#
Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)_{2}X Studied by Fluctuation Spectroscopy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Organic Charge-Transfer Salts—Quasi Two-Dimensional Molecular Metals

#### 2.2. Fluctuation (’Noise’) Spectroscopy

#### 2.2.1. Definitions and Basic Relations

#### 2.2.2. Measuring Resistance Fluctuations and Hooge’s Law for Organic Charge-Transfer Salts

## 3. Results

#### 3.1. Example (1): Superconducting Percolation in Phase Coexistence Regions

#### 3.2. Example (2): Glass-Like Structural Ordering

#### 3.3. Example (3): Charge-Cluster Glass

#### 3.4. Example (4): Mott Metal-Insulator Transition

#### Sample Dependences

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CO | charge ordering |

DDH model | model by Dutta, Dimon and Horn for $1/f$-type noise in metals |

EEG | ethylene endgroups |

ET | BEDT-TTF (bis-ethylenedithio-tetrathiafulvalene) |

FDT | fluctuation-dissipation theorem |

$\kappa $-Br | $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Br |

$\kappa $-Cl | $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Cl |

MIT | metal-insulator transition |

NMR | nuclear magnetic resonance |

PSD | power spectral density |

RRN | random resistor network |

VFT | Vogel-Fulcher-Tammann |

## References and Notes

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**Figure 1.**(

**a**) crystal structure (left) of $\kappa $-(BEDT-TTF)${}_{2}$Cu[N(CN)${}_{2}$]Z with Z = Br, Cl, which can be simply viewed as alternating blocks (right) of conducting and insulating layers with the donor ET and polymeric chain-forming acceptor Cu[N(CN)${}_{2}$]Z molecules, respectively. The box represents the unit cell hosting four formular units; (

**b**) spatial arrangement of the ET molecules in the $\kappa $-phase viewed along the long axis of the molecules. Two ET molecules form a dimer (shaded ellipse), whereas two dimers are arranged almost orthogonal to each other. Considering the dimers as the basic structural units, the conduction band is effectively half-filled and the charger carriers reside on a triangular lattice with two transfer integrals t and t′.

**Figure 2.**Schematic temperature–pressure phase diagram of the $\kappa $-(BEDT-TTF)${}_{2}$X salts with polymeric anions, after Refs. [17,18,19,20,21]. Indicated is the position of systems with different anions X at ambient pressure. Hydrostatic pressure can be mapped on an increasing ratio of $W/U$ (see text). PI, PM, AFI, FE and SC stand for the paramagnetic insulating, paramagnetic metallic, antiferromagnetic insulating, ferroelectric and superconducting phase, respectively. The red line marks the first-order Mott metal-insulator transition, which terminates in a finite-temperature critical point. At elevated temperatures, a structural glass-like transition of the ET molecules’ ethylene endgroup rotational degrees of freedom occurs (dashed yellow line). The green dotted line marks the crossover to an anomalous metallic (’bad metal’) phase above the Fermi liquid.

**Figure 3.**Schematics of a voltage (resistance) noise measurement. (

**a**) time-train of the voltage signal and (

**b**) histogram of the voltage distribution fitted by a Gaussian function; (

**c**) noise power spectral density (PSD) of the signal in (

**a**) fitted by a power law with frequency exponent $\alpha \simeq 1$. Reprinted with permission from [9]. Copyright (2011) John Wiley and Sons.

**Figure 4.**Typical noise spectra. (

**a**) representative resistance noise PSD of an organic charge transfer salt measured at different temperatures. Lines are fits to ${S}_{R}\propto 1/{f}^{\alpha}$. Reprinted with permission from [60]. Copyright (2009) by the American Physical Society; (

**b**) voltage noise PSD with a Lorentzian contribution superimposed in the $1/f$-type background.

**Figure 5.**Noise at the percolative superconducting transition. (

**a**) resistance of two samples of $\kappa $-(ET)${}_{2}$Cu[N(CN)${}_{2}$]Cl. Inset shows a schematic phase diagram, cf. Figures 2 and 10b. Arrows indicate the positions of the samples measured at ambient conditions ($\kappa $-Cl) and at a finite pressure ($\kappa $-Cl${}^{\ast}$); (

**b**) normalized resistance noise PSD ${S}_{R}/{R}^{2}(T,f=1\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz})$ of $\kappa $-Cl${}^{\ast}$. Arrow indicates the superconducting transition (onset of ${T}_{c}$). Inset: PSD of the resistance noise at 90 K in a log-log plot revealing a clean ${S}_{R}\propto 1/{f}^{\alpha}$ spectrum with $\alpha =1.09$. Reprinted with permission from [68]. Copyright (2009) by the American Physical Society.

