess a common pole at $\mathbf{q}=\omega=0$. This means that spin and charge instabilities are degenerate at $\kappa_0^2=0$, fig. 3. It is thus reasonable to approximate $T^0$ in the neighbourhood of the transition and to write

$$ T^0(\mathbf{q} + \mathbf{Q}, \omega) \approx \frac{V/N}{2\left(1 - \frac{V}{N}\chi_0\right)} = \frac{1}{2} T^1(\mathbf{q} + \mathbf{Q}, \omega). \quad \dots \quad (22) $$

Therefore, the total contribution of fluctuations to the electrical resistance becomes

$$ \Delta\rho = \frac{5}{4} \Delta\rho_{sp} = \frac{5}{384} \frac{k_B^2 m^4}{e^2 \hbar^5 N} \left(\frac{V}{\hbar N}\right)^2 \frac{1}{\kappa_0^2 (\kappa_0^2 + 4ak_F^2)} T^2. \quad \dots \quad (23) $$

Equation (23) is valid far from the band crossing transition ($\kappa_0^2 \neq 0$) and within the limit of vanishing temperatures.

## § 4. EXPERIMENTAL RESULTS

The experiments have been conducted in a hydrostatic high-pressure equipment with a low-temperature cryostat described elsewhere (Jerome, Rieux and Achard 1970, Delplanque, Malfait, Rieux and Jerome 1970). By means of an isopentane medium the pressure is applied to the sample in a pressure vessel at room temperature. The vessel is then cooled down. Between 4·2°K and 35°K the temperature of the vessel can be stabilized with the help of a temperature regulation which includes a feedback device. A stability of the order of 0·01°K can easily be achieved over a period of 10 min. The temperature is measured at the outer surface of the vessel. Data are taken after a period long enough for thermal equilibrium to be reached in the vessel.

Recent work has shown that f.c.c. phase of Yb is not stable at low temperature below 2·2 kbar (Kayser 1970, Bucher, Schmidt, Jayaraman, Andres, Maita and Dernier 1970, Rieux and Jerome 1971). However, under isobaric (at $P > 1$ atm) cooling the f.c.c. phase is retained down to helium temperature. Data presented in this paper refer to the single f.c.c. phase. The resistivity data are summarized in fig. 4 between 2°K and 300°K at different pressures.
<figure><img src="image_5.png" /><figcaption>Fig. 4</figcaption></figure>

Resistivity of ytterbium versus temperature under isobaric conditions.

<figure><img src="image_6.png" /><figcaption>Fig. 5</figcaption></figure>

Variation of the exhaustion temperature against pressure. Experimentally the exhaustion temperature is taken as the temperature corresponding to the local maximum of ρ versus T in fig. 4.
Yb is a semimetal with an overlap decreasing with pressure, as has been shown by various authors (McWhan, Rice and Schmidt 1969, Jerome *et al.* 1970). A band-crossing transition has been located around 12 kbar. In the region of small overlaps we notice the existence of a low temperature resistivity mechanism ($\partial\rho/\partial T > 0$) which is saturated above an exhaustion temperature $T_{ex}$. This exhaustion temperature decreases rapidly with pressure in the vicinity of the band-crossing transition and is extrapolated to zero around 12 kbar, fig. 5. The behaviour of $\rho(T)$ is different at $T < T_{ex}$, since $\partial\rho/\partial T < 0$.

At $T < T_{ex}$, the resistivity follows a quadratic law at very low temperatures, of the form

$$ \rho = \rho_0 + B_{ex} T^2 \quad \dots \quad (24) $$

as is summarized in fig. 6.

The coefficient $B_{ex}$ in (24) is plotted against pressure in fig. 7, and $\rho_0$ is the residual resistivity at a given pressure. It should be noticed that $B_{ex}$ increases strongly with pressure, and shows a maximum around 11 kbar. At higher pressure $B_{ex}$ suddenly decreases. At 11.4 kbar $B_{ex}$ is negative, fig. 6.

The behaviour of $B_{ex}$ in the neighbourhood of the pressure corresponding to its maximum can be attributed to the thermal transfer of carriers from one band into the other.

