#
Effect of Co-Doping on the Magnetic Ground State of the Heavy-Fermion System CeCu_{2}Ge_{2} Studied by Neutron Diffraction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Si, Q.; Steglich, F. Heavy Fermions and Quantum Phase Transitions. Science
**2010**, 329, 1161–1166. [Google Scholar] [CrossRef] [PubMed] - Keimer, B.; Kivelson, S.A.; Norman, M.R.; Uchida, S.; Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature
**2015**, 518, 179–186. [Google Scholar] [CrossRef] [PubMed] - Uemura, Y.J. Commonalities in phase and mode. Nat. Mater.
**2009**, 8, 253–255. [Google Scholar] [CrossRef] [PubMed] - Stewart, G.R. Non-Fermi-liquid behavior in d- and f-electron metals. Rev. Mod. Phys.
**2001**, 73, 797–855. [Google Scholar] [CrossRef] - Löhneysen, H.v.; Rosch, A.; Vojta, M.; Wölfle, P. Fermi-liquid instabilities at magnetic quantum phase transitions. Rev. Mod. Phys.
**2007**, 79, 1015–1075. [Google Scholar] [CrossRef] - Gegenwart, P.; Si, Q.; Steglich, F. Quantum criticality in heavy-fermion metals. Nat. Phys.
**2008**, 4, 186–197. [Google Scholar] [CrossRef] - Pfleiderer, C. Superconducting phases of f-electron compounds. Rev. Mod. Phys.
**2009**, 81, 1551–1624. [Google Scholar] [CrossRef] - Brando, M.; Belitz, D.; Grosche, F.M.; Kirkpatrick, T.R. Metallic quantum ferromagnets. Rev. Mod. Phys.
**2016**, 88, 025006. [Google Scholar] [CrossRef] - Mathur, N.; Grosche, F.; Julian, S.; Walker, I.; Freye, D.; Haselwimmer, R.; Lonzarich, G. Magnetically mediated superconductivity in heavy fermion compounds. Nature
**1998**, 394, 39–43. [Google Scholar] [CrossRef] - Sparn, G.; Deppe, M.; Donnevert, L.; Geibel, C.; Hellmann, P.; Heuser, K.; Köppen, M.; Lang, M.; Laube, F.; Link, A.; et al. Multiple magnetic phase transitions in Ce(Cu
_{1-x}Ni_{x})2Ge_{2}. Phys. B Condens. Matter**1997**, 230–232, 317–320. [Google Scholar] [CrossRef] - Singh, D.; Thamizhavel, A.; Lynn, J.W.; Dhar, S.; Rodriguez-Rivera, J.; Herman, T. Field-induced quantum fluctuations in the heavy fermion superconductor CeCu
_{2}Ge_{2}. Sci. Rep.**2011**, 1, 1–6. [Google Scholar] [CrossRef] [PubMed] - Knebel, G.; Eggert, C.; Engelmann, D.; Viana, R.; Krimmel, A.; Dressel, M.; Loidl, A. Phase diagram of CeCu
_{2}(Si_{1-x}Ge_{x})_{2}. Phys. Rev. B**1996**, 53, 11586–11592. [Google Scholar] [CrossRef] - Jaccard, D.; Wilhelm, H.; Alami-Yadri, K.; Vargoz, E. Magnetism and superconductivity in heavy fermion compounds at high pressure. Phys. B Condens. Matter
**1999**, 259–261, 1–7. [Google Scholar] [CrossRef] - de Boer, F.; Klaasse, J.; Veenhuizen, P.; Böhm, A.; Bredl, C.; Gottwick, U.; Mayer, H.; Pawlak, L.; Rauchschwalbe, U.; Spille, H.; et al. CeCu
_{2}Ge_{2}: Magnetic order in a Kondo lattice. J. Magn. Magn. Mater.**1987**, 63–64, 91–94. [Google Scholar] [CrossRef] - Krimmel, A.; Loidl, A.; Schober, H.; Canfield, P.C. Single-crystal neutron diffraction studies on CeCu
