Evidence of weak antilocalization in epitaxial TiN thin films

Journal Pre-proofs Research articles Evidence of weak antilocalization in epitaxial TiN thin films Siddharth Gupta, Ritesh Sachan, Jagdish Narayan PII: DOI: Reference:

S0304-8853(19)32884-7 https://doi.org/10.1016/j.jmmm.2019.166094 MAGMA 166094

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Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

16 August 2019 25 September 2019 1 November 2019

Please cite this article as: S. Gupta, R. Sachan, J. Narayan, Evidence of weak antilocalization in epitaxial TiN thin films, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166094

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Evidence of weak antilocalization in epitaxial TiN thin films Siddharth Gupta, 1, * Ritesh Sachan, 1, 2 and Jagdish Narayan1, * 1 Department 2

of Materials Science and Engineering, Centennial Campus

Materials Science Division, Army Research Office, Research Triangle Park, NC 27709, USA

North Carolina State University, Raleigh, NC 27695-7907, USA *Corresponding Author E-mail: [email protected] (Siddharth Gupta); [email protected] (Jagdish Narayan)

Keywords: Weak antilocalization, ferromagnetism, epitaxy, thin films, Raman spectroscopy, domain matching epitaxy, electron energy-loss spectroscopy

1

Abstract Defect engineering provides a tremendous opportunity to impart novel functionality to nanomaterials. This report is focused on TiN metallic system, where the spin structure and electron-transport are controlled by injecting nitrogen vacancies (VN). The TiN films are epitaxial, with the TiN/Al2O3 epitaxial relationship given by: (111) TiN//(0001) Al2O3 as out-of-plane, and <110>TiN//<1010>Al2O3 and <112>TiN//<1120> Al2O3 as in-plane, after 30o rotation. Epitaxy in such a large misfit system (~9.24%) is rationalized to arise via domain matching epitaxy (DME) paradigm. Following the report of room-temperature ferromagnetism[1] in TiN1-x films formed by injecting nitrogen vacancies, we provide direct experimental evidence of weak antilocalization (WAL) effects by plugging VN using nitrogen annealing of TiN films. This evidence with simultaneous loss of magnetization in nitrogen annealed TiN films is the tell-tale sign of VN acting as magnetically active defects in TiN, as their removal facilitates Berry’s phase formation and generation of time-reversal symmetry. Through detailed EELS and Raman analysis, we have explicitly shown the absence of Ti+2 polarons in TiN films. The resistivity minima in these films are attributed to the WAL effect with persistent log T behavior under 0-7 Tesla magnetic fields. The temperature-dependent coherence length analysis also highlights the emergence of WAL under the two-dimensional localization theory. The WAL effect in TiN is similar to topological insulators, quenching on the introduction of magnetically active defects, while stable against nonmagnetic defects. Our findings demonstrate the prime importance of nitrogen vacancies in tuning the magentotransport characteristics in epitaxial nitride films for optoelectronic device applications.

2

1. Introduction The magnetic properties of nanomaterials play a pivotal role in designing heterostructures for spinbased devices. The discovery of weak ferromagnetism in traditionally nonmagnetic materials by injection of magnetic nanoparticles has created an upsurge of interest to create multifunctional device architectures. These magnetic properties can manifest due to direct/indirect coupling and mediation via a host.[2, 3] More recently, due to rapid progress in spintronics and quantum information, it has become desirable to gain more tunable spin control in strongly correlated materials. Classically, there is a noticeable resistance minimum associated with metals on the introduction of dilute magnetic impurities. This occurrence was attributed to spin-polarized tunneling across grain boundaries, Kondo effect arising from spin disorder and quantum interference effects (QIE). In case of tunneling across grain boundaries, the resistivity minima are found to shift towards lower temperatures on the application of perpendicular magnetic field, subsequently disappearing on further increment in the magnetic field. The Kondo effect is triggered by the interaction of itinerant electrons with the magnetically-active impurities, resulting in a realignment of electron spins below the characteristic Kondo temperature.[4] This interaction restricts the free motion of itinerant electrons, thereby screening the associated magnetic moment of host impurity.[4] This scattering of conduction electrons is mediated via antiferromagnetic (AFM) interactions which trigger a spin-flip in the order of femtoseconds, resulting in the resistance minimum. Due to the susceptibility of magnetic impurities towards Kondo scattering, it has become a deterministic measure of the magnetic nature of impurity/defect in a conductor. This has led to a recent surge of interest in the role of Kondo scattering in weak-ferromagnetic (FM) systems resulting in cooperation between the two opposing effects of long-ranged FM ordering and Kondo screening in the Kondo lattice systems.[5],[4] Generally, QIEs lead to correction to the 3

