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# Enhancement of 14N quadrupolar relaxation in the one dimensional conductor KCP due to lattice fluctuations above the Peierls transition

G. Soda, Claude Bourbonnais, Denis Jerome

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G. Soda, Claude Bourbonnais, Denis Jerome. Enhancement of 14N quadrupolar relaxation in the one dimensional conductor KCP due to lattice fluctuations above the Peierls transition. Journal of the Physical Society of Japan, 1987. hal-04989875

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# Enhancement of (¹⁴N) Quadrupolar Relaxation in the One Dimensional Conductor K₂Pt(CN)₄Br₀.₃3.2H₂O Due to Lattice Fluctuations above the Peierls Transition

G. SODA,† C. BOURBONNAIS and D. JEROME

Laboratoire de Physique des Solides Université de Paris-Sud,
91405 Orsay, France

(Received February, 2, 1987)

The spin-lattice relaxation $T_1$ of ¹⁴N($I=1$) nuclei in a single crystal of the mixed-valency platino-cyanate K₂Pt(CN)₄Br₀.₃3.2H₂O(KCP) quasi-one dimensional conductor has been measured in the temperature domain from room temperature down to the 3-D Peierls ordering below 120 K. A rapid increase of the quadrupolar relaxation is observed below 220 K which follows the power law $1/T_1\alpha r^{-1/2}$ where $r=(T-T_p)/T_p$. This behaviour is explained in terms of the critical slowing down of lattice fluctuations with a Landau-Ginzburg theory. The square root critical divergence of $T_1⁻¹$ is the signature of the harmonic regime for the order parameter fluctuations. The enhancement of $T_1⁻¹$ and the very wide 3-D critical width can be explained by the strong anisotropy of correlation lengths in this Q-1-D conductor.

## §1. Introduction

Spin lattice nuclear relaxation time measurement has proved to be a rewarding experimental technique for the study of precursor regimes developing above the onset of three-dimensional (3D) ordering in quasi-1-D (Q-1-D) conductors.¹

Most of the NMR work has been devoted so far to the experimental investigation of the fluctuation regimes in the organic cation radical salts belonging to the D₂X series.² This family of 1-D conductors (D=organic cation radical, X=pseudo halogen anion) displays a variety of ground states (Spin-Peierls, Spin Density Wave, Superconductivity) depending on parameters such as chemical species, high pressure or the value of the external magnetic field.¹ However, for these Q-1-D conductors the electron-phonon coupling is rarely the dominant interaction for the establishment of 3-D ordering at low temperature (with the exception of TMTTF₂PF₆-like materials) and precursor effects in a Peierls system could not

be studied.

Quite generally, spin-lattice relaxation is due to the exchange of energy between nuclear spins and the energy reservoir (conduction electrons, lattice, etc.). This process is achieved through local random fields seen by nuclei coupled to either magnetic moments or quadrupolar moments. The relaxation rate $T_1⁻¹$ is thus given by:³

$$T_1^{-1} = C^{\text{st}} \sum_q \frac{kT}{\omega_n} \chi''(q, \omega_n), \quad (1)$$

where $\chi''(q, \omega_n)$ is the imaginary part of the susceptibility related to the random field and $\omega_n$ is the nuclear Larmor frequency. For 1-D conductors, excitations contributing most to $T_1⁻¹$ are those accompanied by electron-hole excitations of low energy¹ (i.e. $q=0$ and $2k_F$ in the $q$-summation of eq. (1)). As already discussed in previous studies of 1-D conductors the $2k_F$ component contributes dominantly at low temperature. For organic conductors of the D₂X series the major source of intrinsic spin-lattice relaxation (besides the extrinsic CH₃-reorientation contribution) comes from the modulation of the hyperfine coupling between itinerant electrons and nuclear spins of the molecules (¹H, ⁷⁷Se, ¹³C). $T_1⁻¹$ data have

<footnote>† This article is dedicated to the memory of Gen Soda who started this study while he was staying at Orsay (1974-1976). Gen Soda died in an accident 19 February 1980.</footnote>
been used to show that the 2k<sub>F</sub> component of
the spin susceptibility of conductors such as
TMTSF<sub>2</sub>PF<sub>6</sub> or TMTSF<sub>2</sub>ClO<sub>4</sub> exhibits a
power-law temperature divergence which
becomes clearly visible below 30 K. The expo-
nent is typical of a 1-D behaviour (y<sub>1D</sub> ~ 1).<sup>4)</sup>

The study of the Peierls precursor regime by
NMR has not received that much attention.
The purpose of the present article is to show
how lattice precursor effects can be detected
by NMR investigations and how the interpreta-
tion fits into the framework of the theory of
critical effects at low dimensionality.

