Control of the metal-insulator transition in NdNiO$_3$ thin films through the interplay between structural and electronic properties
Y. E. Suyolcu, K. Fürsich, M. Hepting, Z. Zhong, Y. Lu, Y. Wang, G. Christiani, G. Logvenov, P. Hansmann, M. Minola, B. Keimer, P. A. van Aken, E. Benckiser
Heteroepitaxy offers a new type of control mechanism for the crystal structure, the electronic correlations, and thus the functional properties of transition-metal oxides. Here, we combine electrical transport measurements, high-resolution scanning transmission electron microscopy (STEM), and density functional theory (DFT) to investigate the evolution of the metal-to-insulator transition (MIT) in NdNiO$_3$ films as a function of film thickness and NdGaO$_3$ substrate crystallographic orientation. We find that for two different substrate facets, orthorhombic (101) and (011), modifications of the NiO$_6$ octahedral network are key for tuning the transition temperature $T_{\text{MIT}}$ over a wide temperature range. A comparison of films of identical thickness reveals that growth on [101]-oriented substrates generally results in a higher $T_{\text{MIT}}$, which can be attributed to an enhanced bond-disproportionation as revealed by the DFT+$U$ calculations, and a tendency of [011]-oriented films to formation of structural defects and stabilization of non-equilibrium phases. Our results provide insights into the structure-property relationship of a correlated electron system and its evolution at microscopic length scales and give new perspectives for the epitaxial control of macroscopic phases in metal-oxide heterostructures.
We have synthesized epitaxial NdNiO$_{3}$ ultra-thin films in a layer-by-layer growth mode under tensile and compressive strain on SrTiO$_{3}$ (001) and LaAlO$_3$ (001), respectively. A combination of X-ray diffraction, temperature dependent resistivity, and soft X-ray absorption spectroscopy has been applied to elucidate electronic and structural properties of the samples. In contrast to the bulk NdNiO$_{3}$, the metal-insulator transition under compressive strain is found to be completely quenched, while the transition remains under the tensile strain albeit modified from the bulk behavior.
We report the electronic properties of the NdNiO3, prepared at the ambient oxygen pressure condition. The metal-insulator transition temperature is observed at 192 K, but the low temperature state is found to be less insulating compared to the NdNiO3 prepared at high oxygen pressure. The electric resistivity, Seebeck coefficient and thermal conductivity of the compound show large hysteresis below the metal-insulator transition. The large value of the effective mass (m* ~ 8me) in the metallic state indicate the narrow character of the 3d band. The electric conduction at low temperatures (T = 2 …
Modulation of the charge carrier density in a Mott material by remote doping from a highly doped conventional band insulator is proposed to test theoretical predictions of band filling control of the Mott metal-insulator transition without introducing lattice distortions or disorder, as is the case for chemical doping. The approach is experimentally tested using ultrathin (2.5 nm) NdNiO3 films that are epitaxially grown on La-doped SrTiO3 films. We show that remote doping systematically changes the charge carrier density in the NdNiO3 film and causes a moderate shift in the metal-insulator tra…
Bulk NdNiO3 exhibits a metal-to-insulator transition (MIT) as the temperature is lowered that is also seen in tensile strained films. In contrast, films that are under a large compressive strain typically remain metallic at all temperatures. To clarify the microscopic origins of this behavior, we use position averaged convergent beam electron diffraction in scanning transmission electron microscopy to characterize strained NdNiO3 films both above and below the MIT temperature. We show that a symmetry lowering structural change takes place in case of the tensile strained film, which undergoes a…
Through CaH2 chemical reduction of a parent R3+Ni3+O3 perovskite form, superconductivity was recently achieved in Sr-doped NdNiO2 on SrTiO3 substrate. Using density functional theory (DFT) calculations, we find that stoichiometric NdNiO2 is significantly unstable with respect to decomposition into 1/2[Nd2O3 + NiO + Ni] with exothermic decomposition energy of +176 meV/atom, a considerably higher instability than that for common ternary oxides. This poses the question if the stoichiometric NdNiO2 nickelate compound used extensively to model the electronic band structure of Ni-based oxide analog …
We investigate the behavior of the spectral weight near the Fermi level of NdNiO3 thin films as a function of temperature across the metal-to-insulator transition (MIT) by means of ultraviolet photoelectron spectroscopy. The spectral weight was found to exhibit thermal hysteresis, similar to that of the dc conductivity. A detailed analysis of the temperature dependence reveals two distinct regimes of spectral loss close to the Fermi level. The temperature evolution of one regime is found to be independent from the MIT.
