Nonmagnetic-Doping Induced Quantum Anomalous Hall Effect in Topological Insulators

TL;DR

This paper proposes a nonmagnetic doping method using nitrogen to induce ferromagnetism in topological insulators, enabling the quantum anomalous Hall effect at significantly higher temperatures than previous methods.

Contribution

It introduces a novel nonmagnetic doping strategy with nitrogen that achieves high-temperature QAHE in topological insulators, overcoming limitations of magnetic doping.

Findings

Nitrogen doping induces ferromagnetism in Sb₂Te₃.

Only nitrogen-doped Sb₂Te₃ shows long-range ferromagnetism.

QAHE observed at 17-29 Kelvin, much higher than previous systems.

Abstract

Quantum anomalous Hall effect (QAHE) has been experimentally observed in magnetically doped topological insulators. However, ultra-low temperature (usually below 300 mK), which is mainly attributed to inhomogeneous magnetic doping, becomes a daunting challenge for potential applications. Here, a \textit{nonmagnetic}-doping strategy is proposed to produce ferromagnetism and realize QAHE in topological insulators. We numerically demonstrated that magnetic moments can be induced by nitrogen or carbon substitution in Bi$_2$Se$_3$, Bi$_2$Te$_3$, and Sb$_2$Te$_3$, but only nitrogen-doped Sb$_2$Te$_3$ exhibits long-range ferromagnetism and preserve large bulk band gap. We further show that its corresponding thin-film can harbor QAHE at temperatures of 17-29 Kelvin, which is two orders of magnitude higher than the typical temperatures in similar systems. Our proposed \textit{nonmagnetic} doping scheme may shed new light in experimental realization of high-temperature QAHE in topological insulators.

Tables

Table 1. Table 1: Total magnetic moments (M tot ) and magnetic moments contributed by C ( M C subscript 𝑀 C M_{\rm C} ) or N ( M N subscript 𝑀 N M_{\rm N} ) atom in C- or N-doped bulk Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 . X represents Se or Te atom. The unit of magnetic moment is μ B subscript 𝜇 𝐵 \mu_{B} .

	C-doped TIs		N-doped TIs
TIs	X1	X2	X1	X2
	$M_C$ $M_{\mathrm{tot}}$	$M_C$ $M_{\mathrm{tot}}$	$M_N$ $M_{\mathrm{tot}}$	$M_N$ $M_{\mathrm{tot}}$
Bi2Se3	$0.19$ $0.38$	$0.00$ $0.00$	$0.62$ $1.00$	$0.30$ $0.97$
Bi2Te3	$0.24$ $0.56$	$0.38$ $1.64$	$0.56$ $1.00$	$0.18$ $0.69$
Sb2Te3	$0.06$ $0.13$	$0.00$ $0.00$	$0.00$ $0.00$	$0.14$ $0.58$

Equations

$$\Delta E_{\mathrm{F}}=E_{\mathrm{tot}}^{\mathrm{D}}-E_{\mathrm{tot}}-\sum n_i{\mu }_i,$$

Full text

Quantum anomalous Hall effect (QAHE) Haldane is the counterpart of quantum Hall effect in the absence of Landau levels from strong magnetic field. Due to the topological protection of spatial separation, the chiral edge modes of QHE/QAHE is believed to have immense application potential in future dissipationless quantum devices he2018topological . Since its initial predication in a honeycomb-lattice model Haldane , numerous effort has been made to exploring new platforms for realizing QAHE in related material systems proposal1 ; proposal2 ; proposal3 ; proposal4 ; proposal5 ; proposal51 ; proposal52 ; proposal6 ; proposal7 ; proposal8 ; proposal9 ; proposal10 ; proposal11 ; proposal12 . Because of the virtue of inherent strong spin-orbit coupling in $\mathbb{Z}_{2}$ topological insulators (TIs) TopologicalInsulator1 ; TopologicalInsulator2 , it has narrowed the search for suitable materials possessing intrinsic magnetism. One natural approach to induce magnetism in TIs is to dope magnetic elements DMS ; proposal4 ; TI-magnetism1 ; TI-magnetism2 ; TI-magnetism3 . This has indeed resulted in the first experimental realization of QAHE in Cr/V-doped (Bi,Sb)2Te3 thin films ChangCuiZu ; ExperimentalQAHE1 ; ExperimentalQAHE2 ; 2015NatMat . So far, all the experimentally observed QAHEs in doped TIs were achieved at extremely low temperatures, typically lower than 300 mK, making drastically increase the observation temperature a daunting challenge for potential applications.