**Figure 6.**Percolation models in superconductors. (

**a**) scaling of the normalized noise ${S}_{R}/{R}^{2}\propto {R}^{{l}_{rs}}$ of $\kappa $-Cl${}^{\ast}$ in a representation $log{S}_{R}/{R}^{2}$ vs. $logR$, where the slope reveals the power-law exponent ${l}_{rs}$, for data taken at $T=5$ K and different magnetic fields up to 4 T as an implicite tuning parameter of the sample resistance (see (

**c**)); (

**b**) schematics of classical (bottom) and novel (top) percolation noise. In the former case (${l}_{rs}=0.9$), the noise originates from the resistor elements, where the short-circuits represent superconducting subvolumes. The latter case of p-noise (${l}_{rs}=2.74$) originates from switching elements controlled by random processes ${w}_{i}\left(t\right)$ representing subvolumes with unstable superconductivity, after [72]. Reprinted with permission from [9]. Copyright (2011) John Wiley and Sons; (

**c**) magnetoresistance $R\left(B\right)$ at $T=5$ K. Light and dark gray areas denote different percolation regimes characterized by the scaling exponent ${l}_{rs}$. In the white area, Lorentzian spectra are superimposed on the $1/f$ background noise (see (

**d**)), where the inset shows the shift of the corner frequency ${f}_{c}\left(B\right)$. (

**a**,

**d**) reprinted with permission from [68]. Copyright (2009) by the American Physical Society.

**Figure 7.**Noise spectroscopy and DDH analysis (see text) on fully-deuterated $\kappa $-(D${}_{8}$-ET)${}_{2}$Cu[N(CN)${}_{2}$]Br. (

**a**) resistance (blue squares) and integrated noise PSD (red circles) of a fully-deuterated sample $\kappa $-(D${}_{8}$-ET)${}_{2}$Cu[N(CN)${}_{2}$]Br; (

**b**) corresponding frequency exponent $\alpha \left(T\right)$ (green circles) and model calculation within the generalized DDH model (Equation (16)) (red line); (

**c**) distribution of activation energies $D\left(E\right)$ at $f=1$ Hz calculated using Equation (18). Arrows indicate the two strongest maxima at 260 meV and 90 meV; (

**d**) contour plot of the relative noise level ${a}_{R}=f\times {S}_{R}/{R}^{2}$ vs. frequency f and temperature T in an Arrhenius representation. The slope of the white line corresponds to an activation energy of ${E}_{a}=260$ meV, corresponding to the maximum in $D\left(E\right)$ in (

**c**) as extracted from the DDH model. Reproduced from [92,93]. Copyright IOP Publishing.

**Figure 8.**Glass-like structural EEG ordering in $\kappa $-(ET)${}_{2}$X. (

**a**) normalized resistance noise PSD of $\kappa $-D${}_{8}$-Br for different frequencies. Lines are fits to a Gaussian function. Black circles connect the maxima at each frequency; (

**b**) Arrhenius plot of the noise peak frequency for selected systems of $\kappa $-(ET)${}_{2}$X (see legend). Light grey lines are fits of thermally activated (Arrhenius) behavior to the high-frequency data, whereas dark grey lines are Vogel-Fulcher-Tammann (VFT) fits after Equation (19) to the low-frequency data. The horizontal line equals $\tau =100$ s and may be used to define the dynamic glass-transition temperature; (

**c**) occupation probability $p\left(E\right)$ of the eclipsed (E) ground state conformation (see right inset) calculated from a simple two-level model (left inset) for various cooling rates. ’Pulse rate’ refers to cooling after a heat pulse is applied resulting in cooling rates of order 1000 K/min. Reprinted with permission from [23]. Copyright (2014) by the American Physical Society; (

**d**) preferred conformations of the EEGs for $\kappa $-Br in relation to the nearby anion layer. Dashed lines indicate close EEG⋯Br contacts, while blue arrows indicate soft vibrational degrees of freedom of the terminal Br ligands suggested to couple to the EEG rotation. In $\kappa $-Br, all EEGs are crystallographically equivalent. (