The prototype material showing all signs of
the Peierls distorsion and of a giant Kohn
anomaly in the phonon spectrum is the mixed
valency platinum complex K₂Pt(CN)₄Br₀.₃
3.2H₂O(KCP).⁵) The main feature of KCP
is the columnar stacking along the c-axis of
square planar (CN)₄ ligands comprising each
a Pt atom in the center of the square. The
distance of 2.88 Å between Pt atoms along the
stacking axis is only slightly larger than the
Pt-Pt distance in metal (2.77 Å). Conse-
quently, d²z² orbitals of Pt atoms overlap
strongly along the c-direction and form an
energy band. In principle such a band should
be filled by two electrons given by each Pt
atom. However, it is only 85% filled up to the
wave vector k₇=0.85c* on account of the pre-
sence of bromine atoms giving rise to Br⁻ ions.
Abundant X-ray and neutron scattering ex-
periments have been devoted to the study of
this material.⁶) They have shown the onset of a
periodic lattice distorsion (PLD) below 120 K
characterized by a wave vector (π/a, π/a,
2k₇). Furthermore, X-ray diffuse scattering
experiments have revealed the existence at
T=300 K of a diffuse streaking perpendicular
to the c*-axis which has been attributed to a 1-
D modulation of the underlying lattice.⁷) X-
ray are not able to provide informations rela-
tive to the dynamics of the modulation. The
distance in q-space between diffuse lines and
the lines of Bragg spots allows a very precise
determination of the Fermi wave vector,
leading to 2k₇=0.3(π/c) (c=2.88 Å). This
value is in agreement with the band filling
inferred from chemical considerations and
established the proof for the Peierls
mechanism in this 1-D conductor. In addition,

intense elastic neutron scattering has been ob-
served at q=(0, 0, 2k<sub>F</sub>) even at room tempera-
ture.⁶) On cooling down the growth of the
elastic intensity at (π/a, π/a, 2k<sub>F</sub>) has indi-
cated the occurence of transverse coherence.
Intensities scattered around (0, 0, 2k<sub>F</sub>) and
(π/a, π/a, 2k<sub>F</sub>) are about equal at 200 K.
Below 200 K the (π/a, π/a, 2k<sub>F</sub>) scattering is
dominant but the 1-D(0, 0, 2k<sub>F</sub>) scattering
which diminishes below 120 K (the 3-D order-
ing temperature) remains finite. The absence
of truly established long range order has been
attributed to lattice imperfections. Two
origins can be given to the elastic scattering:⁶) a
Peierls distortion existing at room tempera-
ture or the existence of low frequency critical
Peierls fluctuations. The limited resolution of
neutron experiments prevented the exact deter-
mination of the origin. The study of the
quadrupolar relaxation of <sup>14</sup>N nuclei belong-
ing to the (CN)<sub>4</sub> square planes is “a priori” a
good way to check for the latter possibility.
No itinerant electron density exists on
nitrogen atoms, thus <sup>14</sup>N spins (I=1) should
relax to the lattice via the modulation by lat-
tice vibrations of the electric field gradients on
nitrogen sites. Since (CN)<sub>4</sub> ligands are rigidly
coupled to Pt of the conducting chains the
measurement of <sup>14</sup>N spin-lattice relaxation
should provide a very direct probe for the ex-
istence of critical fluctuations.⁸) The role of
Peierls critical fluctuations on the quadrupolar
relaxation has already been reported in an
other Q-1-D conductor,⁹) Rb<sub>0.3</sub>MoO<sub>3</sub> showing
a Peierls transition at 181 K. However, the
situation is slightly more complicated with this
conductor as two mechanisms contribute
together to the relaxation of <sup>87</sup>Rb(I=3/2)
spins: the hyperfine and the quadrupolar relax-
ation.