In magnetically-doped (Bi,Sb)2Te3 thin films, inhomogeneous ferromagnetism is regarded as one of the main factors that leads to the unexpected low QAHE observation temperature PNAS . Another possible factor is the appearance of the dissipative conduction channels due to metallization of the magnetically doped TI’s bulk-like region APL . Empirically, codoping is an effective way to improve ferromagnetic order, as confirmed in dilute magnetic semiconductors (DMSs) DMS . Our theoretical study has approved that the observed temperature of the QAHE can be largely improved by codoping Sb2Te3 with $p$-type vanadium and $n$-type iodine dopants Qi2016 . Indeed, an increased QAHE temperature was observed in Cr and V-codoped (Bi,Sb)2Te3 thin films CodopingEXP . At the optimal Cr/V ratio, full quantization was achieved at 300 mK, an order of magnitude higher than that with single dopants. The obtained Hall hysteresis loop is more square-like, suggesting a reduced magnetic inhomogeneity in Cr- and V-codoped (Bi,Sb)2Te3 thin films CodopingEXP . In the study of DMSs, it is known that homogeneous DMS by magnetic doping is unrealistic DMS . By carefully tuning crystal growth conditions or utilizing codoping method, magnetic inhomogeneity can only be suppressed more or less. This is because strong attractive interaction from doped magnetic impurities are difficult to suppress their aggregation in DMSs DMS2 .

Alternatively, many materials by nonmagnetic dopants have exhibited robust high-temperature ferromagnetism. For example, carbon(C)-doped ZnO films deposited by pulsed-laser deposition showed ferromagnetism with Curie temperature higher than 400 K CZnO ; CZnO2 ; Above room-temperature ferromagnetism was also observed in C-doped hexagonal boron nitride CBN ; Doping nonmagnetic fluorine in two-dimensional antimonene also realizes robust above-room-temperature ferromagnetism Antimonene . More importantly, no strong attractive interactions exists between nonmagnetic dopants, naturally avoiding the formation of inhomogeneous ferromagnetism and dissipative conduction channels. Therefore, it is of great interest to explore the possibility of realizing high-temperature QAHE in nonmagnetic-doped TIs.

In this Letter, we showed that high-temperature QAHE can be realized in nonmagnetically-doped TIs, e.g., by doping nonmagnetic C or N atoms in Sb2Te3 TI thin films. We took three representative TIs (i.e., Bi2Se3, Bi2Te3, and Sb2Te3) as examples, and found that their magnetic moments can be induced via C or N substitution. For all doping concentrations considered in our study, only N-doped Sb2Te3 can give rise to a long-range ferromagnetic order. Moreover, an estimated Curie temperature of 17-29 Kelvin can be reached at the concentration of 11-22% N-doped Sb2Te3. Aside from the bulk properties, the band gaps of the corresponding Sb2Te3 thin films can also be as large as 17 meV, much larger than what is needed to sustain ferromagnetic order at the high Curie temperature. Explicit Berry curvature calculation further confirms that the N-doped Sb2Te3 thin film can harbor QAHE. Our proposed nonmagnetic doping approach thus offers a versatile route to realize QAHE in TIs and related materials.