Rare earth nickelates RENiO3 which attract interest due to their sharp metal-insulator phase transition, are instable in bulk form due to the necessity of an important oxygen pressure to stabilize Ni in its 3+ state of oxidation. Here, we report the stabilization of rare earth nickelates in [(SmNiO3)t/(NdNiO3)t]n thin film multilayers, t being the thickness of layers alternated n times. Both bilayers and multilayers have been deposited by Metal-Organic Chemical Vapour Deposition. The multilayer structure and the presence of the metastable phases SmNiO3 and NdNiO3 are evidenced from by X-ray an…
Optically induced metallic state with persistent monoclinic symmetry in NdNiO$_3$
University of Zurich（苏黎世大学）；Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy（马克斯·玻恩非线性光学和短脉冲光谱研究所）；Harvard University（哈佛大学）；Institute for Physics and Astronomy, TU Berlin（柏林工业大学物理与天文研究所）
Active memristor elements, also called neuristors, are self-oscillating devices that are very good approximations to biological neuronal functionality and are crucial to the development of low-power neuromorphic hardware. Materials that show conduction mechanisms that depend superlinearly with temperature can lead to negative differential resistance (NDR) regimes, which may further be engineered as self-oscillators. Thermal runaway, insulator to metal phase transitions (IMT) can lead to such superlinearity and are being extensively studied in systems such as TaO$_x$, NbO$_x$ and VO$_2$. Howeve…
We have investigated possible spin and charge ordered states in 3d transition-metal oxides with small or negative charge-transfer energy, which can be regarded as self-doped Mott insulators, using Hartree-Fock calculations on d-p-type lattice models. It was found that an antiferromagnetic state with charge ordering in oxygen 2p orbitals is favored for relatively large charge-transfer energy and may be relevant for PrNiO$_3$ and NdNiO$_3$. On the other hand, an antiferromagnetic state with charge ordering in transition-metal 3$d$ orbitals tends to be stable for highly negative charge-transfer e…
Polycrystalline Nd$_{1-x}$Eu$_{x}$NiO$_3$ ($0 \leq x \leq 0.5$) compounds were synthesized in order to investigate the character of the metal-insulator (MI) phase transition in this series. Samples were prepared through the sol-gel route and subjected to heat treatments at $\sim$1000 $^\circ$C under oxygen pressures as high as 80 bar. X-ray Diffraction (XRD) and Neutron Powder Diffraction (NPD), electrical resistivity $ρ(T)$, and Magnetization $M(T)$ measurements were performed on these compounds. The results of NPD and XRD indicated that the samples crystallize in an orthorhombic distorted pe…
The rare-earth nickelates possess a diverse set of collective phenomena including metal-to-insulator transitions, magnetic phase transitions, and, upon chemical reduction, superconductivity. Here, we demonstrate epitaxial stabilization of layered nickelates in the Ruddlesden-Popper form, Nd$_{n+1}$Ni$_n$O$_{3n+1}$, using molecular beam epitaxy. By optimizing the stoichiometry of the parent perovskite NdNiO$_3$, we can reproducibly synthesize the $n = 1 - 5$ member compounds. X-ray absorption spectroscopy at the O $K$ and Ni $L$ edges indicate systematic changes in both the nickel-oxygen hybrid…
Neodymium nickelate, NdNiO3 attracts attraction due to the simultaneous occurrence of several phase transitions around the same temperature. The electronic properties of NdNiO3 are extremely complex as structural distortion, electron correlation, charge ordering, and orbital overlapping play significant roles in the transitions. We report the effects of electron and hole injection via doping a single 3d metal, V, in the NdNiO3 nanostructures to understand the variations in the electronic properties without any structural distortion. A reversible resistivity modulation more than five orders of …
Controlling phase transitions in transition metal oxides remains a central feature of both technological and fundamental scientific relevance. A well-known example is the metal-insulator transition which has been shown to be highly controllable while a less well understood aspect of this phenomenon is the length scale over which the phases can be established. To gain further insight into this issue, we have atomically engineered an artificially phase separated system through fabricating epitaxial superlattices consisting of SmNiO$_{3}$ and NdNiO$_{3}$, two materials undergoing a metal-to-insul…
Rare-earth nickelates are strongly correlated oxides displaying a metal-to-insulator transition at a temperature tunable by the rare-earth ionic radius. In PrNiO$_3$ and NdNiO$_3$, the transition is very sharp and shows an hysteretic behavior akin to a first-order transition. Both the temperature at which the transition occurs and the associated resistivity change are extremely sensitive to doping and therefore to off-stoichiometry issues that may arise during thin film growth. Here we report that strong deviations in the transport properties of NdNiO$_3$ films can arise in films grown consecu…