**a**,

**b**,

**d**) reproduced from [93]. Copyright IOP Publishing.

**Figure 9.**Charge-cluster glass in $\theta $-(ET)${}_{2}$RbZn(SCN)${}_{4}$. (

**a**) temperature dependence of the resistivity during cooling for different sweeping rates ${q}_{c}$. Insets indicate the crystal structures and charge distribution of the high-temperature (charge-liquid) phase (lower inset) and the slowly cooled low-temperature (charge-ordered) phase (upper inset); (

**b**) resistance noise PSD for various temperatures as ${f}^{\alpha}\times {S}_{R}/{R}^{2}$ vs. f with fits (red lines) to the distributed Lorentzian model (see text); (

**c**) temperature profiles of the fitting parameters ${f}_{{c}_{1}}$ and ${f}_{{c}_{2}}$; (

**d**) slowing of the centre frequency and (

**e**) concomitant growth of the dynamic heterogeneity, respectively. Reprinted with permission from [113]. Copyright (2013) Springer Nature.

**Figure 10.**Tuning a partially-deuterated sample $\kappa $-[(H${}_{8}$-ET)${}_{1-x}$(D${}_{8}$-ET)${}_{x}$]${}_{2}$Cu[N(CN)${}_{2}$]Br with $x=0.8$ through the Mott transition. (

**a**) resistance measurements for various cooling rates q; (

**b**) generic phase diagram after [21]. Open circles indicate values taken from the literature of the second-order critical end point of the first-order Mott MIT (thick solid curve). The arrow indicates the position of the slowly cooled, pristine sample in the generic phase diagram. The change of the sample position with q is schematically indicated by the colored lines. For comparison, the blue color represents fully-deuterated $\kappa $-D${}_{8}$-Br shown in Figure 7. Reprinted with permission from [53]. Copyright (2015) by the American Physical Society.

**Figure 11.**Critical slowing down of charge fluctuations. (

**a**) resistance noise PSD ${S}_{R}/{R}^{2}(T,f=1\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz})$ of slowly-cooled $\kappa $-H${}_{8}$/D${}_{8}$-Br (yellow triangles) in comparison to the data of $\kappa $-D${}_{8}$-Br (blue circles) normalized to the value of the local maximum around 100 K. The black arrow indicates a peak in the noise level, cf. data of the same sample shown in Figure 7a. (

**b**,

**c**) measurements for increasing cooling rates q. Note the rescaling of the ordinate in (

**c**); (

**d**) frequency exponent $\alpha \left(T\right)$ for $q=5$ K/min, where the noise level shown in (

**c**) is largest. The line represents the DDH model calculation; (

**e**) relative noise level ${a}_{R}\equiv f\times {S}_{R}/{R}^{2}$ vs. f vs. T for the same cooling rate. Reprinted with permission from [53]. Copyright (2015) by the American Physical Society.

**Figure 12.**Resistance and noise measurements at the Mott transition in partially-deuterated $\kappa $-[(H${}_{8}$-ET)${}_{1-x}$(D${}_{8}$-ET)${}_{x}$]${}_{2}$Cu[N(CN)${}_{2}$]Br with $x=0.75$. (

**a**) resistance measurements of some selected warming cycles generated by using different thermal relaxation protocols described in the text. The black points mark the temperatures of the entrance and re-entrance half-maximum resistance jump, indicating the crossing of the Mott transition line; (

**b**) contour plot of the resistance vs. temperature and warming cycle. The grey area marks the region where the DDH model shows strong deviations; the white circle indicates the critical point; (

**c**) contour plot of the normalized resistance noise PSD taken at $1\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ in dependence of temperature and warming cycle (note different scales).

**Figure 13.**Temperature dependent frequency exponent $\alpha $ for some selected warming cycles. The black line corresponds to the values determined with the DDH model; the grey shaded area emphasizes the temperature region where the DDH model deviates from the experimental data.

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Müller, J.; Thomas, T.
Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)_{2}X Studied by Fluctuation Spectroscopy. *Crystals* **2018**, *8*, 166.
https://doi.org/10.3390/cryst8040166

**AMA Style**

Müller J, Thomas T.
Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)_{2}X Studied by Fluctuation Spectroscopy. *Crystals*. 2018; 8(4):166.
https://doi.org/10.3390/cryst8040166

**Chicago/Turabian Style**

Müller, Jens, and Tatjana Thomas.
2018. "Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)_{2}X Studied by Fluctuation Spectroscopy" *Crystals* 8, no. 4: 166.
https://doi.org/10.3390/cryst8040166