<figure><img src="image_7.png" /><figcaption>Fig. 6</figcaption></figure>

A plot of $\rho$ versus $T^2$ at given pressures.
§ 5. DISCUSSION

We have seen in the previous section that Yb shows an anomalously large $T^2$ contribution to the low-temperature resistivity, and that the latter contribution becomes larger as the density of carriers decreases. In this section we would like to emphasize that the spin and charge fluctuations theory (23) leads to a qualitative agreement with experiment.

At a pressure of 8 kbar, which is sufficiently small for the coefficient $B_{ex}$ to be equal to $B$, the density of carriers obtained by galvanomagnetic data is $10^{20} \text{ cm}^{-3}$ (Jerome *et al.* 1970). According to fig. 7, $B=10^{-2} \mu\Omega \text{ cm K}^{-2}$ at 8 kbar. Equation (23), together with the previous data, leads to a value of the interaction 0.17 ev or 0.06 ev with enhancement factors

$$ \frac{1}{\kappa_0^2(\kappa_0^2 + 4ak_F^2)} $$

of 10 or 1 respectively. In agreement with (23) the contribution $B$ increases as the overlap decreases.

The theory of Jerome (1970) gives a relation between the interaction and the critical gap at $T=0^\circ\text{K}$, $|G_0| \sim 3V \log |1-\gamma|$. With a large critical gap, namely a few hundredths of one electron volt, we would expect some effect to be detectable on the low-temperature transport properties (Jerome, Rice and Kohn 1967). In pressure experiments no singularities have been observed up to 12 kbar at low temperature. This makes us think that $|G_0|$ is much smaller than 0.06 ev. The excitonic phase is eventually reduced to an exceedingly small domain in the temperature-pressure plane.

We should not forget the plausible possibility that both Fermi surfaces in Yb are strongly anisotropic, which would give rise to a very bad nesting with a $\gamma$ value far from unity. In this case the derivation of $|G_0|$ in Jerome (1970) valid for $\gamma \approx 1$ is doubtful.

It would be helpful to measure the unknown enhancement factor through another physical effect such as the electronic specific heat under pressure. Recently, Bucher *et al.* (1970) have published the data of $\gamma_{el}=8.36 \text{ mJ/}\kappa^2$ mole in plastically deformed Yb. The authors pointed out that this value for $\gamma_{el}$ is high, especially if one considers the very low density of carriers existing in Yb. This fact leads us to think that $\gamma_{el}$ is enhanced by correlation effects similar to those contributing to the resistivity at low temperature. A weakness of the theory is that it is based on the existence of a constant interaction $V$. In the limit of a low density of carriers this assumption is presumably very restrictive because the interaction is no longer strongly screened by the cloud of free carriers.

For large overlaps, the mechanism of resistivity we have calculated is equivalent to an extension of the Baber (1937) scattering process to semimetalts. Far from the excitonic instability, the enhancement effect is negligible and the resistivity process calculated in this paper resembles
the regular Baber scattering. In Baber's treatment only s-electrons scattered by d-electrons contribute to the current, whereas we have assumed both families of carriers to contribute to the total conductivity.

We have seen in § 2 that the small Fermi surfaces of Yb must satisfy certain symmetry requirements. The vector joining a hole pocket to an electron pocket must be equal to $\pm \frac{1}{2}g$, where $g$ is a reciprocal lattice vector.

To our knowledge there are not yet any clear experimental data about the Fermi surface for the true f.c.c. phase of Yb. According to theoretical predictions, Fermi surfaces probably pass through symmetry points in the Brillouin zone and therefore the requirements needed in this model for Umklapp processes may not be too restrictive.

In conclusion, we wanted to show in this paper that charge and spin fluctuations could have a significant effect at low temperature in the vicinity of an excitonic transition. According to the theory, the scattering of current-carrying carriers against the fluctuations gives rise to a $T^2$ resistivity variation. Resistivity measurements have been performed in ytterbium because it is a system which undergoes a band-crossing metal to insulator transition under pressure. We have found a contribution to the resistivity between 0 and 12 kbar varying as $BT^2$. The coefficient $B$ is quasi-divergent in the neighbourhood of the transition. Although the usual Baber scattering would yield such a result, the theory of nearly excitonic systems leads to an enhancement factor which is in reasonable quantitative agreement with the experimental data.

