# Negative Magnetoresistivity in Highly Doped n-Type GaN

^{1}

^{2}

^{*}

## Abstract

**:**

_{φ}for heavily doped n-type GaN. The obtained τ

_{φ}value is proportional to T

^{−1.34}, indicating that the electron–electron interaction is the main dephasing mechanism for the free carriers.

## 1. Introduction

^{17}to mid-10

^{19}cm

^{−3}. However, in view of some structural problems occurring in heavily doped films, germanium (Ge) became, over time, the second competitor of n-type doping. While doping GaN to low and intermediate concentrations using Si and Ge has become routine, compensation mechanisms are activated under very high donor doping, limiting the maximum electron concentration achievable with either dopant. The metal-insulator transition in GaN occurs at the critical concentration of uncompensated donors equal to approximately 1.6 × 10

^{18}cm

^{−3}[1]. So, n-type GaN material, with free carrier concentrations higher than this critical value can be considered as a disordered system because of the high concentration of dopant atoms randomly distributed in the lattice.

_{H}≥ 8 × 10

^{17}cm

^{−3}. The study was carried out for materials obtained with the use of five different crystal growth technologies and doped with different donor dopants. It was found that all the samples under study exhibit nMR at a low temperature (10 K < T < 50 K) and for some of them this effect could be observed up to unexpected high temperatures (up to 100 K). For the first time, the free carriers scattering mechanisms in GaN were analyzed by applying the weak-localization theory. An understanding of the conductivity processes in a highly doped material is essential for device development.

## 2. Experimental Methods and Room Temperature Characterization

_{2}ambient at 1000 K. For low-temperature measurements a He-free cryostat enabling measurements at temperatures ranging from 10 K was used. The temperature was stabilized with a precision better than 0.1 K. The resistivity measurements were performed using the van der Pauw method taking the average of all current configurations. Likewise, the van der Pauw approach [24] was used for the Hall Effect experiments. During all the electrical measurements, the current I

_{s}through the sample was kept adequately low to ensure ohmic conditions. As an example of typical behavior, Figure 1 presents the results of the Hall Effect (Hall voltage V

_{H}) measurements as a function of magnetic field at different temperatures for sample H1. The Hall resistance R = V

_{H}/I

_{s}vs. B dependences at T = 10.6 K, T = 30 K, T = 50 K and at room temperature (T = 290 K) are practically identical. This dependence on temperature is characteristic for highly doped material [23] and indicated that for the all samples under study; the conduction process, in the whole investigated temperature range, is related to only one type of carriers, i.e., the free electrons in the conduction band.

## 3. Magnetoresistivity as a Function of Temperature

^{−4}(T

^{−2}). Taking into account the mobility of sample M5 at T = 100 K equal to 175 cm²/Vs, the fitting result corresponds to A = 1.35, which is an expected value between an ionized impurity scattering mode and acoustic phonon modes.

_{o}and inelastic scattering time τ

_{φ}(phase coherence time after which the coherent backscattering disappears). Using these two parameters, Kawabata derived an expression for the variation of the conductivity as a function of magnetic field in the absence of spin-orbit and magnetic interactions. Depending on the parameter δ defined as:

_{ε}elastic and inelastic mean free path, respectively, he proposed two possible types of behavior of Δσ(B,T) [10]:

_{o}is the conductivity in the absence of magnetic field, e is the electron charge, m*—the carrier effective mass (here m* = 0.22 m

_{o}), and ℏ, π have their usual meanings. In the case where Equation (4) may be applied, the experimental results can be analyzed treating the phasing coherence time τ

_{φ}as the fitting parameter. The experimentally observed low-temperature changes in the conductivity with the magnetic field Δσ(B) = σ(B) − σ

_{o}for different carrier concentrations are presented in Figure 3. Although the investigated nMR effect is weak (in the best case it does not exceed 1%), the observed trends do not raise any doubts. As can be seen from the figure, the criteria that allow for applying Equations (3) or (4) to describe the conductivity corrections Δσ(B) strongly depend on the carrier concentration in the sample.