resistivity from two different sources: (i) electron-electron (e-e) interaction and subsequent modification of the density of states (DOS) at the Fermi energy (EF); and (ii) weak localization (WL) effect arising from the self-interference of the wave pockets as they are backscattered coherently by the impurities or other defects. Notably, this approach of rendering materials ferromagnetic is limited by non-uniform dopant dispersion, resulting in short-range ferromagnetic ordering. [6, 7] TiN presents a compelling case for the study of magnetic impurity/defect interactions with the itinerant electron gas due to its inherent paramagnetism[8, 9] and high electrical conductivity. The high ratio of metal/non-metal radii imparts covalency to the TiN lattice, making it robust and stable as the composition varies from TiN1.2 up to TiN0.57.[10] As most transition-metal nitrides are metallic and non-magnetic, their potential applications are limited. However, there have been reports on the introduction of bandgap[11] and ferromagnetism upon doping metallic materials. Ferromagnetism in TiN can add a new dimension to their current applications as metallic interconnects[12], epitaxial diffusion-barrier buffer layers[13], and coatings.[14] Introduction of defects/dopants can generate unpaired spin states in metallic nitrides, making them suitable for integration in spin-based devices. However, spin injection by dopant introduction creates a significant drawback due to their precipitation and clustering, which inhibits the creation of FM ordering for longer-lasting magnetization-demagnetization cycles.[6, 7] The coupling of conduction electrons with localized magnetic moments, a central problem of condensed-matter physics, has not been realized in nitrides due to their magnetically inactive behavior. In an earlier report, we had shown that vacuum annealing in epitaxially grown TiN films results in ferromagnetic ordering with Curie temperature of ~700 K and saturation magnetization at absolute zero of 13.6 emu g-1. Although quantum interactions and ferromagnetic ordering in 4

non-magnetic nitrides have been a subject of active investigations, all the contributions to unpaired spin states and their quantum interactions were not considered comprehensively.[15-17] In this work, low-temperature transport characteristics of nitrogen annealed TiN films are investigated. These films exhibit persistent resistivity minimum under a magnetic field as high as 7 Tesla. Detailed NRG fits revealed the departure from Kondo scattering on nitrogen annealing, essentially proving the non-magnetic nature of remaining defects. In N2 atmosphere at high temperature, the formation of non-magnetic (VTi) defects (1.88 eV) is more energetically favorable as compared to VN (1.70 eV) due to the nitrogen-rich conditions.[18, 19] As both of these defect have high formation energy, prominently defect-free film growth occurs.

These static non-magnetic

impurities create time-reversal invariant perturbations, resulting in weak localization (WL) or weak anti-localization (WAL) characteristics depending on the spin-orbit coupling (SOC). In case of high SOC, the electron’s spin is arrested with momentum, resulting in a phase difference of 2π on time-reversal for backscattering trajectories. This 2π phase difference creates destructive interference, effectively suppressing the backscattering, leading to increased conductance for WAL behavior. The results provide direct evidence of weak antilocalization in TiN films, explicitly proving that VN are magnetically active defects in TiN and plugging them results in reentrant 𝜋 Berry’s phase. Moreover, the power-law fit of phase-coherent length versus temperature further confirms the WAL effect of our TiN films is within a two-dimensional (2D) localization paradigm. Our results thus call for revisiting the previously published data on the emergence of defect induced ferromagnetism and quantum interference effects in nitrides.[15, 17, 20-22]

5

2. Experimental section 2.1. Sample growth The Krypton Fluoride Excimer laser (λ=248 nm; 25 ns pulse duration) at an energy density of 3.0 J/cm2 was utilized to ablate a 99.99% pure TiN polycrystalline target. The depositions were limited to grow 40 nm thick films, under 1x10-6 Torr vacuum at 700oC, inside the pulsed laser deposition chamber. After film deposition, the thin films were annealed under exposure to 99.99% pure N2 at 𝑃𝑁2of 2 Torr for 30 mins with a gradual cool down to form stoichiometric TiN thin films. The films were steadily cooled to ambient conditions at 10 K/minute. The constraint of fabricating high-quality heteroepitaxial films is the high deposition temperature (700oC) required for forming single-crystalline epitaxial films. On thermal annealing under such conditions, the VN concentration has Arrhenius temperature dependence on growth temperature, making it near impossible to achieve films with discrete values of non-stoichiometry (X). At 700oC in vacuum, the formation energy(Ef) of VN is 0.91 eV, while that of VTi is 4.78 eV2,3 resulting in the nonstoichiometry of 0.07. On using lower growth temperatures, the films start exhibiting other defects and high-angle grain boundaries, while higher temperature growth increases surface reactivity of atoms at the substrate/film interface which can lead to formation of pseudomorphic TiNxOy layer.4 Extreme care was taken to avoid oxygen contamination by fabricating the films under ultra-high vacuum and ultrapure N2 atmospheric conditions.