## §2. Experimental Results

The single crystal KCP sample was oriented
with the c-axis parallel to the external
magnetic field. Temperature between 300 and
120 K were achieved using a dry N₂-gas flow
system. The ¹⁴N NMR signal was obtained by
a conventional spin echo technique using an
analog box-car integrator. The spectrometer
was tuned at a frequency of 9.9 MHz and the
center of the resonance occured at 36.2 kOe.
The observed NMR corresponds to the high field quadrupolar satellite of the ¹⁴N(I=1) doublet. Given the orientation of the static magnetic field and the cylindrical symmetry of electric field gradients, *q*, the shift of the line leads to (e²qQ/h)=1650 kHz. The temperature independent inhomogeneous broadening (ΔH=300 Oe) of the line can be understood in terms of the static disorder introduced by Br⁻ ions and frozen H₂O molecules.¹⁰) Spin-lattice relaxation was obtained by a π/2 comb-τ-(π/2), π pulse sequence varying the delay τ. The data for the half recovery rate T₀⁻¹ instead of T⁻¹ are displayed in Fig. 1. The minimum occuring around 220 K indicates a change of regime for the nuclear quadrupole relaxation. At high temperature the relaxation rate is increasing with temperature. This temperature dependence is expected for quadrupolar relaxations induced by a spin-phonon coupling. Since this temperature domain is higher than the Debye temperature of KCP the second order Raman process leading to T⁻¹αT² should dominate.³) The data on Fig. 1 are obviously not accurate enough to ex-

tract a well characterized T-dependence between 220 and 280 K. However, they are in qualitative agreement with the existence of the Raman process. Below 220 K the T⁻¹ temperature dependence cannot be explained by spin-phonon coupling in regular solids. The divergence observed when approaching the Peierls transition must be ascribed to a critical slowing down of the lattice fluctuations.⁸)

### §3. Discussion

In the following we will be mainly concerned with the origin of the strong enhancement of T⁻¹ that emerges below 200 K. The log-log plot of the T⁻¹ data versus T-T_c in figure 1 clearly supports a power law behaviour of the form T⁻¹α((T-T_c)/T_c)⁻⁻⁻. The best fit of the data gives ν=1/2 ± .05 with T_c ≈ 117 K. It is well established from several experiments that the temperature of the Peierls "ordering" in KCP occurs around T_p ≈ 110-120 K, showing the possible relevance of the 3D "ordering" on the enhancement of T⁻¹. In fact, such a critical profile for the quadrupolar relaxation is reminiscent of what is commonly observed near incommensurate structural phase transitions,⁸,¹¹) like the one taking place in the ferroelectric crystal Rb₂ZnCl₄ for example.¹¹) The exponent ν=1/2 is mean field like and is well known to result from strongly damped 3D harmonic fluctuations which affect the relaxation via the so-called direct process. One must note however that the critical width associated with this type of enhancement namely, ΔH = ((T_f - T_c)/T_c) is usually of the order of 10⁻² or so in these compounds.¹¹) This is clearly much smaller than the one observed here for KCP. Indeed, from the figure 1 T_f ≈ 200 K (i.e. ΔH ~ 1). It is well known that strong spatial anisotropies in the correlation lengths can significantly increase the observable width of critical fluctuations around T_c.¹²,¹³) This is especially relevant here since the Peierls transition in KCP is known to be a classical example of a system of weakly coupled incommensurate charge density wave chains.⁶) Indeed, the 1D-2k_F phonon coupling to the 2k_F charge density wave instability of the 1D electron gas already dominates above room temperature with very weak transverse correlations that level off at T_p.⁶) In