_{2}Ge_{2}sand CeCu_{1.9}Ni_{0.1}Ge_{2}s. Phys. Rev. B**1997**, 55, 6416–6420. [Google Scholar] [CrossRef] - Zwicknagl, G. Kondo Effect and Antiferromagnetism in CeCu
_{2}Ge_{2}: An Electronic Structure Study. J. Low Temp. Phys.**2007**, 147, 123–134. [Google Scholar] [CrossRef] - Knopp, G.; Loidl, A.; Knorr, K.; Pawlak, L.; Duczmal, M.; Caspary, R.; Gottwick, U.; Spille, H.; Steglich, F.; Murani, A. Magnetic order in a Kondo lattice: A neutron scattering study of CeCu
_{2}Ge_{2}. Z. Phys. B Condens. Matter**1989**, 77, 95–104. [Google Scholar] [CrossRef] - Tripathi, R.; Das, D.; Geibel, C.; Dhar, S.K.; Hossain, Z. Non-Fermi-liquid behavior at the antiferromagnetic quantum critical point in the heavy-fermion system Ce(Cu
_{1-x}Co_{x})_{2}Ge_{2}. Phys. Rev. B**2018**, 98, 165136. [Google Scholar] [CrossRef] - Tripathi, R.; Das, D.; Biswas, P.K.; Adroja, D.T.; Hillier, A.D.; Hossain, Z. Quantum Griffiths phase near an antiferromagnetic quantum critical point: Muon spin relaxation study of Ce(Cu
_{1-x}Co_{x})_{2}Ge_{2}. Phys. Rev. B**2019**, 99, 224424. [Google Scholar] [CrossRef] - Loidl, A.; Krimmel, A.; Knorr, K.; Sparn, G.; Lang, M.; Geibel, C.; Horn, S.; Grauel, A.; Steglich, F.; Welslau, B.; et al. Local-moment and itinerant antiferromagnetism in the heavy-fermion system Ce(Cu
_{1-x}Ni_{x})_{2}Ge_{2}. Ann. Phys.**1992**, 504, 78–91. [Google Scholar] [CrossRef] - Chapon, L.C.; Manuel, P.; Radaelli, P.G.; Benson, C.; Perrott, L.; Ansell, S.; Rhodes, N.J.; Raspino, D.; Duxbury, D.; Spill, E.; et al. Wish: The New Powder and Single Crystal Magnetic Diffractometer on the Second Target Station. Neutron News
**2011**, 22, 22–25. [Google Scholar] [CrossRef] - Rodriguez-Carvajal, J. Rietveld Analysis of Neutron or X-ray Powder Diffraction Data. Available online: https://www.ill.eu/sites/fullprof/ (accessed on 23 March 2023).
- Straumanis, M.; Yu, L. Lattice parameters, densities, expansion coefficients and perfection of structure of Cu and of Cu-In α phase. Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr.
**1969**, 25, 676–682. [Google Scholar] [CrossRef] - Krimmel, A.; Loidl, A. The phase diagram of CeCu
_{2}(Si_{1-x}Ge_{x})_{2}. Phys. B Condens. Matter**1997**, 234-236, 877–879. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Neutron powder diffraction patterns of Ce(Cu${}_{0.95}$Co${}_{0.05}$)${}_{2}$Ge${}_{2}$ from one of the detector banks of the WISH diffractometer at T = 2 K. The solid line represents the Rietveld refinement profile fit for the ThCr${}_{2}$Si${}_{2}$-type body-centered tetragonal (space group $I4/mmm$) structure. A small amount of impurities of copper ($Fm-3m$), from the Cu-sample can, is included in the refinement. (