6

2.2. Characterization The out-of-plane orientation of TiN was analyzed using X-ray diffraction (XRD) θ-2θ Rigaku diffractometer (CuKα source; λav =0.154 nm). The in-plane alignment and thin film growth characteristics were analyzed using φ scan measurements on Phillips X’Pert Pro diffractometer. The point defects in TiN crystallites trigger Raman-active vibrational modes which were probed via WITec Raman confocal microscope system (Alpha 3000M) with the source wavelength of 532 nm. The spectral acquisitions were calibrated with single-crystal Silicon exhibiting the primary Raman peak at 520.6 cm-1. The impact of thermal annealing on transport characteristics and charge carrier concentration was studied using an Ecopia HMS-3000 Hall measurement system. The modifications in magnetic characteristics of TiN films post thermal annealing was studied on the Quantum design Superconducting Quantum Interference Device (SQUID) magnetic property measurement system (MPMS) in the VSM magnetometry mode which exhibits sensitivity ≤10−8 emu. The isothermal magnetization data was acquired as a function of magnetic field varying from -1 Tesla to +1 Tesla using samples of ∼4 mm×5 mm geometry under the application of magnetic field applied parallel to the film surface. The electronic transport and magneto-transport measurements were performed on the same sample inside the Quantum design Physical Property Measurement System (PPMS3) with the magnetic field applied perpendicular to the in-plane electronic transport. Electrodes were deposited on the samples with conductive silver paint under the four-probe configuration for DC resistivity measurements. To study the atomically resolved structural characteristics, the scanning transmission electron microscopy (STEM) studies were performed. The TEM specimen lamellae of 40 nm thickness was prepared using an FEI Quanta 3D equipped with a field emission gun (FEG) and dual (electron and Ga ion) beams. Post formation of the TEM lamellae, the ion beam damage was cleaned up 7

using a low-energy (5 kV, 10 pA) cleaning procedure. The atomically resolved high-angle annular dark-field (HAADF) image acquisitions were performed at 100 keV using an aberration-corrected Nion UltraSTEM (STEM, scanning transmission electron microscope) under the beam convergence and collection angles of 30 and 86 mrad, respectively. The probe current used in the STEM experiments was 28±2 pA. The nitride stoichiometry was analyzed by performing electron energy-loss spectroscopy (EELS) using the Gatan Digiscan II STEM controller equipped with a 2k 794IF/20 MegaScan CCD. The Pixel-by-pixel EELS analysis performed at collection angle of 48 mrad. 3. Results and Discussion 3.1. High-quality TiN films. For this study, TiN thin films with a thickness of 40 nm were deposited using the PLD technique on c-Al2O3. After the deposition, exposure to pure N2 during thermal annealing formed pure TiN thin films. At 700oC in N2 atmosphere, the VN formation energy is very high (1.88 eV), which is comparable to that of VTi (1.70 eV) limiting the total number of point defects introduced in TiN lattice.[18, 19] The stoichiometry of metal nitride thin films in non-equilibrium conditions of PLD is primarily controlled by deposition parameters, and to a certain extent lattice misfit with the substrate. X-ray diffraction studies are performed to probe the structure of thin films and changes that occur after nitrogen and vacuum annealing. Fig. 1(a) shows the out-of-plane alignment for TiN/ c-Al2O3 heterostructure is analyzed with θ-2θ X-ray diffraction scans. It confirms the highly textured TiN film growth with <111> out-of-plane direction, aligning with the <0001> Al2O3. The d111 lattice spacing for the deposited TiN films is 0.2474 nm. On comparing with the ideal d111 spacing for the TiN crystal (0.2448 nm), the residual out-of-plane strain is determined as 1.06% (tensile). The thermal strain contributions to in-plane stain 𝜀𝑇 = (𝛼𝑠 ― 𝛼𝑓)𝛥𝑇, ∼0.06% is extremely low on comparing against the total residual strain. 8

To ascertain if the deposited films are epitaxial in nature, XRD φ scan measurements are performed. The reflections shown in Fig. 1(b) are (200) for TiN and (1004) plane for Al2O3 confirming the three-dimensional alignment and epitaxial nature of TiN film. From the φ analysis, it is clear that Al2O3 exhibits 3-fold symmetry with 120° spacing between the diffraction peaks, while TiN exhibits 6-fold symmetry with an azimuthal rotation of 30° on comparing with Al2O3. As no other family of reflections are observed across the complete 360o rotation cycle, it confirms the TiN/ Al2O3 heteroepitaxy. To gain further insights regarding the in-plane epitaxial relationships, the TiN/c-Al2O3 heterostructure is further probed by atomically resolved high-angle annular dark-field (HAADF) imaging along the [112] zone axis. Figure 2 reveals an epitaxially grown TiN film devoid of stacking faults or Ti precipitates, suggesting that these films may contain only point defects. Fig. 2(a) shows the ~40 nm thick singal crystalline TiN film on c-Al2O3 substrate. The <110>TiN//<10 10>Al2O3 heterostructure has a large misfit of ∼9.24%. This form of epitaxial growth triggers the formation of periodic dislocations at the film/substrate interface, with integral multiples of lattice planes matching to accommodate the large-misfit. These results are also consistent with the XRD results (Fig. 1(a) and (b)) where high counts and sharp <111> reflections were observed in θ-2θ and φ scans. The fast Fourier diffractograms (FFT) acquired from TiN and Al2O3 in Fig. 2(c) and (d) reveal the alignment of (111) spot of TiN with (006) spot for Al2O3 and (110) FFT spot of TiN with (330) spot for Al2O3. This atomically-resolved observation is in agreement with the global XRD φ-scan results, thereby establishing the heteroepitaxy in the large misfit system following the domain matching epitaxy (DME) paradigm. [23] The linear dislocation density at the interface is calculated as 0.805/nm from Fig. 2(b). The average dislocation periodicity can be calculated by matching 11 planes of TiN with 12 planes of Al2O3 creating an average dislocation 9