<figure><img src="image_3.png" /><figcaption>Fig. 1. Temperature dependence of the ¹⁴N magnetization half recovery rate (bottom). Log-log plot of the half recovery rate versus T-T_c with the value of T_c=117 K (top).</figcaption></figure>
addition, from neutron scattering experi-
ments,⁶⁾ the correlation lengths associated to
the superstructure wave vector (π/a, π/a,
2k<sub>F</sub>) at T<sub>fl</sub> are highly anisotropic namely, (ξ<sub>x,y</sub><sup>0</sup>, ξ<sub>y</sub><sup>0</sup>, ξ<sub>z</sub><sup>0</sup>)=(3 Å, 3 Å, >300 Å). From resolution
limitations, only a lower bound of ξ<sub>z</sub><sup>0</sup> can be
given. The small ratio ξ<sub>x,y</sub><sup>0</sup>/ξ<sub>z</sub><sup>0</sup> ≪ 1 is consistent
with a large temperature domain where
1D-2k<sub>F</sub> lattice precursors along the z direc-
tion should dominate. Actually, NMR Knight
shift measurements on <sup>195</sup>Pt have shown the ex-
istence of a 1D pseudo-gap in the density of
states at the Fermi level.<sup>14</sup> This had led to in-
fer that the characteristic or the mean field tem-
perature T<sub>p</sub><sup>0</sup> for the apparition of 1D fluctua-
tions should be around 600 K or so!. As the
non singular renormalized 2k<sub>F</sub> phonon fre-
quency ω<sub>2k<sub>F</sub></sub> ~ 80 K<sup>6</sup>) we have 2πT<sub>p</sub><sup>0</sup> ≫ ω<sub>2k<sub>F</sub></sub>
which means that a strong adiabaticity
prevails between the lattice and the electron
gas system. It follows that a classical fluc-
tuating Peierls order parameter is a well de-
fined quantity below T<sub>p</sub><sup>0</sup>.

The Peierls instability is incommensurate in
KCP and the order parameter has two com-
ponents thereby allowing amplitude and phase
fluctuations. In the purely 1D regime, the
amplitude fluctuations are strongly anhar-
monic. The repulsive potential (Coulombic) in-
terchain coupling, though weak, in amplitude
will favor an antiphase ordering of the order
parameter in the transverse direction. The

growth of the correlation length ξ<sub>x,y</sub> will mark
the onset of a dimensionality crossover for
fluctuations which in turn restores the har-
monicity of the order parameter fluctuations.
The harmonicity persists as long as ξ<sub>x</sub> or ξ<sub>y</sub> are
not too large otherwise non-mean field like
critical effects take place within the so-called
Ginzburg critical width Δ<sub>G</sub>.<sup>15)</sup>

The latter will always satisfy the inequality
Δ_G < Δ_H.

Given the strong adiabaticity of the elec-
ron-phonon system precursor to the transi-
tion, the lattice fluctuations can be considered
classically and an important contribution to
the NQR relaxation will come from the direct
process. According to Zummer and Blinc¹ for
example, the contribution to T⁻¹ becomes in
this case:

$$
T_{-1}^{-1} = \text{cst} |T'(l)|^2 2T \int d_q^d \frac{\chi''(q, \omega)}{\omega}. \quad (2)
$$

T'(l) is the first derivative of the electric field
gradient tensor at the site l · χ''(q, ω) is the im-
aginary part of the retarded correlation func-
tion for the classical lattice displacement at
wave vector q and frequency ω. Applying the
dynamic scaling hypothesis<sup>2,8,16)</sup> to the three
quantities ∫ d<sub>q</sub><sup>d</sup> χ''(q, ω) and ω, the latter will
scale with the coherence length ξ to some
characteristic power (exponent). We will get
for q=q<sub>0</sub>=(π/a, π/a, 2k<sub>F</sub>) and d=3 dimen-
sions:

$$
\begin{align}
T_{\text{lcr}}^{-1} &= 2\text{Cst} |T'(l)|^2 T_p \Gamma \frac{r^{3\nu-\gamma-Z\nu}}{(\xi_x^0 \xi_y^0 \xi_z^0)} \int_0^{c_i} \frac{d(\xi_x q_x) d(\xi_y q_y) d(\xi_z q_z) \chi''(q \cdot \xi, \Gamma r^{-Z\nu} \omega)}{(\Gamma r^{-Z\nu} \omega)}, \nonumber \\
T_{\text{lcr}}^{-1} &\propto \frac{r^{3\nu-\gamma-Z\nu}}{(\xi_x^0 \xi_y^0 \xi_z^0)} \quad (r<1), \tag{3}
\end{align}
$$

where r=(T-T<sub>p</sub>)/T<sub>p</sub>, Γ is the order
parameter damping constant and c<sub>i">1 (i=x,
y, z) for r<1. A Peierls order parameter is a
non-conserved quantity and therefore Γ will
be q independent. ν, γ and Z are respectively
the critical exponents of the correlation length
(ξ<sub>i</sub>=ξ<sub>i</sub><sup>0</sup>r<sup>-γ</sup>) susceptibility (χ"αr<sup>-γ</sup>) and fluctua-
tion relaxation rate (ωαr<sup>Zν</sup>). Given the ob-
served anisotropies for the ξ<sub>i</sub><sup>0</sup>, (ξ<sub>i</sub>=ξ<sub>i</sub><sup>0</sup> at
T=T<sub>fl</sub>), an harmonic time-dependent Landau-
Ginzburg approach is justified below T<sub>fl</sub>. This