_{H}> 1 × 10

^{20}cm

^{−3}(sample P6), the experimental dependence Δσ(B) is perfectly described by the universal Equation (3) with the magnetic field dependence~√B. However, as the carrier concentration is lowered, this dependence changes its character so that for n

_{H}~1 × 10

^{19}cm

^{−3}the Δσ(B) dependence becomes similar to a parabolic dependence (Δσ~B²) and consequently can be analyzed under Equation (4).

_{φ}. For these samples, the magnetoresistance data vs. magnetic field were fitted using Kawabata’s Equation (4), treating the phasing coherence time as the fitting parameter. As an example, Δσ(B) for sample M4 with carrier concentration n

_{H}= 6.9 × 10

^{19}cm

^{−3}is presented in Figure 5. While the results at low temperature do not allow for the fitting of the parabolic dependence of Δσ(B) (Equation (4)), this condition is met by increasing the temperature.

_{φ}(T) determined this way for several samples are presented in Figure 6. In Figure 6a are presented the results for MOVPE, PAMBE, and HVPE samples. In the temperature range T > 30 K, the best power-law fit of this set of data is of the form:

_{φ}(T) = G × 10

^{−11}× T

^{−1.34}(s)

_{φ}(T)∝ T

^{−0.85}. The results for Am and MFS samples are presented in Figure 6b. For these samples the phasing coherence time τ

_{φ}could be determined at a lower temperature range T ≤ 40 K and its temperature dependence can be described by:

_{φ}(T) = 2 × 10

^{−11}× T

^{−0.85}(s),

_{φ}was determined only for one sample (P15) and in the highest temperature range T > 60 K. This may indicate that the surrounding of the Si dopant is different than in the case of Ge or O doping.

_{φ}(T) ∝ T

^{–}

^{p}allows for identifying the scattering mechanism dominating in the sample. The electron-phonon scattering τ

_{e-ph}and the electron-electron scattering τ

_{e}

_{-e}are the two main mechanisms responsible for phase-breaking [13]. In the case of the electron–phonon interaction, the standard theoretical concept scales τ

_{e}

_{-ph}with temperature as T

^{−3}. In the presence of strong impurity scattering, however, the situation is less straightforward and different values of the exponent of temperature p, ranging from p = 2 to p = 4, have been predicted. In particular, the T

^{−2}dependence of the relaxation rate is widely observed in experiments [13]. As can be seen in Figure 6, the best power-law fit of the experimental results correspond to τ

_{φ}∝ T

^{−1.34}. This value of the temperature exponent is significantly lower than (p ≈ 2÷4) established for the electron–phonon interaction. However, this exponent value is close to τ

_{e}

_{-e}∝ T

^{−3/2}, the formula for electron–electron interaction in the case of heavily doped material. This scattering mechanism could also explain the observed decrease of the exponent p as the temperature decreases below 30 K. In this temperature range, the phasing coherence time τ

_{φ}could only be measured in the less doped samples, and according to paper [27] for 3D systems approaching a metal-insulator transition a weaker temperature dependency τ

_{φ}∝ T

^{−1}can be expected. A similar effect of decreasing of the exponential term with decreasing temperature was also observed for the GaN nanowall network [16].

_{φ}can be written as:

_{φ}=D +1/τ

_{i}

_{i}is the relevant inelastic electron scattering time in question. As is evident from this equation, in the higher temperature range (1/τ

_{i}> D) the temperature dependence of τ

_{φ}is controlled by the temperature dependence of τ

_{i}, while D determines the saturated value of the dephasing time in the limit of very low temperatures. This effect can be related to the inhomogeneities in the samples, which may be of particular importance near the Mott transition.

## 4. Conclusions

_{φ}was determined for 3D electron gas in GaN. The determined values are coherent with the results for other 3D semiconductor materials.