with a spacing of ~11 atomic columns of the film, consistent with the previous DME predictions.[13, 24] As the film grows over time; the surface energy becomes higher than the dislocation formation energy. It results in instantaneous dislocation nucleation and film relaxation, within a few monolayers.[25] This form of epitaxial thin film growth via DME paradigm in largemisfit systems results in formation of periodic dislocations at the film/substrate interface.[23] The dislocation separation at the interface is determined to be 1.61 nm. The 9.24% misfit at the interface relaxes by matching of 11 planes of TiN with 12 planes of Al2O3 creating an average dislocation with a spacing of ~11 atomic columns of the film.[24] The FFT spot alignment and XRD φ scan results confirm the epitaxial relationship between TiN and c-Al2O3 as (111)TiN ∥ (0001) Al2O3 out-of-plane, <110>TiN ∥ <1010> Al2O3 and <112>TiN ∥ <1120> in-plane. Under the increased lattice misfit, the dislocations nucleate within growth of couple of monolayers, confining misfit strain at the interface, resulting in subsequent relaxed film growth.[13] Hence, the only defects introduced in TiN films are point defects, making the direct interpretation of modifications in magnetic and electronic properties feasible. 3.2. TiN bonding characteristics. To inspect the ionicity and the nature of chemical entities inside the TiN films, atomically resolved EELS analysis is performed across the TEM specimen. Figure 3(a) reveals the core-loss N-K edge profiles for TiN and non-stoichiometric TiN1-x thin films after background subtraction.This is done for detailed analysis of energy-loss near-edge structure which is highly sensitive to VN formation. Using TiN and Ti spectra as the extremes, the nitrogen-vacancy content in TiN1-x films formed after vacuum annealing is determined by the multiple linear leastsquares fitting approach [26] as~12±2 at. %. From equilibrium-based considerations, the number of VN is: 𝑉𝑁 = 𝑁𝑜𝑒 ― 𝐸𝑓/𝑘𝑇. From EELS quantification we attain, VN/No = ~0.12 for TiN which results in Ef = 0.72 eV. Performing a similar analysis on TiN films, with Ef = 1.88 eV provides a 10

vacancy concentration of 2x10-3 vacancies per unit cell. Based on the EELS quantification, each nitrogen-vacancy defect results in ~1.2 µB (Bohr magneton) in TiN1-x films. From this quantification, the non-stoichiometry in TiN films is estimated to be ~10-3 per unit cell which has the same order as thermodynamic considerations. On comparing the N-K edge for TiN and TiN0.88 films, we found that (i) the N-K edge pre-peak in TiN0.88 shifts to higher energy in comparison with pure TiN, with a partial merging of the N-K prepeak and main peak. VN formation generates new energy states near EF, altering the Ti-L32 and N-K EEL spectra, reflecting changes in the local Ti environment. Under equilibrium considerations, VN formation follows the following equations: 1

𝑁𝑁↔𝑉𝑁 +0.5 𝑁2 +3𝑒′; 𝐾1 = [𝑉𝑁]𝑒′ 𝑃 2𝑁 3

2

(1)

Here, XX and 𝑉𝑋 denote an atom (X) at its lattice site and a vacancy(X) with a positive charge, respectively. The HMS measurements suggest a decrease in carrier density from -9.464x1022/𝑐𝑐 in TiN to -3.164x1022/𝑐𝑐 in TiN1-x films, highlighting that electrons released on VN formation are localized in nature, not adding to the carrier density. Instead, they trigger the formation of Ti+2 polarons by charge compensation, following: 𝑇𝑖 +3 + 𝑒 ― →𝑇𝑖 +2 equation. Overall, VN causes reduction of Ti+3 into Ti2+in TiN. The unpaired electrons associated with these Ti2+ polarons generate unpaired spin ordering in TiN. To further analyze the TiN bonding characteristics, Raman spectroscopy is performed. In TiN, the primary Raman modes are rendered forbidden due to the NaCl lattice symmetry. It is a Ramaninactive metallic material with first-order acoustic and optical Raman active vibrational modes arising from the inclusion of defects. These Raman active vibrational modes can be traced back to the defects that make the crystal imperfect. The generation of VN is reflected by Ti-Ti vibrational 11