leads to exponents<sup>16)</sup> ν=1/2, γ=1, Z=
2(Γ=cst) and to the classical profile

$$
T_{-1}^{-1} \propto \frac{r^{-1/2}}{(\xi_x^0 \xi_y^0 \xi_z^0)}, \quad (r < 1) \quad (4)
$$

for r<1. Owing to the strong anisotropies of
ξ<sub>i</sub><sup>0</sup> in KCP, the amplitude of T<sub>i</sub><sup>-1</sup> will be en-
hanced compared to the isotropic case by the
factor (ξ<sub>i</sub><sup>0</sup>)<sup>2</sup>/(ξ<sub>y</sub><sup>0</sup>ξ<sub>x</sub><sup>0</sup>) ~ 10<sup>4</sup>. Such a value can
explain why Δ<sub>H</sub> is much larger than the
harmonic width found in more isotropic sys-
tems such as the ferroelectric compound Rb₂ZnCl₄¹¹) and other similar systems.⁸,¹⁵,¹⁷) Roughly speaking, if we assume that the differences between Δ<sub>H</sub> come essentially from the ξᵢ⁰'s, then this implies the value ξᵢ,y,z ≈ 30 Å or so for a 3D system like Rb₂ZnCl₄. This characteristic length is consistent with the typical Ginzburg critical width Δ<sub>G</sub> ~ 10⁻⁴ found in ferroelectric systems (see ref. 15).

The anisotropy in the ξᵢ⁰'s does not only affect the direct process-type of quadrupolar relaxation but also the relaxation contribution coming from the second order Raman process.⁸ According to refs. 8 and 11, the Raman contribution is non-singular as we approach the transition namely, $T_1^{-1}\alpha(\xi^0)^{-5}T^2$. However, for an anisotropic system like KCP, this contribution is enhanced by a factor [$\xi_x^0\xi_x^0/\xi_z^0\xi_z^0$]$^{-5/3} \approx 10^7$ at $T_{fl}$. Such a huge factor may be responsible for the increase of $T_1^{-1}$ at $T>T_{fl} \sim 200$ K (see Fig. 1). This will mask the 1D fluctuation effects on the quadrupolar relaxation that should take place above $T_{fl}$. Actually, 1D amplitude and phase Peierls fluctuations would give, via direct processes, contributions of the form $T_1^{-1} \sim |(T-T_0)/T_0|^{-3/2} (T>T_0)$ and $T_1^{-1} \sim (T/T_0)^{-1} (T<T_0)$. Here, $T_0 \sim T_p^0/3$ is a characteristic temperature for the apparition of phasons.¹ None of these profiles seem to be relevant to the observation. One must note however, that in KCP, $T_0 \approx 200$ K $\approx T_{fl}$, which means that only 1D amplitude fluctuations are relevant for $T_{fl} < T < T_p^0$.

The existence of precursor effects above the Peierls transition is also suggested by the early measurements of ¹⁹⁵Pt(I=1/2) spin-lattice relaxation rates.¹⁴) The ¹⁹⁵Pt relaxation rate is anomalously large and the enhancement relative to a Korringa mechanism derived from Knight shift data increases when T decreases. Therefore, the divergence of the Korringa enhancement factor (1/T₁)exp/(1/T₁)Korringa observed at low temperature¹⁴) looks quite similar to the divergence of the quadrupolar relaxation rate presented in the present study. ¹⁹⁵Pt and ¹⁴N resonances probe respectively the

electronic and lattice effects of the precursor fluctuations.

In conclusion, the study of the quadrupolar relaxation of ¹⁴N nuclei in the 1-D conductor KCP has demonstrated that Peierls lattice fluctuations strongly contribute to the spin-lattice relaxation in the temperature domain above the Peierls transition. The square root divergence which is revealed below 220 K is fairly well understood in terms of the effect of the critical slowing down of the fluctuations on the spin-phonon direct process.

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