Metal-to-insulator phase transitions in complex oxide thin films are exciting phenomena which may be useful for device applications, but in many cases the physical mechanism responsible for the transition is not fully understood. Here we demonstrate that epitaxial strain generates local disproportionation of the NiO6 octahedra, driven through changes in the oxygen stoichiometry, and that this directly modifies the metal-to-insulator phase transition in epitaxial (001) NdNiO3 thin films. Theoretically, we predict that the Ni-O-Ni bond angle decreases, while octahedral tilt and local disproporti…
We report a temperature-dependent Raman scattering investigation of thin film rare earth nickelates SmNiO3, NdNiO3 and Sm0.60Nd0.40NiO3, which present a metal-to-insulator (MI) transition at TMI and an antiferromagnetic-paramagnetic Neel transition at TN. Our results provide evidence that all investigated samples present a structural phase transition at TMI but the Raman signature across TMI is significantly different for NdNiO3 (TMI = TN) compared to SmNiO3 and Sm0.60Nd0.40NiO3 (TMI =/ TN). It is namely observed that the paramagnetic-insulator phase (TN < T < TMI) in SmNiO3 and Sm0.60Nd0.40Ni…
Polaron-mediated metal-insulator transition and proton conduction in hydrogenated nickelate perovskites
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理学院）；Key Laboratory of Multiscale Spin Physics (Beijing Normal University), Ministry of Education（教育部多尺度自旋物理重点实验室（北京师范大学））；College of Physics, Hebei Normal University（河北师范大学物理学院）；Hebei Advanced Thin Films Laboratory（河北省先进薄膜实验室）
Nucleation processes of mixed-phase states are an intrinsic characteristic of first-order phase transitions, typically related to local symmetry breaking. Direct observation of emerging mixed-phase regions in materials showing a first-order metal-insulator transition (MIT) offers unique opportunities to uncover their driving mechanism. Using photoemission electron microscopy, we image the nanoscale formation and growth of insulating domains across the temperature-driven MIT in NdNiO3 epitaxial thin films. Heteroepitaxy is found to strongly determine the nanoscale nature of the phase transition…
Recent discoveries of a new type of quantum criticality arising from Yb-valence fluctuations in Yb-based metal in periodic crystal and quasicrystal have opened a new class of quantum critical phenomena in correlated electron systems. To clarify whether this new concept can be generalized to other rare-earth-based semimetal and insulator, we study SmS which exhibits golden-black phase transition under pressure. By constructing the model for SmS, we show that Coulomb repulsion between 4f and 5d orbitals at Sm drives first-order valence transition (FOVT) and semimetal-to-insulator transition (MIT…
The mechanism responsible for the extraordinary interface conductivity of LaAlO$_{3}$ on SrTiO$_{3}$ and its insulator-metal transition remains controversial. Here, using density functional theory calculations, we establish a comprehensive and coherent picture that the interplay of electronic reconstructions, lattice distortions, and surface oxygen vacancies fully compensates the polarization potential divergence in LaAlO$_{3}$/SrTiO$_{3}$, explaining naturally the experimental observations under different conditions. While lattice distortions and a charge redistribution between LaO and AlO$_2…
Oxides RNiO3 (R = rare-earth, R # La) exhibit a metal-insulator (MI) transition at a temperature TMI and an antiferromagnetic (AF) transition at TN. Specific heat (CP) and anelastic spectroscopy measurements were performed in samples of Nd1-xEuxNiO3, 0 <= x <= 0.35. For x = 0, a peak in CP is observed upon cooling and warming at essentially the same temperature TMI = TN ~ 195 K, although the cooling peak is much smaller. For x >= 0.25, differences between cooling and warming curves are negligible, and two well defined peaks are clearly observed: one at lower temperatures, that define TN, and t…
We use density functional theory plus dynamical mean-field theory (DFT+DMFT) to study multiple control parameters for tuning the metal-insulator transition (MIT) in CaVO$_3$ thin films. We focus on separating the effects resulting from substrate-induced epitaxial strain from those related to the reduced thickness of the film. We show that tensile epitaxial strain of around 3-4% is sufficient to induce a transition to a paramagnetic Mott-insulating phase. This corresponds to the level of strain that could be achieved on a SrTiO$_3$ substrate. Using free-standing slab models, we then demonstrate…