modes (LA and TA) below 300 cm-1 while VTi results in N-N vibrational modes (LO and TO) near 550 cm-1.[27] Utilizing this, the TO/TA intensity ratio for normalized Raman spectra can qualitatively provide estimates regarding number of VN in TiN. Figure 3(b) shows the normalized Raman acquisitions from TiN and TiO2 (rutile) epitaxial thin films. Notably, in TiN Raman spectrum, the Ti-O vibrational characteristics are not observed, suggesting that diffusion of O in TiN has not occured during thermal annealing. The TO/TA ratio is 0.8 in TiN to 0.33 in TiN1-x, suggesting a considerably low number of VN. Typical fitting routine[28] (Fig. 3(b) inset) reveals the TA mode peak is centered at 202 cm-1 in TiN. Notably, pure TiN films have TA peak at 200 cm-1[27], which shifts to higher frequencies with an increase in VN, corroborating the purity and low VN concentration in nitrogen annealed TiN films. 3.3. Magnetic ordering. Before performing the magnetic studies, isothermal magnetization loops are generated for Al2O3 substrate, as shown in the inset of Fig. 4(a). The straight line in the M-H loop corresponds to diamagnetic nature. Figure 4(a) compares the room-temperature magnetization loops for TiN and TiN1-x films. The saturation magnetization (Ms) decreases from 11.4 emu g-1 in (TiN1-x) to 0.1 emu g-1 (TiN) on N2 annealing. The positive slope for spontaneous 𝑑𝑀

magnetization ( 𝑑𝑇 ) from 5-300 K in TiN films, reveals the weak magnetic coupling in Fig. 4(b). The spontaneous magnetization for ferromagnetic materials following continuous spin-wave model is given by: 𝑀(𝑇) = 𝑀(0)(1 ― 𝑏𝑇𝑛)

(2)

where M(0) is the magnetization at absolute zero and n is Bloch exponent[29]. On fitting the spontaneous field-cooled (FC) magnetization measurements at 100 Oe Fig. 4(b) with the spinwave equation, Bloch exponent of 1.977 is extracted for TiN1-x films. The magnetic ordering in TiN originates from introduction of VN, as pristine TiN is paramagnetic. These vacancies are 12

physically isolated and interact with each other via the itinerant electron gas, a manifestation of the many-body effect known as the RKKY interaction.[30] The localized spin states generated at defects/impurities are equivalent to localized spin perturbing the conduction electrons. The response of electron gas under localized spin perturbations is reflected in the wavevector susceptibility (χq). Its real space value is be calculated by taking inverse Fourier transform of χq: 𝜒(𝑟) =

2𝑘3𝐹 𝜒𝑃 𝜋

𝐹(2𝑘𝐹𝑟)

(3)

here kF is the Fermi wavevector, 𝜒𝑃 is the dielectric permeability with the distance between unpaired spin states as r. Here the 𝐹(𝑥) is a sinusoidal given by: 𝐹(𝑥) =

―𝑥𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 𝑥4

(4)

with 𝑥 as 2𝑘𝐹𝑟. The 𝜒(𝑟) is an oscillatory function and diverges at low 𝑟 due to the localized electronic perturbations. This mechanism is invoked for localized magnetic impurities in metal systems, where the oscillatory magnetic response on the electron gas interacts with the magnetic moment of the neighboring localized magnetic impurity. Depending on distance between the two moments, the changeover from ferromagnetic to anti-ferromagnetic interaction occurs, as shown in the F(r) plots in Fig. 4(c). This can lead to the simultaneous existence of both FM and AFM ordering, noted in many systems where resultant magnetism is governed by RKKY interactions, leading to the emergence of complex spin-glass-like-states. A single VN releases 3e-, which may populate the empty 3d states near EF, increasing the eDOS. The 2p and 3d eg orbitals are directed towards the nearest neighbor N sites resulting in them getting strongly impacted on discharge of free electrons during VN formation. This creates unpaired localized spin states below EF, generating ferromagnetic ordering.[31]

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The Hall measurements reveal a carrier concentration (𝑛𝑒)/𝑐𝑐 of -9.46 x1022/𝑐𝑐 for TiN and -3.16 x1022/𝑐𝑐 at 300 K for TiN1-x films, at room temperature. The Fermi wavevector (kF) in case of TiN is calculated as 1.4/Å which lowers to 0.9716/Å for TiN1-x. In TiN the VN are far apart, leading to low exchange coupling. Considering a uniform distribution of Ti+2 polarons in TiN lattice, the exchange-interaction remains positive in the range of 5.0-6.2 Å for r, as revealed in Fig. 4(d). At x<0.05, the exchange interaction term is negative, leading to oscillatory FM-AFM behavior, which is a direct observation from magnetic measurements in Fig. 4(a) and (b). 3.4. Low-temperature resistivity minima. The temperature-dependent resistivity plots for TiN films reveal linearity at high temperatures (>50 K) due to the inelastic electronic scattering. Rather than flattening off to ρ0 at low temperatures, the curve exhibits a resistivity minimum ~30 K. This persistent resistivity minimum is observed for the measurements performed under a constant magnetic field of 0-7 Tesla as shown in Fig. 5(a). Notably, there are negligible changes observed in the resistivity near minima, with and without the application of the magnetic field. There have been reports on the emergence of anomalous resistivity minimum in amorphous magnetic materials[32], semiconductors[33] and lately, in strong ferromagnets[34, 35]. In granular alloys and polycrystalline disordered systems, the resistivity upturn can arise from the Coulomb blockade (CB) effect[36], electron-electron scattering[37], or Kondo scattering. For the CB effect, resistivity is given by: 𝜌 = 𝜌0 + 𝜌𝐶𝐵 𝑒𝑥𝑝

