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Observation of long phase-coherence length in epitaxial La-doped CdO thin films

The search for long electron phase coherence length, which is the length that an electron can keep its quantum wave-like properties, has attracted considerable interest in the last several decades. Here, we report the long phase coherence length of ~ 3.7 micro meters in La-doped CdO thin films at 2 K. Systematical investigations of the La doping and the temperature dependences of the electron mobility and the electron phase coherence length reveal contrasting scattering mechanisms for these two physical properties. Furthermore, these results show that the oxygen vacancies could be the dominant scatters in CdO thin films that break the electron phase coherence, which would shed light on further investigation of phase coherence properties in oxide materials.


I. INTRODUCTION
The electron phase-coherence length and time are the length and time that an electron can travel before losing its phase coherence, and are among the most important quantum, wave-like properties for one electron [1,2]. In low-dimensional mesoscopic systems, the interference of the electron phases leads to very interesting physical phenomena, including Aharonov-Bohm oscillations, universal conductance fluctuation, weak localization effects, etc. [3][4][5][6]. The search for long electron phase-coherence length, which is critical for quantum-coherent electronic devices, has attracted considerable interest in the last several decades. One of the central issues of fundamental importance is to understand the underlying mechanisms responsible for the loss of phase coherence [1,2,7]. For example, several important scattering mechanisms including electron-electron interaction, electron-phonon scattering, and magnetic or spin-orbit scattering have been taken into account to explain the loss of phase coherence [2,7,8]. To experimentally probe the phase-coherence properties, weak localization, a quantum increase of the electron resistivity, has been extensively used in various systems including metals, semiconductors, superconductors, and the emergent Dirac materials [2,[7][8][9][10][11][12][13][14][15].
Recently, oxide electronics has attracted much scientific and technological attention since the discovery of the high-mobility two-dimensional electron gas (2DEG) in SrTiO3 and its heterostructures [16][17][18][19][20]. Despite the high mobility, the electron phase-coherence length is only ~160 nm at 1.3 K, and ~280 nm at 12 mK [21,22], which limits the potential use of oxide materials for quantum-coherent electronic devices.
Here, we report the observation of long phase-coherence length in lanthanum-doped cadmium oxide thin films (LaxCd1-xO) and the identification of the oxygen vacancies as the major phase-scattering sources in oxide thin films. A long phase coherence length ( ) of ~ 3.7 μm is observed in La0.04Cd0.96O thin film at 2 K via the weak localization measurement, which is comparable to the value reported in ultra-high-mobility GaAs/GaAlAs 2DEG [10,23,24].
Contrasting scattering mechanisms for the electron mobility and the phase coherence are identified via the systematical studies of the La doping and temperature dependences. We show that the oxygen vacancies could be the dominant scatters in CdO thin films that break the electron phase coherence.

II. EXPERIMENTAL DETAILS
The LaxCd1-xO thin films are epitaxially grown on (001) MgO substrates by co-evaporating La and Cd in diluted ozone using oxide molecular beam epitaxy (MBE-Komponenten GmbH; Octoplus 400) system with a base pressure lower than 1×10 -10 mbar. Prior to the film growth, the MgO substrates were annealed at 600 ºC for 1 h to clean the surface. Then ~5nm MgO buffer layer was grown by e-beam deposition with the substrate temperature holding at 250ºC and a growth rate of ~0.18 nm/min. This thin MgO buffer layer was used to achieve better crystalline quality LaxCd1-xO films. La and Cd elements were co-deposited in diluted ozone pressure of ~1×10 -6 mbar from La and Cd thermal effusion cells with the substrate temperature at 200ºC.
During growth, in situ reflective high energy electron diffraction (RHEED) is used to characterize the crystalline quality. To avoid degradation in air during the electron transport and phase-coherence measurements, an ∼5 nm MgO thin film is deposited as the capping layer. The electron transport properties of the LaxCd1-xO thin films were measured via the van der Pauw method using the Physical Properties Measurement System (PPMS; Quantum Design). La0.040Cd0.960O epitaxial thin film. Sharp RHEED patterns ( Fig. 1(b)) and x-ray diffraction results ( Fig. 1(c)) indicate good crystalline properties of the La0.040Cd0.960O thin film. The thickness of the film ~15 nm is determined from the fringe peaks around the CdO (002) main peak. The rootmean-square roughness is ~0.15 nm, obtained from an area of 2×2 µm 2 obtained using ex situ atomic force microscopy ( Fig. 1(c) inset). As shown in Fig. 1(d), the electron mobility ( ) of the La0.040Cd0.960O thin film increases as temperature decreases, while it remains almost constant below 50K. This behavior could be well reproduced by a scattering-dependent theoretical model, which includes scatterings from ionized impurity and longitudinal optical (LO) phonon [25]. The carrier concentrations exhibit little variation as the temperature decreases from 300 to 2 K, as shown in Fig. 1(e).