We combine low energy muon spin rotation (LE-$μ$SR) and soft-X-ray angle-resolved photoemission spectroscopy (SX-ARPES) to study the magnetic and electronic properties of magnetically doped topological insulators, (Bi,Sb)$_2$Te$_3$. We find that one achieves a full magnetic volume fraction in samples of (V/Cr)$_x$(Bi,Sb)$_{2-x}$Te$_3$ at doping levels x $\gtrsim$ 0.16. The observed magnetic transition is not sharp in temperature indicating a gradual magnetic ordering. We find that the evolution of magnetic ordering is consistent with formation of ferromagnetic islands which increase in number …
We report an infrared spectroscopy study of the axion topological insulator candidate EuIn$_2$As$_2$ for which the Eu moments exhibit an A-type antiferromagnetic (AFM) order below $T_N \simeq 18 \mathrm{K}$. The low energy response is composed of a weak Drude peak at the origin, a pronounced infrared-active phonon mode at 185 cm$^{-1}$ and a free carrier plasma edge around 600 cm$^{-1}$. The interband transitions start above 800 cm$^{-1}$ and give rise to a series of weak absorption bands at 5\,000 and 12\,000 cm$^{-1}$ and strong ones at 20\,000, 27\,500 and 32\,000 cm$^{-1}$. The AFM transit…
Some correlated materials display an insulator-to-metal transition as the temperature is increased. In most cases this transition can also be induced electrically, resulting in volatile resistive switching due to the formation of a conducting filament. While this phenomenon has attracted much attention due to potential applications, many fundamental questions remain unaddressed. One of them is its characteristic lengths: what sets the size of these filaments, and how does this impact resistive switching properties. Here we use a combination of wide-field and scattering-type scanning near-field…
In this work we present a model system built out of artificially layered materials, allowing us to understand the interrelation of magnetic phases with that of the metallic-insulating phase at long length-scales, and enabling new strategies for the design and control of materials in devices. The artificial model system consists of superlattices made of SmNiO$_3$ and NdNiO$_3$ layers -- two members of the fascinating rare earth nickelate family, having different metal-to-insulator and magnetic transition temperatures. By combining two complementary techniques -- resonant elastic x-ray scatterin…
Thin films of the correlated transition-metal oxide LaNiO$_3$ undergo a metal-insulator transition when their thickness is reduced to a few unit cells. Here, we use angle-resolved photoemission spectroscopy to study the evolution of the electronic structure across this transition in a series of epitaxial LaNiO$_3$ films of thicknesses ranging from 19 to 2 u.c. grown in situ by RF magnetron sputtering. Our data show a strong reduction of the electronic mean free path as the thickness is reduced below 5 u.c. This prevents the system from becoming electronically two-dimensional, as confirmed by t…
The role of orbital ordering on metal-insulator transition of transition-metal oxides is investigated by the cluster self-consistent field approach in the strong correlation regime. A clear dependence of the insulating gap on the orbital order parameter is found in the single-particle excitation spectra. The thermal fluctuation drives the orbital order-disorder transition, diminishes the gap and leads to the metal-insulator transition. The interplay between spins and orbits results in unusual temperature dependence of the orbital polarization in the orbital insulator, which can be seen in the …
Resistances that exceed the Mott-Ioffe-Regel limit, known as bad metal behavior, and non-Fermi liquid behavior are ubiquitous features of the normal state of many strongly correlated materials. Here we establish the conditions that lead to bad metal and non-Fermi liquid phases in NdNiO3, which exhibits a prototype, bandwidth-controlled metal-insulator transition. We show that resistance saturation is determined by the magnitude of the Ni eg orbital splitting, which can be tuned by strain in epitaxial films, causing the appearance of bad metal behavior under certain conditions. The results shed…
We predict a new class of topological crystalline insulators (TCI) in the anti-perovskite material family with the chemical formula A$_3$BX. Here the nontrivial topology arises from band inversion between two $J=3/2$ quartets, which is described by a generalized Dirac equation for a "Dirac octet". Our work suggests that anti-perovskites are a promising new venue for exploring the cooperative interplay between band topology, crystal symmetry and electron correlation.