( ) + 𝜌𝐶𝐵𝑇𝑃 Δ 𝑇

(5)

where Δ is the Coulomb energy required for tunneling of itinerant electrons across the grain boundary. As the TiN films are epitaxial, there are no grain boundaries in these single-crystal films. Hence, there is minimal possibility of CB effect triggering a resistivity upturn. Figure 5(b) suggests that the resistivity at temperatures below resistivity minimum exhibits a logarithmic 14

behavior under magnetic fields as high as 7 Tesla. The slope of each curve below the resistivity minima roughly remains the same, revealing that there is a negligible change in the interaction between the unpaired spin states and itinerant electrons on the introduction of the magnetic field. Due to the weak magnetic coupling in TiN, quantum interference effects (QIE) like e-e interactions and weak localization can become dominant at low temperature, inducing resistivity upturn. In the case of disordered systems, below the resistivity minimum, the elastic electron-electron interactions[38] induce an upturn in resistivity taking the form: 𝜌(𝑇) = ― 𝜌𝑒 𝑇 + 𝜌0 + 𝜌𝑛𝑇𝑛

(6)

For TiN films, n~1 at high temperatures. Interestingly, on fitting the Eq. (6) to TiN resistivity measurements, n is extracted to be 5, which is expected at low temperatures for metallic materials. The electronic scattering below this anomalous resistivity upturn is further probed by the numerical renormalization group (NRG)[39, 40] calculations where the Kondo scattering contribution to

[

1

)

resistivity can be expressed as 𝜌 = 𝜌𝑠 1 + (2𝑆 ― 1 (𝑇/𝑇𝐾)2]

―𝑆

. Here, S is the unpaired spin states

and TK is the characteristic Kondo temperature. The NRG fitting analysis provides the best possible fits for TiN films with S=1/2 and TK of 26 K, as shown in Fig. 5(c). The experimental data decidedly falls above NRG fitting at low temperatures for TiN films, suggesting the incompatibility of NRG Kondo fitting, ruling out Kondo effect as an origin for the resistivity upturn. 3.5. Quantum interference effects. According to localization theory[41], the resistivity for strongly correlated materials in two dimensions becomes: 𝜌(𝑇) = λln ( 𝑇) + 𝜌0 + 𝜌𝑛𝑇𝑛

15

(7)

With the first term (λ) arising from weak anti-localization. Fitting the resistivity plots acquired under 0 Tesla with equation 7 reveals the coefficient of determination (COD) value of 0.9993 (weak anti-localization) and 0.9963 (e-e interaction) at zero-field, as shown in Fig. 5(d). Under high magnetic field (7 Tesla), the COD values change to 0.9909 (WAL) and 9.9989 (e-e interactions) reflecting the weakening of Berry phase under high magnetic fields. See Table 1 for the detailed tabulation of fitting parameters and COD values. These fits reveal persistent 𝑙𝑛(𝑇) behavior below the resistivity minima, suggesting increased contributions from weak antilocalization effects. The classical Kondo minimum has two salient features: namely, the depth of minima and TK, which are proportional to the number of impurity states, and that the minimum itself disappears under magnetic field. As the resistivity minimum is observed even at 7 Tesla, it is suggested to manifest from weak anti-localization effects. The electronic scattering from non-magnetic impurities is a time-reversal invariant perturbation, which creates a time-reversal symmetric path with identical directional probability. [42] In the case of weak antilocalization, the momentum and spin for the electron are locked as a consequence of strong spin-orbit coupling, causing in a phase difference of 2 between spin states (forming π Berry phase[43]). This leads to the destructive interference of the time-reversed loops, effectively suppressing the backscattering, thereby leading to increased magnetoconductance (MC). To probe the emergence of WAL states in TiN films, magnetoresistance (MR) measurements are performed in the PPMS with the magnetic field applied perpendicular to the electronic transport, with the magnetic field varying from -1 to 1 Tesla. Figure 6(a) reveals that at 5 K under zero magnetic 1

field, a sharp cusp is noted for magnetoconductance (MC) data with negative MC (∆𝐺(𝐵) = 𝑅(𝐵) ― 1 𝑅(0)