**b**) The expanded view shows the differences between 2 K (green) and 0.28 K (red) data, where the magnetic Brag peaks are quite visible. The blue curve at the bottom shows the temperature difference data 0.28–2 K.

**Figure 2.**Magnetic Bragg peaks (

**a**) (0 0 0) $\pm \phantom{\rule{0.166667em}{0ex}}k$, (

**b**) (0 0 2) $-\phantom{\rule{0.166667em}{0ex}}k$, and (

**c**) (0 −1 1)/(−1 0 1) $+\phantom{\rule{0.166667em}{0ex}}k$ at various temperatures, and (

**d**) the integrated intensity of the peaks (0 0 0) $\pm \phantom{\rule{0.166667em}{0ex}}k$, (0 0 2) $-\phantom{\rule{0.166667em}{0ex}}k$, and (0 −1 1)/(−1 0 1) $+\phantom{\rule{0.166667em}{0ex}}k$ versus temperature for x = 0.05. The solid lines are fit to the data with power-law behavior.

**Figure 3.**Rietveld refinement of the magnetic intensities obtained by subtraction of high-temperature (2 K) paramagnetic data from the low-temperature (0.28 K) data of Ce(Cu${}_{1-x}$Co${}_{x}$)${}_{2}$Ge${}_{2}$ with x = 0.05. The solid black lines show the fit. The difference between the experimental and calculated intensities is shown by the blue curves at the bottom. The olive vertical ticks show the position of the magnetic Bragg peaks (bottom) and the structural Bragg peaks (top).

**Figure 4.**The incommensurate magnetic structure of Ce(Cu${}_{1-x}$Co${}_{x}$)${}_{2}$Ge${}_{2}$ (x = 0.05) with magnetic propagation vector k = (0.2852, 0.2852, 0.4495) obtained from the refinement of ND pattern at 0.28–2 K. The solid lines show the unit cell, and the magnetic moments are shown by red arrows at the Ce atom position presented by red balls.

**Figure 5.**(

**a**) Neutron powder diffraction patterns of Ce(Cu${}_{0.8}$Co${}_{0.2}$)${}_{2}$Ge${}_{2}$ at T = 50 mK. The solid line represents the Rietveld refinement profile fit for the ThCr${}_{2}$Si${}_{2}$-type body-centered tetragonal (space group $I4/mmm$) structure. The refinement includes a small amount of copper impurities ($Fm-3m$). (

**b**) No magnetic Brag peak could be seen in the expanded view of the difference between 0.9 K and 50 mK data.

**Table 1.**Crystallographic parameters obtained from Rietveld refinement of powder XRD (room temperature) and ND data of Ce(Cu${}_{1-x}$Co${}_{x}$)${}_{2}$Ge${}_{2}$ (x = 0.05, 0.2) at different temperatures with the body-centered tetragonal ThCr${}_{2}$Si${}_{2}$-type structure (space group $I4/mmm$).

$\mathit{x}=0.05$ | $\mathit{x}=0.2$ | |||||
---|---|---|---|---|---|---|

XRD | ND | ND | XRD | ND | ND | |

(300 K) | (5 K) | (0.28 K) | (300 K) | (0.9 K) | (50 mK) | |

Lattice parameter | ||||||

a(Å) | 4.1759(1) | 4.1689(5) | 4.1680(1) | 4.1744(4) | 4.1729(5) | 4.1708(2) |

c(Å) | 10.1860(7) | 10.1702(3) | 10.1711(1) | 10.1225(1) | 10.1230(8) | 10.1149(2) |

V(Å${}^{3}$) | 177.6249 | 176.7518 | 176.6930 | 176.3908 | 175.2727 | 175.9545 |

Atomic coordinate | ||||||

${z}_{Ge}$ | 0.3767 | 0.3776 | 0.3777 | 0.3753 | 0.3766 | 0.3766 |

Refinement quality | ||||||

${\chi}^{2}$ | 1.67 | 19.4 | 20.3 | 2.13 | 17.4 | 19.3 |

${R}_{P}$(%) | 16.5 | 17.93 | 15.01 | 19.6 | 17.93 | 18.82 |

${R}_{WP}$(%) | 20.4 | 15.32 | 14.91 | 26.3 | 19.72 | 21.11 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tripathi, R.; Khalyavin, D.; Sharma, S.; Adroja, D.T.; Hossain, Z.
Effect of Co-Doping on the Magnetic Ground State of the Heavy-Fermion System CeCu_{2}Ge_{2} Studied by Neutron Diffraction. *Magnetochemistry* **2023**, *9*, 115.
https://doi.org/10.3390/magnetochemistry9050115

**AMA Style**

Tripathi R, Khalyavin D, Sharma S, Adroja DT, Hossain Z.
Effect of Co-Doping on the Magnetic Ground State of the Heavy-Fermion System CeCu_{2}Ge_{2} Studied by Neutron Diffraction. *Magnetochemistry*. 2023; 9(5):115.
https://doi.org/10.3390/magnetochemistry9050115

**Chicago/Turabian Style**

Tripathi, Rajesh, Dmitry Khalyavin, Shivani Sharma, Devashibhai Thakarshibhai Adroja, and Zakir Hossain.
2023. "Effect of Co-Doping on the Magnetic Ground State of the Heavy-Fermion System CeCu_{2}Ge_{2} Studied by Neutron Diffraction" *Magnetochemistry* 9, no. 5: 115.
https://doi.org/10.3390/magnetochemistry9050115