B. Long phase-coherence length in La0.040Cd0.960O
Weak localization measurements are carried out to investigate as a function of the temperature in this La0.040Cd0.960O thin film. Fig. 2(a) shows the representative magneto- conductance curves (open circles) at 2, 10, 30, and 50 K, respectively. The ratio of the resistance correction due to the weak localization over the total resistance (ΔR/R) increases as the temperature decreases, as shown in Fig. 2(b) inset. This temperature dependence is in good agreement with the theoretical expectation of ln (T) dependence of the ΔR (red dashed line) for weak localization [1,5,26,27]. The Hikami-Larkin-Nagaoka theory is used to extract , which successfully describes the conductance correction due to quantum interference in twodimensional systems [26]: where Δ ( ) is the magneto-conductance that is due to weak localization, e is the electron charge, α is the weak localization coefficient, ℏ is the reduced Plank constant, is the digamma function, and B is the magnetic field. The fitted curves (the solid lines in Fig. 2(a)) based on Eq.
(1) are in good agreement with our experimental results. As shown in Fig. 2(b), increases as the temperature decreases, and it reaches 3.7 µm at 2 K. Despite its extremely low electron mobility less than 350 cm 2 /Vs, the electron phase coherence length in La0.04Cd0.96O thin film is comparable to that in the ultrahigh-mobility GaAs/GaAlAs 2DEG and graphene (Table Ⅰ) [10, 13,14,23,24], which indicates that La0.04Cd0.96O thin film might be a promising candidate for future quantum-coherent electronic devices. Furthermore, the electron phase-coherence length is much longer than that in high-mobility SrTiO3 and its heterostructures (Table Ⅰ) [21,22], presenting the fundamental question of the phase-scattering mechanisms. 3.7

C. The role of La doping in the electron transport properties
To determine the physical origin for this long phase-coherence length, we systematically vary the La doping in these films and examine the role of La doping for the electron phase coherence. LaxCd1-xO thin films of ~15 nm with various La doping (x) from 0 to 0.062 are grown using the oxide molecular-beam epitaxy. RHEED and x-ray diffraction measurements indicate good crystalline properties for all these thin films except x=0.062 [25]. Fig. 3(a) shows the temperature-dependence of the mobility for these films, and all temperature dependent behaviors can be well explained by ionized impurity and LO phonon scatterings [25]. Fig. 3 cm -3 , which might be related to slightly poor crystal quality compared to other doping films [25].
The sheet resistance at 2 K exhibits a strong decrease as the La doping increases, as shown in the inset of Fig. 3(d).

D. The role of La doping in the electron phase coherence properties
The electron phase-coherence properties are investigated in these LaxCd1-xO thin films using the weak localization measurements. Fig. 4(a) shows the representative magneto-conductance curves at 2 K (open circles) for La doping with x=0, 0.008, 0.012, and 0.040, respectively.
Clearly, a narrower dip is observed for a higher La doping, indicating a longer . To quantitatively determine as a function of the La doping, we numerically fit the experimental results based on Eq. (1), indicated by solid lines in Fig. 4(a). The obtained at 2 K is summarized in Fig. 4 5(b)). For the LaxCd1-xO films with x=0 and 0.003, increases as the temperature decreases from 50 to ~10 K, and then it saturates below ~ 10 K (Fig. 5(a)), whereas, for the LaxCd1-xO films with x=0.040 and 0.028, increases monotonically as the temperature decreases from 50 to 2 K (Fig. 5(b)). is estimated from the electron phase-coherence length and the diffusion constant (D) based on the relation = . Meanwhile, the diffusion constant can be obtained from the relation [37]: where is the Fermi velocity, is the mean-free path, RS is the sheet resistance, and * is the effective electron mass, which is ~0.21me (me: free electron mass) for the La-doped CdO thin films in our study [38]. Detailed information about the transport properties, the sheet resistance, mean-free path, 2D carrier density, and Fermi velocity can be found in the Supplemental Materials [25]. For the LaxCd1-xO films with x=0 and 0.003, increases as the temperature decreases from 50 to ~10 K, and then it saturates below ~10 K (Fig. 5(c)). The saturation of indicates the presence of strong phase-scattering centers. For the LaxCd1-xO films with x=0.028 and 0.040 (Fig. 5(d)), increases monotonically as the temperature decreases from 20 to 2 K following the relation ~T -1 (red dashed line), which could be attributed to electron-electron scattering mechanism [1,2,39,40]. When the temperature is higher (20 -50 K), the temperature dependence of follows the relation ~T -2 (magenta dashed line), which could be attributed to electron-phonon scattering or large-energy-transfer electron-electron scattering mechanisms [2,32]. The crossover of from the linear to quadratic temperature dependence is at ~ 20 K [25].
One major difference between the low-doping films and the high doping films is that carriers have different origins. Oxygen vacancies are the main source of carriers in relatively low-doping films, while most of the carriers in relatively high-doping films come from the La 3+ replacing Cd 2+ . Oxygen vacancies could be the scattering centers that could break the electron phase coherence. For example, for nondoped CdO film (x=0), a tiny weak anti-localization feature is indeed observed below 10 K (Fig. 5(e)), while it is not shown for higher doping of La (Fig. 5(f)).
The weak anti-localization results for the nondoped CdO film indicate the presence of strong spin-orbit scattering centers that break the phase coherence. This observation agrees well with the saturation of the phase-coherence time and length below 10 K in Fig. 5(a) and 5(c). It also explains the slight increase of the magneto-conductance for higher temperature in Fig. 5(e).
Furthermore, post-annealing in oxygen of the non-doped CdO film is performed to decrease the oxygen vacancies. An increase of the phase-coherence length and the mobility is observed, which further supports that the oxygen vacancies are the major phase-coherence scattering centers [25].

