# Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds

## Abstract

**:**

## 1. Introduction

## 2. The Kondo Lattice Model and the Transversal Dynamical Susceptibility

## 3. Special Cases

#### 3.1. Korringa Relaxation

#### 3.2. The Kondo Impurity

#### 3.3. Paramagnetic Kondo lattice

#### 3.4. Kondo Lattice with Antiferromagnetic Order

#### 3.5. Kondo Lattice with Ferromagnetic Order

#### 3.6. Summary

## 4. Antiferroquadrupolar Ordered CeB${}_{6}$

#### 4.1. g-Factor for ESR in Phase II of CeB${}_{6}$

#### 4.2. Ferromagnetic Correlations in Phase II of CeB${}_{6}$

#### 4.3. Line Width of ESR in Phase II of CeB${}_{6}$

#### 4.4. Second Resonance at High Fields in Phase II of CeB${}_{6}$

#### 4.5. Inelastic Neutron Scattering in CeB${}_{6}$

- (1)
- A
**resonant magnetic exciton mode**[59], similar to the ones found in unconventional superconductors [60], including heavy fermion superconductors (CeCu${}_{2}$Si${}_{2}$, CeCoIn${}_{5}$, and CeRu${}_{2}$Al${}_{10}$) [61,62], was observed at $R[{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}}]$, but in phase III. The mode is non-dispersive, sharply peaked and associated with the opening of a spin-gap at low energies. The spin-gap is the consequence of the magnetic order, since for $T>{T}_{N}$ the resonance peak shifts to $\omega =0$ and becomes the quasi-elastic peak of the paramagnetic state. A theoretical interpretation of the resonant exciton mode was provided by Akbari and Thalmeier [63]. - (2)
- At the $\mathsf{\Gamma}$-point (zone center), a strong
**FM soft mode**was observed [64], These ferromagnetic fluctuations are large in phase III but also present (although weak) in phase II. However, no dispersive magnon excitations were found in the AFQ phase. The intensity of the magnetic excitations collapses into a broad central peak at zero energy (quasi-elastic peak) just above ${T}_{N}$. The INS line width is smallest at the $\mathsf{\Gamma}$-point. The ferromagnetic fluctuations are expected to be enhanced in a magnetic field and are the reason for an accessible ESR signal in CeB${}_{6}$. - (3)
**Spin-wave modes**emanate from the AFM wave-vectors ${Q}_{AF{M}_{1}}$ and ${Q}_{AF{M}_{2}}$ below ${T}_{N}$. They display a spin-gap of about 0.3 to 0.4 meV and, at the zone boundary (M point), the modes reach up to 0.7 meV. Hence, the spin-gap and the band width are comparable.All the above excitations merge to form a continuous dispersive magnon band in a narrow energy range. The band is more dispersive in the AFQ phase.- (4)
- In unconventional superconductors, a strong magnetic field splits the resonant magnetic exciton mode into two components. This is not the case for CeB${}_{6}$, where a second field-induced magnon mode emerges whose energy increases with magnetic field [58]. At the FM zone center ($\mathsf{\Gamma}$-point) only a single mode is found with a non-monotonic field dependence in phase III. Inside the hidden order phase, it agrees well with the ESR resonance energy (Figure 6). INS measurements in the field range of the second (high-field) ESR resonance have not been carried out. It is interesting to point out that this secondary ISN response occurs also at the R-point, which is not accessible by ESR.

#### 4.6. Summary

## 5. Longitudinal Dynamical Susceptibility

## 6. Conclusions

## Conflicts of Interest

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**Figure 1.**Relaxation rate over T of $A{u}^{171}$Yb for 9 GHz (X-band) as a function of T. Open symbols denote the ${m}_{I}=+1/2$ transitions and closed symbols the ${m}_{I}=-1/2$ transitions. Triangles correspond to a sample with 280 ppm and squares to a sample with 670 ppm. The solid curve is Equation (19) for ${T}_{K}=0.5\times {10}^{-8}$ K. Adapted with permission from Ref. [3], American Physical Society, 2018.

**Figure 2.**Sketch of the relaxation rate as a function of T for AFM or Kondo correlations, FM fluctuations and no interactions (Korringa). Close to a magnetic transition (either AFM or FM), there are additional relaxation mechanisms due to collective excitations (spin-waves or magnons) which have not been taken into account here. The horizontal dashed line schematically indicates the resonance energy. Only below that line is a signal observable.

**Figure 3.**The low temperature H vs. T phase diagram for CeB${}_{6}$ displays four phases. Phase I is the paramagnetic Kondo phase. Before the Kondo effect can compensate the internal degrees of freedom, there is a second order transition to the antiferro-quadrupolar phase (II) with the ${Q}_{AFQ}$ at the R point of the Brillouin zone. At lower T in the phases III and III’ antiferromagnetism kicks in. Phase diagram taken from Ref. [35].

**Figure 4.**${g}_{eff}$ as a function of $\theta $ for the rotation of the magnetic field in the $\langle 011\rangle $ plane for two temperatures ($T=1.8$ K and 2.65 K) from Ref. [44]. The solid curve is the fit to theory [40,43] for $T=2.65$ with $\phi =0.18\pi $ and ${K}^{\ast}=0.48$. Stronger antiferromagnetic fluctuations are expected at 1.8 K.

**Figure 5.**Line width W of CeB${}_{6}$ as a function of temperature for the three principal axis from Ref. [44]: The dark triangles correspond to [100], the open squares to [110] and the open circles to [111]. The straight line represents a Korringa relaxation for intermediate T and the [110] and [111] directions. The dashed line is a parabolic fit.

**Figure 6.**Magnetic field dependence of the two resonances. One is observed for all fields (open circles) and the other one (open squares) is only seen at high magnetic fields above 12 T [57]. The dashed straight lines correspond to a pure Zeeman splitting with ${g}_{eff}\approx 1.7$ and ${g}_{eff}\approx 1.3$, respectively. Data reproduced with permission from Ref. [58], American Physical Society, 2018.

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Schlottmann, P. Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds. *Magnetochemistry* **2018**, *4*, 27.
https://doi.org/10.3390/magnetochemistry4020027

**AMA Style**

Schlottmann P. Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds. *Magnetochemistry*. 2018; 4(2):27.
https://doi.org/10.3390/magnetochemistry4020027

**Chicago/Turabian Style**

Schlottmann, Pedro. 2018. "Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds" *Magnetochemistry* 4, no. 2: 27.
https://doi.org/10.3390/magnetochemistry4020027