Obviously, new experiments such as specific heat or susceptibility measurements would bring more confidence into the hypothesis of the existence of critical fluctuations in Yb under pressure.

## ACKNOWLEDGMENTS

The authors are indebted to Professor A. Blandin and D. L. Mills for fruitful discussions.

## APPENDIX

Inserting (8) into (14) with the expression (16) $\chi_0$, the function $G(\omega)$ becomes

$$G(\omega) = \left(\frac{V}{N}\right)^2 \int_0^{2k_F} \frac{q dq}{(\kappa_0^2 + aq^2 + b\omega^2)^2 + C_0\omega^2(1 - \alpha q^2)^2} \quad . \quad (A 1)$$

After the change of variable $u=q^2$, the integration in (A 1) can be performed exactly and gives

$$G(\omega) = G_1(\omega) - G_2(\omega), \quad \dots \quad (A 2)$$
where

$$G_1(\omega) = iz \log \left| \frac{4k_F^2 - u'}{4k_F^2 - u''} \right|,$$

$$G_2(\omega) = -iz \log \left| \frac{u'}{u''} \right|$$

and

$$z = - \left( \frac{V}{N} \right)^2 \frac{1}{4C_0\omega(a + \alpha(\kappa_0^2 + b\omega^2))}$$

$u'$ and $u''$ are given by

$$u' = (u'')^+ = A - iB$$

and

$$A = -a(\kappa_0^2 + b\omega^2) + C_0^2\omega^2\alpha,$$

$$B = C_0\omega\{\alpha(\kappa_0^2 + b\omega^2) + a\}.$$

The previous values of $u'$ and $u''$ enable us to calculate $G_1(\omega)$ and $G_2(\omega)$ and then

$$G(\omega) = - \left(\frac{V}{N}\right)^2 \frac{1}{2C_0\omega\{a + \alpha(\kappa_0^2 + b\omega^2)\}} \\ \times \left\{ \arctg \frac{C_0\omega\{a + \alpha(k_0^2 + b\omega^2)\}}{a(\kappa_0^2 + b\omega^2) - C_0^2\omega^2\alpha + 4k_F^2(a^2 + C_0^2\omega^2\alpha^2)} \right. \\ \left. - \arctg \frac{C_0\omega\{a + \alpha(\kappa_0^2 + b\omega^2)\}}{a\kappa_0^2 + b\omega^2} - C_0^2\omega^2\alpha \right\} . \quad (A3)$$

(1) Let us first look at a simplified version of (A 3) when $\alpha=0$. If the condition

$$C_0\omega < \kappa_0^2 \quad \dots \quad \text{A 4}$$

is fulfilled we can expand the arctg in $\omega^2$:

$$G(\omega) = \left(\frac{V}{N}\right)^2 \frac{2k_F^2}{(\kappa_0^2 + b\omega^2 + 4ak_F^2)(\kappa_0^2 + b\omega^2)} \quad \dots \quad \text{A 5}$$

(2) The general situation is $\alpha \neq 0$. However, this case goes back to the $\alpha=0$ situation when both conditions

$$\alpha(\kappa_0^2 + b\omega^2) < a = 1/12k_F^2 \quad \dots \quad \text{A 6}$$

and

$$C_0^2\omega^2\alpha < a(\kappa_0^2 + b\omega^2) \quad \dots \quad \text{A 7}$$

are fulfilled.

A necessary condition resulting from (A 6) and (A 7) is

$$C_0\omega\alpha < a. \quad \dots \quad \text{A 8}$$
(A 8) expressed in terms of (19) leads to a limitation on the temperature

$$T < \frac{(1 - \gamma)^2(1 - \gamma^{1/3})|\log(1 - \gamma)|}{12\pi\sqrt{3\gamma}} T_F. \quad \dots \quad (A 9)$$

For $T \ll T_F$ the $q$ dependence of $\chi''_0$ may be neglected in lowest order in $q$ without particular condition for the value of $\gamma$.

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