Topological insulators are insulating in the bulk but possess metallic surface states protected by time-reversal symmetry. Here, we report a detailed electronic transport study in high quality Bi2Se3 topological insulator thin films contacted by superconducting (In, Al and W) electrodes. The resistance of the film shows an abrupt and significant upturn when the electrodes become superconducting. In turn, the Bi2Se3 film strongly weakens the superconductivity of the electrodes, significantly reducing both their transition temperatures and critical fields. A possible interpretation of these resu…
Topological insulators are bulk semiconductors that manifest in-gap massless Dirac surface states due to the topological bulk-boundary correspondence principle [1-3]. These surface states have been a subject of tremendous ongoing interest, due both to their intrinsic properties and to higher order emergence phenomena that can be achieved by manipulating the interface environment [4-11]. Here, angle resolved photoemission (ARPES) spectromicroscopy and supplementary scanning tunneling microscopy (STM) are performed on the model topological insulator Bi2Se3 to investigate the interplay of crystal…
In intrinsic magnetic topological insulators, Dirac surface state gaps are prerequisites for quantum anomalous Hall and axion insulating states. Unambiguous experimental identification of these gaps has proved to be a challenge, however. Here we use molecular beam epitaxy to grow intrinsic MnBi2Te4 thin films. Using scanning tunneling microscopy/spectroscopy, we directly visualize the Dirac mass gap and its disappearance below and above the magnetic order temperature. We further reveal the interplay of Dirac mass gaps and local magnetic defects. We find that in high defect regions, the Dirac m…
The quasi-one-dimensional metallic system BaVS3 with a metal-insulator transition at T_MI=70 K shows large changes of the optical phonon spectrum, a central peak, and an electronic Raman scattering continuum that evolve in a three-step process. Motivated by the observation of a strongly fluctuating precursor state at high temperatures and orbital ordering and a charge gap at low temperatures we suggest a concerted action of the orbital, electronic, and lattice subsystems dominated by electronic correlations.
The possibility of realizing topological insulators by spontaneous formation of electronic superstructure is theoretically investigated in a minimal two-orbital model including both the spin-orbit coupling and electron correlations on a triangular lattice. Using the mean-field approximation, we show that the model exhibits several different types of charge ordered insulators, where the charge disproportionation forms a honeycomb or kagome superstructure. We find that the charge ordered insulators in the presence of strong spin-orbit coupling can be topological insulators showing quantized spin…
Band inversion, one of the key signatures of time-reversal invariant topological insulators (TIs), arises mostly due to the spin-orbit (SO) coupling. Here, based on ab initio density-functional calculations, we report a theoretical investigation of the SO-driven band inversion in isostructural bismuth and antimony chalcogenide TIs from the viewpoint of its interplay with the crystal-field effect. We calculate the SO-induced energy shift of states in the top valence and bottom conduction manifolds and reproduce this behavior using a simple one-atom model adjusted to incorporate the crystal-fiel…
Topological insulators are a novel materials platform with high applications potential in fields ranging from spintronics to quantum computation. In the ongoing scientific effort to demonstrate controlled manipulation of their electronic structure by external means, stoichiometric variation and surface decoration are two effective approaches that have been followed. In ARPES experiments, both approaches are seen to lead to electronic band structure changes. Such approaches result in variations of the energy position of bulk and surface-related features and the creation of two-dimensional elect…
Orbital degrees of freedom can have pronounced effects on the fundamental properties of electrons in solids. In addition to influencing bandwidths, gaps, correlation strength and dispersion, orbital effects have also been implicated in generating novel electronic and structural phases, such as Jahn-Teller effect and colossal magnetoresistance. In this work, we show for the first time how the orbital nature of bands can result in non-trivial effects of strain on the band structure. We use scanning tunneling microscopy and quasiparticle interference imaging to study the effects of strain on the …