_{φ}∝ T

^{−1.34}agrees well with that predicted by theories based on electron–electron interactions. This result is very surprising and unexpected because at such high temperatures the role of electron phonons scattering should be significant and even dominant.

_{φ}∝ T

^{−0.85}was observed, which can be the signature for 3D systems approaching a metal-insulator transition. To study this phenomenon in more detail, it would be advisable to extend the measurements to the lowest possible temperature range. If it turns out that by lowering the temperature, the relaxation time tends to a constant value, conclusions can be made regarding disorder in the system. Then, the nMR phenomenon could be used to estimate the quality of the material.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The Hall resistance R = V

_{H}/I

_{s}vs. magnetic field B at T = 10.6 K, 30 K, 50 K and at room temperature (T = 290 K). Results for sample H1.

**Figure 2.**Relative variation of resistivity Δρ(B)/ρ(0) vs. magnetic field B. (

**a**) Results for sample M5, respectively, at T = 100 K (circles) and T = 11 K (diamonds). (

**b**) Zoom on 100 K results. Solid line: Gauss model Δρ(B)/ρ(0) = β × B² with the fitting parameter β = 4.1 × 10

^{−4}(T

^{−}²).

**Figure 3.**Δσ vs. magnetic field B. Results at T = 11 K for samples with different, increasing carrier concentration: -n

_{H}= 3.1 × 10

^{18}cm

^{−3}(diamonds, sample H6); -n

_{H}= 1.1 × 10

^{19}cm

^{−}

^{3}(stars, sample H8); -n

_{H}= 1.2 × 10

^{20}cm

^{−}

^{3}(circles, sample P6). Solid line—Kawabata’s universal Equation (3).

**Figure 4.**Relative variation of conductivity σ

_{o}(T)/σ

_{o}(T

_{min}) vs. temperature. Results for samples with different, increasing carrier concentrations: -n

_{H}= 3.1 × 10

^{18}cm

^{−3}(diamonds, sample H6); -n

_{H}= 1.1 × 10

^{19}cm

^{−}

^{3}(stars, sample H8); -n

_{H}= 1.2 × 10

^{20}cm

^{−}

^{3}(circles, sample P6).

**Figure 5.**Δσ vs. magnetic field B at different temperature. Results for sample M4 with n

_{H}= 6.9 × 10

^{18}cm

^{−3}. Solid lines—fitting procedure. (

**a**): complete set of results. For T < 30 K, the parabolic equation for Δσ(B) dependency cannot be fitted. (

**b**): The fitting procedure can be applied to the high-temperature part of the results with the fitting parameter τ

_{φ}equal to: T = 30 K: τ

_{φ}= 1 × 10

^{−12}s; T = 50 K: τ

_{φ}= 5.3 × 10

^{−13}s; T = 60 K: τ

_{φ}= 4 × 10

^{−13}s; T = 70 K: τ

_{φ}= 3.3 × 10

^{−13}s.

**Figure 6.**The phasing coherence time τ

_{φ}derived from the nMR data versus temperature for different samples. (

**a**): MOVPE samples: ∇ M2; Δ M3; ◊ M4; PAMBE samples: ☆ P6; ○ P15; HVPE samples: ◁ H1; □ H3; The data points are given with dashed and dot parallel lines as a guide to the eye. Solid line, fitting for ∇ and Δ samples: τ

_{φ}(T) = 6.2 × 10

^{−11}× T

^{−1.34}(s). (

**b**): Am samples: ○ A1; ◊ A2; □ A4; Δ A5; ∇ A6; ◁ A7; A8; MFS samples: Ψ F1; F2; ⊕F3; ◊ F4; Solid line: τ

_{φ}(T) = 2 × 10

^{−11}× T

^{−0.85}(s).