) on both sides of the cusp, which is attributed to WAL. The MC data in Fig. 6(b) is then fitted

to the Hikami-Larkin-Nagaoka (HLN) equation[44] modified for the strong spin-orbit coupling 16

conditions, following the theory of quantum correction in the 2D diffusive transport regime given ―𝛼𝑒2

ℏ

1

ℏ

by: ∆𝜎 = 𝜎(𝐵) ―𝜎(0) = 2𝜋2ℏ [ln 4𝐵𝑒𝑙2 ―𝜓(2 + 4𝐵𝑒𝑙2 )], with e as the electronic charge, ℏ is the 𝜙

𝜙

Planck’s constant, 𝑙𝜙 is phase coherence length, and 𝜓 (x) is the digamma function. The coefficient α is −0.5 per transport channel for WAL, and 1 for WL. [44] The characteristic localization length is determined using 𝐵𝜙=ℏ/4e𝐿2𝜙 where 𝑙𝜙 is the distance traversed by an electron till the phase coherence is lost. As can be noted from Fig. 6(a) the data fits well to the HKN equation resulting in 𝐵𝜙=655 Oe at 5 K. Similar fits are performed on MC data acquired at 30 K and 2 K using the HKN equation in Fig. 6(b) and Fig. 6(c), respectively. Above 30 K, the MC data was too noisy to obtain any meaningful analysis. Figure 6(c) highlights the WAL fit for the MC data acquired at 2 K. The power law fit of the phase coherence length vs. temperature suggests that the WAL effect in TiN thin films is governed by the two-dimensional (2D) localization theory. The TiN films exhibit a superconducting transition at the temperature of ~2.4 K, with the resistance dropping down to 0 Ω. This transition is also noted in the MR plots, with a sharp deviation near 3000 Oe magnetic field due to the onset of superconducting states in TiN, as shown in the inset of Fig. 6(d), eventually leading to zero resistance. This sudden reaction to the magnetic field with fluctuations in resistivity reflects the free motion of carriers, suggesting that TiN films fabricated as such, are clean superconductors. Notably, in TiN1-x films, the robust FM ordering destroys the π Berry’s phase at the low-temperature WAL limit[45] resulting in broad positive MC. 3.6. High-temperature transport characteristics: The above fitting results illustrate resistivity upturn below 28 K in case of near-stoichiometric TiN films. From the e-e interaction dependence on resistivity (equation 6), the m factor (𝜌𝑒/𝜌20) can be calculated, which is ~2400 (ΩmK1/2)−1for TiN, deviating from the near-universal value[29, 38] of ~600 (ΩmK1/2)−1 for amorphous and crystalline alloys. This suggests that the resistivity upturn in TiN films is not affected by e-e 17

interactions. As shown above, the fits to WAL equation remain satisfactory even under high magnetic fields. With an increase in the magnetic field, WAL effects are known to weaken as can be noted form the fits at 7 Tesla for TiN films. For the scattering of conduction electrons by lattice phonons, precise fits are again performed using the Bloch–Grüneisen formula for the hightemperature range (100-300 K) in Figure 7:

𝜌𝑙 = 𝐴

𝑇 𝑛

( ) 𝜃𝐷

𝜃𝐷

𝑧𝑧𝑑𝑧

∫0𝑇 (𝑒𝑧 ― 1)(1 ― 𝑒 ―𝑧) 𝑇𝑛

(8)

where 𝜃D=Debye temperature and n=3 for magnetic metals and alloys with a large d-band DOS giving rise to electron-phonon scattering involving s-d transitions. This contribution added to the scattering by ferromagnetic spin waves (𝜌𝑚=BT2) accounts for the total scattering of conduction electrons. The contribution from spin waves (𝜌𝑚) in ferromagnetic metals originates from the exchange interaction between the conduction electrons and the localized magnetic electrons, commonly called s-d interaction. By performing these fits on TiN and TiN1-x films, 𝜃D is calculated to be 568 K and 621 K, respectively. Interestingly, the B term for TiN films is non-existent for best fits, suggesting the absence of long-range ferromagnetic ordering. The Debye temperature for pure titanium nitride is 636 K and for Ti metal is 380 K.[46] Notably, on change in stoichiometry TiN0.89 exhibits Debye temperature of 580 K.[47] 4. Conclusion Magnetism in TiN originates from localized moments created at Ti+2 polarons, which interact via conduction electrons following the RKKY interaction mechanism. The TiN films exhibit weak magnetic coupling due to the considerable average distance between localized unpaired spin states. The electron-transport measurements in TiN reveal the presence of persistent resistivity minima at 28±0.6 K. The non-satisfactory NRG fits below the resistivity minima reflected the non-magnetic 18