E. Electron phase-coherence time vs momentum scattering time
Our results show that the electron mobility and electron phase coherence exhibit totally different behaviors. First, the temperature dependence of these two physical properties are different. The electron mobility saturates below the temperature of ~ 50 K, whereas, the electron phase-coherence length monotonically increases as the temperature declines from 50 to 2 K.
Besides, these two physical properties show contrasting La doping dependence. In the doping region of La from 0 to 0.012, the electron mobility increases from ~150 to ~550 cm 2  Clearly, these observations indicate the mechanisms that determine the electron mobility, one of electron's "particle-like" properties, and the phase coherence, one of electron's "quantum wavelike" properties, are different. To be specific, the electron mobility is determined by the elastic-scattering process, while and are related to the inelastic-scattering process. In the doping region of La from 0.011 to 0.040, the elastic scattering increases due to the La doping as impurities, which results in the decreasing of the electron mobility. However, the decrease of oxygen vacancies by La doping results in the large enhancement electron phase-coherence lengths and times. These results suggest that oxygen vacancies in La-doped CdO with low La doping can break the electron phase coherence and could be the dominant phase-coherence scattering sources in general oxide materials.

IV. CONCLUSION
In summary, we observe a long phase-coherence length of ~3.7μm in La0.04Cd0.96O thin films at 2 K via the weak localization measurement, which is comparable to the value reported in ultrahigh-mobility GaAs/GaAlAs 2DEG [10,23,24]. Contrasting scattering mechanisms for the electron mobility and the phase coherence are identified via the systematical study of the La doping and temperature dependences. The oxygen vacancies are identified to be the dominant scattering sources in CdO thin films that break the electron phase coherence, which provides a unique insight for further investigation of phase-coherence properties in all oxide materials.

S1. Temperature dependence of the electron mobility
The temperature dependent electron mobility can be attributed to ionized impurity and longitudinal optical (LO) phonon scattering mechanisms, as described in the equation below following previous work on SrTiO3 [1,2]: where is the electron mobility due to ionized impurity, and is the electron mobility due to LO phonon scattering, which has a strong temperature dependence, which could be described with a polaron model as previously derived by Low and Pines (42): where is the electron-phonon coupling constant, ℏ is the LO phonon mode energy for scattering, ℏ is the reduced Plank constant, is the electron charge, is the polaron mass, and is the electron effective mass, ( ) is a slowly varying function ranging from 1.0 to 1.4, is Boltzmann constant, and T is the temperature. As shown in Fig. S1, the red solid line shows the fitting results based on equation (S1) and The (002) main peak of CdO shifts leftwards as La doping increases ( Fig. S3(a)). The c-axis lattice parameter could be derived based on the XRD results ( Fig. S3(b)). For the whole doping region, a tiny variation is observed ( . . . × 100% = 0.5%). Hence, the structural defectinduced elastic scattering can be ruled out to explain the variation of the phase coherence properties. S3. Temperature dependence of the sheet resistance, mean free path, and Fermi velocity is estimated from the electron phase coherence length and the diffusion constant (D).
During the calculation, detailed transport properties, the sheet resistance, mean free path, 2D carrier density, and Fermi velocity are shown in Fig. S4.

S4. Post-annealing in oxygen gas
Regarding the role of the oxygen defects on the mobility and the phase coherence length, we have performed additional study on the non-doped films. By annealing the non-doped CdO film in oxygen gas at different temperatures, we observe an increase of both the electron mobility and phase coherence length, as shown in Fig. S5. i.e. electron-phonon scattering or large-energy-transfer electron-electron scattering) for the dephasing rate. The crossover of from the linear to quadratic temperature dependence is at ~ 20 K.