#
Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin

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## Abstract

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## 1. Introduction

## 2. Weak-Antilocalization (WAL) Effect

#### 2.1. Electronic Motion in the Quantum Diffusive Regime

#### 2.2. WAL in Relevant Materials

## 3. WAL in Bi_{2}Se_{3} Thin Films

#### 3.1. Growth Methods

#### 3.2. Magnetotransport Properties and WAL Effect in Bi_{2}Se_{3} Thin Films

## 4. Remarks and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Scheme of a Dirac cone at the surface of a Topological Insulator (TI) showing the spin-momentum locking (spin orientations are indicated by red arrows). (

**b**) Angle-Resolved Photoemission Spectroscopy (ARPES) image of the ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ band structure showing the Dirac cone at the center of the Brillouin zone. Reprinted with permission from Reference [7]. Copyright 2010 American Physical Society.

**Figure 2.**Schemes showing the band structure and the different contributions to transport in TIs: (

**a**) n-type conduction with parallel contributions of the surface and the bulk. (

**b**) Topological regime with pure surface transport, also indicating the possibility of bulk presence due to thermal activation. (

**c**) p-type conduction with surface and bulk contributing. Blue and yellow regions in the bands indicate electron and hole populations, respectively.

**Figure 3.**Schematic representation of the unit cell of rhombohedral ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}\text{}$(blue: Bi, violet: Se1, orange: Se2) showing the Quintuple Layer (QL) arrangement and the c-axis direction.

**Figure 4.**Sketch representing the movement of electrons through scattering centers: (

**a**) two possible paths (1 and 2) for an electron going from A to B. (

**b**) A loop formed by time reversal partners.

**Figure 5.**Simulations within the HLN model of the magnetoconductance $\u2206{\mathrm{G}}_{\mathrm{xx}}\left(\mathrm{B}\right)$ for different values of ${\mathrm{l}}_{\mathsf{\phi}}$ at fixed $\mathsf{\alpha}$: (

**a**) a single coherent channel $\mathsf{\alpha}=-1/2$. (

**b**) Two independent channels $\mathsf{\alpha}=-1$.

**Figure 6.**(

**a**) Hall resistance against magnetic field for ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ films with different thicknesses (indicated in QLs on the right part of the plot) showing the non-linear dependence. (

**b**) Carrier densities obtained from the SdH oscillations against thickness. (

**c**) Mobilities obtained from the two-band analysis against thickness. In (

**b**) and (

**c**), the blue circles represent the bulk data $\left({\mathrm{n}}_{\mathrm{b}},{\text{}\mathsf{\mu}}_{\mathrm{b}}\right)$ and the red squares represent the surface data $\left({\mathrm{n}}_{\mathrm{s}},{\text{}\mathsf{\mu}}_{\mathrm{s}}\right)$, where empty and filled squares represent the two different surfaces. (

**d**) ${\mathrm{G}}_{\mathrm{s}}/{\mathrm{G}}_{\mathrm{tot}}$ against film thickness. (

**e**) 2D Magnetoconductance for different film thicknesses. (

**f**) Value of $\mathsf{\alpha}$ against film thickness. Inset in (

**e**) shows schematic energy bands above and below the critical thickness. Data were taken at $\mathrm{T}=1.6\text{}\mathrm{K}$. Reprinted with permission from Reference [13]. Copyright 2012 American Physical Society.

**Figure 7.**Transport data of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ thin films against thickness: (

**a**) conductance, (

**b**) carrier density, (

**c**) mobility. The solid straight lines are guides for the eyes. The subscript ${\mathrm{SC}}_{1}$ corresponds to the topological surface states whereas the subscript ${\mathrm{SC}}_{2}$ corresponds to the 2DEG. (

**d**) Magnetoconductance for different thicknesses. (

**e**) ${\mathrm{l}}_{\mathsf{\phi}}$ against thickness. (

**f**) $\mathsf{\alpha}$ against thickness. Insets in (

**a**), (

**b**), and (

**c**) show the data for thinner films. Reprinted with permission from Reference [64]. Copyright 2012 American Physical Society.

**Figure 8.**Modulation of $\mathsf{\alpha}$: (

**a**) effect of negative gating voltage ${\mathrm{V}}_{\mathrm{G}}\text{}\mathrm{on}\text{}\mathsf{\alpha}$. The insets on the left and right are band diagrams showing the Fermi level modulation relative to the bands. (

**b**) Effect of Cu doping ($\mathrm{x}$ represent the Cu content) on $\mathsf{\alpha}.$ The numbers i and ii correspond to the coupled and decoupled case, respectively. Reprinted with permission from references [32,40]. Copyright 2011 and 2014 American Physical Society.

**Figure 9.**(

**a**) Mobility against thickness following the conventional dependence of bulk transport in films, which is indicated by the solid line. (

**b**) Magnetoresistance for lower thicknesses. (

**c**) Magnetoresistance for high thicknesses showing the ${\mathrm{B}}^{2}\text{}$ dependence characteristic of bulk dominance. Inset displays the low field region. (

**d**) Parameter $\mathsf{\alpha}$ against thicknesses. (

**e**) ${\mathrm{l}}_{\mathsf{\phi}}$ against thickness. Reprinted with permission from Reference [31]. Copyright 2011 American Physical Society.