**Table 1.**Sample characteristics: Hall concentration at 300 K n

_{H}(300 K) = 1/(eR

_{H}) and Hall mobility µ

_{H}= 1/(eρ n

_{H}) at 300 K and 10 K respectively.

Dopant | N° | n_{H}cm ^{−3} | µ_{H} (300 K)cm²/Vs | µ_{H} (10 K)cm²/Vs |
---|---|---|---|---|

Ge | M1 | 2.0 × 10^{18} | 276 | 108 |

M2 | 2.8 × 10^{18} | 230 | 112 | |

M3 | 4.1 × 10^{18} | 216 | 119 | |

M4 | 6.9 × 10^{18} | 196 | 131 | |

M5 | 7.3 × 10^{18} | 202 | 135 | |

M9 | 4.4 × 10^{19} | 153 | 140 | |

M10 | 1.3 × 10^{20} | 120 | 121 | |

Ge | H1 | 4.1 × 10^{18} | 184 | 85 |

H2 | 8.8 × 10^{18} | 192 | 110 | |

H3 | 1.6 × 10^{19} | 127 | 109 | |

H4 | 6.1 × 10^{19} | 107 | 103 | |

Si | H5 | 1.3 × 10^{18} | 297 | 108 |

H6 | 3.1 × 10^{18} | 246 | 80 | |

H7 | 7.9 × 10^{18} | 144 | 86 | |

H8 | 1.1 × 10^{19} | 153 | 108 | |

Ge | P3 | 1.7 × 10^{18} | 186 | 33 |

P6 | 1.2 × 10^{20} | 88 | 88 | |

P10 | 5.3 × 10^{20} | 47 | 50 | |

P11 | 7.9 × 10^{20} | 33 | 31 | |

Si | P12 | 8.3 × 10^{19} | 140 | 142 |

P15 | 2.7 × 10^{19} | 97 | 88 | |

O | A1 | 5.2 × 10^{18} | 164 | 91 |

A2 | 8.1 × 10^{17} | 336 | 9 | |

A3 | 1.4 × 10^{19} | 122 | 101 | |

A4 | 1.1 × 10^{18} | 249 | 17 | |

A5 | 2.9 × 10^{19} | 129 | 125 | |

A6 | 8.0 × 10^{18} | 187 | 140 | |

A7 | 2.0 × 10^{19} | 153 | 136 | |

A8 | 4.4 × 10^{19} | 123 | 119 | |

O | F1 | 7.4 × 10^{19} | 77 | 78 |

F2 | 5.7 × 10^{19} | 65 | 65 | |

F3 | 9.9 × 10^{19} | 56 | 55 | |

F4 | 1.2 × 10^{20} | 49 | 56 |

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**MDPI and ACS Style**

Konczewicz, L.; Iwinska, M.; Litwin-Staszewska, E.; Zajac, M.; Turski, H.; Bockowski, M.; Schiavon, D.; Chlipała, M.; Juillaguet, S.; Contreras, S. Negative Magnetoresistivity in Highly Doped n-Type GaN. *Materials* **2022**, *15*, 7069.
https://doi.org/10.3390/ma15207069

**AMA Style**

Konczewicz L, Iwinska M, Litwin-Staszewska E, Zajac M, Turski H, Bockowski M, Schiavon D, Chlipała M, Juillaguet S, Contreras S. Negative Magnetoresistivity in Highly Doped n-Type GaN. *Materials*. 2022; 15(20):7069.
https://doi.org/10.3390/ma15207069

**Chicago/Turabian Style**

Konczewicz, Leszek, Malgorzata Iwinska, Elzbieta Litwin-Staszewska, Marcin Zajac, Henryk Turski, Michal Bockowski, Dario Schiavon, Mikołaj Chlipała, Sandrine Juillaguet, and Sylvie Contreras. 2022. "Negative Magnetoresistivity in Highly Doped n-Type GaN" *Materials* 15, no. 20: 7069.
https://doi.org/10.3390/ma15207069