# Negative Magnetoresistance in Hopping Regime of Lightly Doped Thermoelectric SnSe

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}and B

^{2}resistivity dependence is observed, posing a challenge to theoretical comprehension of the underlying physical mechanism.

## 1. Introduction

## 2. Materials and Methods

_{2}atmosphere and slowly flown through a silica capillary into the carbon-coated fused silica ampoule prior to adding the untreated Se granules. A Se excess of 0.05 at.% was chosen instead of exact stoichiometry. The tapered ampoule of 13 mm inner diameter containing a total mass of 29.05 g was evacuated and fused once it reached a nominal pressure of 10

^{−6}mbar. The ampoule was slowly heated in a vertical Bridgman furnace to a temperature slightly above the melting point of 873.7 °C [27] to allow the reaction. After 24 h rest period for homogenization, the ampoule was lowered through a temperature gradient of approx. 12 K/cm using a rate of 0.5 mm/h and finally cooled down to room temperature. The obtained ingot (see Supplementary Materials, Figure S1) was single-crystalline with the cleavage plane almost parallel to the growth direction (see Supplementary Materials, Figure S2). Cutting of the bar-shaped samples with dimensions of ∼2 × 2 × 8 mm

^{3}with the long axis aligned along three perpendicular crystal directions was performed by a wire saw (WS 22, KD UNIPRESS, Warszawa, Poland) using a 50 µm thick tungsten wire and boron carbide powder (800 mesh) in a glycerol suspension to minimize surface damage of the samples. Adjusting the crystallographic orientation was performed using X-ray Laue backscattering technique and a special adapter to transfer the orientation to the wire saw. On as-cut surfaces, sharp reflection spots of the Laue images indicated the high structural quality of the single crystal as well as the gentle cutting process (see Supplementary Materials, Figure S3).

_{α1}radiation using SnSe from the tip of the present crystal with the FullProf program package [30] yielded room temperature cell parameters of a = 11.498(1) Å, b = 4.152(1) Å, and c = 4.446(1) Å. For further details, please refer to the Supplementary Materials, Figure S4.

## 3. Results

#### 3.1. Specific Heat

#### 3.2. Electronic Transport

## 4. Discussion

#### Magneto-Transport

_{3}, Bi

_{2}O

_{2}Se, and SrTiO

_{3}[24,25,49], which are characterized by vanishingly small Fermi surfaces where umklapp-like electron–electron scattering is supposedly not possible. Further theoretical work is needed to fully understand this phenomenon. Likewise, we cannot support a simple Fermi liquid scenario here due to the same reason, i.e., the small Fermi surface pockets. Additionally, Kohler’s rule, which is one important benchmark of a simple Fermi liquid, is not satisfied here. According to Kohler’s rule, the magneto resistivity coefficient should scale as ${T}^{-4}$ [50] which is not observed here, as demonstrated in Figure 7c. This was already indicated in the literature at the slightly higher doping value [20]. Instead, other scattering mechanisms, such as phonon-mediated electron–electron scattering [51], may be worth considering.

## 5. Conclusions

## Supplementary Materials

_{α1}radiation). The agreement values are R

_{p}= 10.7, R

_{wp}= 10.2, R

_{exp}= 5.71 and χ

^{2}= 3.20.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The low-temperature molar specific heat of SnSe in a $c/T$ versus ${T}^{2}$ plot. The red line represents a fit to the data in the region between 2 K and 3 K. The values are calculated per mol of Sn

_{0.5}Se

_{0.5}formula unit.

**Figure 2.**The temperature-dependent electrical resistivity of SnSe along three orthogonal crystallographic directions a, b, and c. Green lines represent fits to the model described by relation (4).

**Figure 3.**The magnetoresistance measured at different temperatures up to 14 T in case of (

**a**) electric current along c-axis and magnetic field along a-axis, and (

**b**) electric current along a-axis and magnetic field along c-axis. (

**c**) The Hall coefficient measured in filed up to 2 T for electric current and field orientations presented in panels (

**a**) and (

**b**).

**Figure 5.**Electrical resistivity presented in the (

**a**) $\mathrm{Ln}\left(\rho \right)\mathrm{vs}.{T}^{-\frac{1}{3}}$ plot, (

**b**) $\mathrm{Ln}\left(\rho \right)\mathrm{vs}.{T}^{-1}$, and (

**c**) $\rho \mathrm{vs}.{T}^{2}$. Green lines are guides for the eye.

**Figure 6.**(

**a**,

**b**) Magnetoresistance in low field ${B}^{2}$ regime along different crystallographic directions. Lines represent fits to the experimental data up to 0.4 T. (

**c**) Temperature scaling of magnetoresistance coefficient. Green lines are results of the fit.

**Figure 7.**(

**a**,

**b**) Magnetoresistance in metallic regime along different crystallographic directions. Purple line represents ${B}^{2}$ dependence. (

**c**) Temperature scaling of magnetoresistance coefficient. Green lines are results of the fit to ${T}^{-2}$ dependence.

**Table 1.**Parameters of the fit for different regimes obtained from relations (1) to (3), while parameters obtained by relation (4) are given in italics for comparison.

Crystal Direction | ${\mathbf{\rho}}_{0}\left(\Omega cm\right)$ | ${T}_{0}\left(K\right)$ | ${\mathbf{\rho}}_{1}\left(\Omega cm\right)$ | ${E}_{a}/{k}_{B}\left(K\right)$ | ${\mathbf{\rho}}_{2}\left(\Omega cm\right)$ | ${A}_{2}\left(\mathsf{\mu}\Omega cm/{K}^{2}\right)$ |
---|---|---|---|---|---|---|

a | 0.108 | 1.6 | 0.024 | 40.7 | −0.007 | 3.02 |

a * | 0.163 | 1.3 | 0.078 | 45.3 | −0.064 | 2.90 |

b | 0.054 | 2.1 | 0.015 | 36.4 | 0.022 | 1.68 |

b * | 0.047 | 2.2 | 0.0037 | 75.9 | 0.013 | 1.74 |

c | 0.048 | 1.4 | 0.015 | 34.4 | 0.015 | 1.49 |

c * | 0.043 | 1.5 | 0.0057 | 75.2 | 0.063 | 1.53 |

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**MDPI and ACS Style**

Zorić, M.; Dhami, N.S.; Bader, K.; Gille, P.; Smontara, A.; Popčević, P.
Negative Magnetoresistance in Hopping Regime of Lightly Doped Thermoelectric SnSe. *Materials* **2023**, *16*, 2863.
https://doi.org/10.3390/ma16072863

**AMA Style**

Zorić M, Dhami NS, Bader K, Gille P, Smontara A, Popčević P.
Negative Magnetoresistance in Hopping Regime of Lightly Doped Thermoelectric SnSe. *Materials*. 2023; 16(7):2863.
https://doi.org/10.3390/ma16072863

**Chicago/Turabian Style**

Zorić, Marija, Naveen Singh Dhami, Kristian Bader, Peter Gille, Ana Smontara, and Petar Popčević.
2023. "Negative Magnetoresistance in Hopping Regime of Lightly Doped Thermoelectric SnSe" *Materials* 16, no. 7: 2863.
https://doi.org/10.3390/ma16072863