**Figure 10.**(

**a**) Temperature dependence of resistivity for three ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ films with different thicknesses showing the metallic behaviour ($\mathrm{d}\mathsf{\rho}/\mathrm{dT}>0)$. (

**b**) Hall resistivity at 2 K against magnetic field for different thicknesses. (

**c**) Mobility at 2 K against film thickness. (

**d**) Normalized magnetoresistance at 2 K for different thicknesses. (

**e**) Normalized magnetoresistance for a 15-nm-thick film at different temperatures. (

**f**) Coherent transport parameters ${\mathrm{l}}_{\mathsf{\phi}}$ and $\mathsf{\alpha}$ against thickness at 2 K.

**Figure 11.**Representation of the phase coherence length ${\mathrm{l}}_{\mathsf{\phi}}$ versus mobility $\mathsf{\mu}$ values for different films. The dashed black lines are guides for the eyes. Red empty squares correspond to data obtained in our samples. The rest have been taken from literature: Red circles, [13]; blue square, [30]; light green upwards triangle, [31]; cyan left-pointing triangle, [42]; downwards magenta triangles, [54]; right pointing orange triangle, [56]; dark green hexagon, [64]; violet pentagon, [69]; maroon star, [70].

**Table 1.**Transport parameters of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ thin films reported in the literature.

Reference | t (nm) | n_{2D}${\left({10}^{13}\mathbf{c}{\mathbf{m}}^{-2}\right)}^{\text{}}$ | n $\left({10}^{19}\mathbf{c}{\mathbf{m}}^{-3}\right)$ | μ $\left(\mathbf{c}{\mathbf{m}}^{2}/\left(\mathbf{V}\mathbf{s}\right)\right)$ | T $\left(\mathbf{K}\right)$ | ${\mathbf{l}}_{\mathsf{\phi}}$ $\left(\mathbf{n}\mathbf{m}\right)$ | $\mathsf{\alpha}$ |
---|---|---|---|---|---|---|---|

[13] | 2–50 | - | 4 | 100–1200 | 1.6 | 100–1000 | −0.5 |

[14] | 20 | - | - | - | 0.3–10 | 80–300 | −1~−0.5 |

[30] | ≈38 | 1 | 390–880 | 1.8 | 306–968 | −0.52 | |

[31] | 1–100 | - | 0.1–6 | 70–1150 | 1.5 | 150–1000 | −0.6~0.5 |

[32] | 5–20 | 0.8–8.6 | - | 20–1000 | 1.2 | 143–∞ | −0.5 |

[42] | 10 | - | 6 | 472 | 0.4–10 | 150–870 | −0.6 |

[44] | 7 | 1.5 | - | - | 2.5 | 55–90 | −0.6~−0.2 |

[51] | 1–6 | 3.5 | - | 31–350 | 1.5 | 75–200 | −0.6~−0.3 |

[54] | 6–22 | - | 3.5–6.5 | 80–530 | 2 | 200–900 | −0.55~−0.35 |

[56] | 30 | - | 1.1 | 954 | 2 | 640 | −0.56 |

[57] | 9–54 | ~100 | - | - | 2–9 | 10–159 | −1.08~0.16 |

[64] | 10–245 | 3 | - | 500 | 1.5 | 750 | −0.6 |

[67] | 30–300 | 0.81–3.25 | - | - | 10 | 318–879 | −0.72~−0.34 |

[68] | 9.8–23 | 0.3 | - | - | 0.3–8 | 300–800 | −0.4 |

[69] | 5 | - | 3 | 27 | 2–20 | 8–20 | −0.5 |

[70] | 12 | - | 4.6 | 37 | 1.6–6 | 65–110 | −0.7~−0.6 |

Our data | 15–60 | 7–50 | 7–50 | 50–150 | 2 | 120–325 | ~−0.5 |

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**MDPI and ACS Style**

Gracia-Abad, R.; Sangiao, S.; Bigi, C.; Kumar Chaluvadi, S.; Orgiani, P.; De Teresa, J.M.
Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin. *Nanomaterials* **2021**, *11*, 1077.
https://doi.org/10.3390/nano11051077

**AMA Style**

Gracia-Abad R, Sangiao S, Bigi C, Kumar Chaluvadi S, Orgiani P, De Teresa JM.
Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin. *Nanomaterials*. 2021; 11(5):1077.
https://doi.org/10.3390/nano11051077

**Chicago/Turabian Style**

Gracia-Abad, Rubén, Soraya Sangiao, Chiara Bigi, Sandeep Kumar Chaluvadi, Pasquale Orgiani, and José María De Teresa.
2021. "Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin" *Nanomaterials* 11, no. 5: 1077.
https://doi.org/10.3390/nano11051077