nature of remaining defects and departure from Kondo scattering. The HKN fitting revealed that this minimum was a result of quantum interaction effects between scarcely located non-magnetic defects and the conduction electrons, governed by 2D weak antilocalization effect. These results shed light on the possible pathways to tune the magnetic nature of transition metal nitrides to integrate novel functionalities for designing multifunctional devices and systems. Acknowledgments This study was performed under the National Science Foundation (NSF) grant DMR- 1735695. Part of the analysis was conducted at the Analytical Instrumentation Facility (AIF) at North Carolina State University, supported by the state of North Carolina. A part of this work was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. Author Contributions J.N. and S.G. initiated the research. S.G. fabricated the samples, performed the Raman spectroscopy, Hall-effect, Magnetic measurements, X-ray θ-2θ, and Phi-scan measurements, and fabricated the TEM specimens. R.S. performed the HAADF imaging and EELS measurements. S.G. wrote the manuscript with inputs from all co-authors. Competing financial interests: The authors declare no competing financial interests . Sample

Equation Magnetic field

TiN

6

0 Tesla

𝝆𝟎

𝝆𝒆

𝝆𝒏

(μΩcm)

(μΩcm K-0.5)

(μΩcm K-5)

21.840

0.0112

19

2.8E-10

COD

0.9963

TiN

7

0 Tesla

21.844

0.4158

1.1E-11

0.9993

TiN

6

7 Tesla

21.8596

0.0126

4.9E-10

0.9989

TiN

7

7 Tesla

21.864

0.045

1.4E-11

0.9909

Table 1: The detailed tabulation of fitting parameters and COD values for equation 6 and 7 regarding the transport characteristics of TiN films.

Figures:

20

Fig. 1. (a) θ-2θ patterns acquired from TiN films grown on (0001) Al2O3 (b) X-ray diffraction φscan measurements acquired from TiN (111)/Al2O3 (0001) heterostructure for TiN (200) and Al2O3 (104) reflections.

21

Fig. 2. (a) The low-magnification HAADF image of N2 annealed TiN/Al2O3 heterostructure with a film thickness of ~40 nm. (b) Cross-sectional STEM HAADF image showing the atomicallysharp interface between TiN and Al2O3. (c), and (d) show the IFFT pattern of TiN and the Al2O3 are highlighted in red, revealing the epitaxial relationships. (e) shows the periodic occurrence of dislocations, revealing the epitaxial growth, following domain matching epitaxy paradigm.

22

Fig. 3. (a) The N-K electron-loss near-edge spectra for TiN and TiN1-x films. (b) Reveals the normalized Raman spectra for TiN, TiN1-x, and TiO2 films with the inset highlighting the rightshift in TA mode with a change in TiN stoichiometry.

23

Fig. 4. (a) Isothermal magnetization measurements acquired from TiN and TiN1-x films at room temperature. The bottom-right inset highlights the diamagnetic nature of sapphire. (b) The fieldcooled magnetization M(T) data acquired at (100 Oe) for TiN and TiN1-x films with Bloch law fit for TiN1-x films. (c) 2-D contour plot revealing RKKY interaction function F(r) dependence on r. (d) F(r) calculations for different Fermi wavevectors corresponding to TiN and TiN1-x films as a function of interaction distance(r) between the unpaired spin states.

24

Fig. 5. (a) Temperature-dependent resistivity of TiN films under a constant magnetic field of 0 to 7 Tesla acquired in the range of 5 K- 100 K. The inset highlights the low-temperature upturn in resistivity. (b) The ρ vs. ln(T) curve at low temperature shows a logarithmic temperature dependence of the resistivity. (c) represents the semi-log graph for fits to the numerical renormalization group (NRG) formalism, for the low-temperature resistivity data acquired at 0 Tesla magnetic field. (d) highlights the accurate fit to WAL profiles for TiN films, revealing the emergence of localization behavior below the resistivity minima.

25

Fig. 6. Weak antilocalization effect in magnetoresistance measurement of epitaxial TiN thin films. The magneto-conductance measurement for TiN films at (a) 5 K, (b) 30 K and (c) 2 K. The inset in Figure (b) highlights the phase-coherence length as a function of temperature, revealing Lφ ∝ T−1/2 power-law fit. (d) Represents the onset of superconducting states at 2.4 K under 0 Oe magnetic field, while the inset reveals the onset of superconducting states in TiN films at ~2500 Oe in TiN films at 2 K, with the resistance dropping to zero resistance in both cases.

26

Fig. 7. Zero-field resistivity is plotted against temperature for TiN and TiN1-x thin films from 5 K up to 300 K.

References 27

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      

TiN films exhibit (<0.5 at.%) VN generating weak magnetic coupling limit. It results in the evolution of Berry’s phase triggering WAL characteristics below 28±0.6 K. Negative MC with sharp cusps confirmed WAL emergence. The MC data follows 2D localization theory with T-1/2 dependence of localization length. VN are instrumental in forming Ti+2 polarons with localized unpaired spin states. The long-ranged magnetic ordering between Ti+2 polarons is actuated by itinerant electron gas, invoking RKKY interactions. Ti+2 polarons are magnetically active defects which generate increased spin-orbit coupling in TiN films.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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