The breaking of time-reversal symmetry by ferromagnetism is predicted to yield profound changes to the electronic surface states of a topological insulator. Here, we report on a concerted set of structural, magnetic, electrical and spectroscopic measurements of \MBS thin films wherein photoemission and x-ray magnetic circular dichroism studies have recently shown surface ferromagnetism in the temperature range 15 K $\leq T \leq 100$ K, accompanied by a suppressed density of surface states at the Dirac point. Secondary ion mass spectroscopy and scanning tunneling microscopy reveal an inhomogene…
We investigate the properties of a single substitutional Mn impurity and its associated acceptor state on the (111) surface of Bi$_2$Se$_3$ topological insulator. Combining ab initio calculations with microscopic tight-binding modeling, we identify the effects of inversion-symmetry and time-reversal-symmetry breaking on the electronic states in the vicinity of the Dirac point. In agreement with experiments, we find evidence that the Mn ion is in the ${+2}$-valence state and introduces an acceptor in the bulk band gap. The Mn-acceptor has predominantly $p$-character, and is localized mainly aro…
We consider the generic problem of a two-level quantum emitter near a two-dimensional Chern insulator in the dipole approximation, and study how the frequency-dependent response and electronic density of states of the insulator modifies the transition rate of the emitter between the ground and excited levels. To this end, we obtain the full real-frequency behavior of the conductivity tensor by performing a tight-binding calculation based on the Qi-Wu-Zhang model and using a Kubo formula, and derive the full electromagnetic Green tensor of the system, which breaks Onsager reciprocity. This enab…
In magnetic topological phases of matter, the quantum anomalous Hall (QAH) effect is an emergent phenomenon driven by ferromagnetic doping, magnetic proximity effects and strain engineering. The realization of QAH states with multiple dissipationless edge and surface conduction channels defined by a Chern number $\mathcal{C}\geq1$ was foreseen for the ferromagnetically ordered SnTe class of topological crystalline insulators (TCIs). From magnetotransport measurements on Sn$_{1-x}$Mn$_{x}$Te ($0.00\leq{x}\leq{0.08}$)(111) epitaxial thin films grown by molecular beam epitaxy on BaF$_{2}$ substra…
The origin of ferromagnetic insulating state of La$_{7/8}$Sr$_{1/8}$MnO$_3$ is investigated. Based on the tight-binding model, it is shown that this state can be attributed to the Peierls instability arisen from the interplay of spin and orbital ordering. The importance of the hole-orbiton-phonon intercoupling in doped manganites is revealed. This picture explains well the recent experimental finding of the reentrance of ferromagnetic metal state at low temperature [Phys. Rev. Lett. 96, 097201 (2006)].
Reply to the Comment by Tikhonov and Khrapai on "Long-range crossed Andreev reflection in a topological insulator nanowire proximitized by a superconductor"
We study the influence of external pressure on the electronic and magnetic structure of EuMnO3 from first-principles calculations. We find a pressure-induced insulator-metal transition at which the magnetic order changes from A-type antiferromagnetic to ferromagnetic with a strong interplay with Jahn-Teller distortions. In addition, we find that the non-centrosymmetric E*-type antiferromagnetic order can become nearly degenerate with the ferromagnetic ground state in the high-pressure metallic state. This situation can be exploited to promote a magnetically-driven realization of a non-centrosy…
Preceded by the discovery of topological insulators, Dirac and Weyl semimetals have become a pivotal direction of research in contemporary condensed matter physics. While easily accessible from a theoretical viewpoint, these topological semimetals pose a serious challenge in terms of experimental synthesis and analysis to allow for their unambiguous identification. In this work, we report on detailed transport experiments on compressively strained HgTe. Due to the superior sample quality in comparison to other topological semimetallic materials, this enables us to resolve the interplay of topo…
Thin layers of topological insulator materials are quasi-two-dimensional systems featuring a complex interplay between quantum confinement and topological band structure. To understand the role of the spatial distribution of carriers in electrical transport, we study the Josephson effect, magnetotransport, and weak anti-localization in bottom-gated thin Bi$_2$Te$_3$ topological insulator films.We compare the experimental carrier densities to a model based on the solutions of the self-consistent Schrödinger-Poisson equations and find excellent agreement. The modeling allows for a quantitative i…