Our first-principles calculations were performed using the projected augmented-wave method PAW as implemented in the Vienna Ab-initio Simulation Package (VASP) VASP1 . The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type was used to treat the exchange-correlation interaction GGA . $1\times 1\times 1$ Bi2Se3, Bi2Te3, and Sb2Te3 cells were chosen in this study. For thin film calculations, the film thickness was chosen to be five quintuple layers (QLs). A vacuum buffer space of 30 Åwas used to prevent coupling between adjacent slabs. The kinetic energy cutoff was set to 400 eV. During structural relaxation, all atoms were allowed to relax until the Hellmann-Feynman force on each atom is less than 0.01eV/Å. The Brillouin-zone integration was carried out by using $15\times 15\times 2$ and $9\times 9\times 1$ Monkhorst-Pack grids for bulk and thin film systems, respectively. Unless mentioned otherwise, spin-orbit coupling was considered in all calculations. The Curie temperature $T_{\rm C}$ was estimated within the mean-field approximation $k_{\rm B}T_{\rm C}=\frac{2}{3}xJ$ CurieTemperature , where $k_{\rm B}$ is the Boltzmann constant, $x$ is the doping concentration, and $J$ is the exchange parameter obtained from the total energy difference between ferromagnetic and antiferromagnetic configurations at different doping concentrations.

Let us first discuss the possibility of doping N and C atoms in Bi2Se3, Bi2Te3, and Sb2Te3 TIs. As a concrete example, the bulk structure of Sb2Te3 is shown in Fig. 1(a). Each QL [see Fig. 1(b)] consists of two equivalent Te atoms (Te1), two equivalent Sb atoms, and a third Te atom (Te2). Atoms within one QL are coupled by a strong chemical bond, whereas weak van der Waals forces hold QLs together. To this end, we place the impurities at Se or Te site in $2\times 2\times 1$ bulk Bi2Se3, Bi2Te3, and Sb2Te3 TIs, and compare their formation energies $\Delta E_{\rm F}$, which can be evaluated by

[TABLE]

where $E_{\rm tot}^{\rm D}$ is the total energy of the supercell including one nonmagnetic dopant, $E_{\rm tot}$ is the total energy of the supercell, $\mu_{i}$ is the chemical potential for the species $i$ (host atoms or dopants) and $n_{i}$ is the corresponding number that has been added to or removed from the supercell. We chose rhombohedral Bi, rhombohedral Sb, hexagonal Se, hexagonal Te, nonmagnetic body centered-cubic C, antiferromagnetic hexagonal N as reference to evaluate the chemical potentials of the elements. The computational details were adopted from a previous study of magnetically doped TIs WGZhu .

As displayed in Figs. 1(d)-1(f), the formation energies indicate that Se or Te substitutional sites are preferred by N atoms, because N possesses less formation energies (about 1.0 eV) in the whole range of the accessible host element chemical potentials. Particularly for Sb2Te3, the formation energy of N substitution is only in the range of 0.25-0.50 eV. It seems that the estimated formation energy of C substituted Se or Te is much larger than that of N substituted Se or Te, which is about 5.0 eV. While it is noteworthy that C-doped ZnO has already been realized in experiments even if the estimated formation energy of C substituted O in ZnO is about 5.3 eV CZnO . Therefore, one can reasonably conclude that N or C substituted Bi2Se3, Bi2Te3, and Sb2Te3 TIs can be experimentally produced.

Next, we focused on whether such nonmagnetic dopants in TIs can induce magnetic moments, and further establish long-range ferromagnetism, as that confirmed in C-doped ZnO CZnO ; CZnO2 . TABLE 1 summarizes the magnetic moments of C- and N-substituted Bi2Se3, Bi2Te3, and Sb2Te3 TIs. For the C-doped systems, our results show that the magnetic moments can always arise when C substitutes at Te1 or Se1 site, with the corresponding total moments being respectively 0.38, 0.56, and 0.10 $\mu_{B}$ for three different systems; While when C substitutes Te2 site, only the Bi2Te3 system gives a non-vanishing magnetic moment of 1.64 $\mu_{B}$. Compared with C-doping, magnetic moments can always be induced by N doping except the case of N-substitution at Te1 site of Sb2Te3.

The resulted magnetic moments in N- and C-doped TIs can be understood from the calculated local density of states (LDOS), see the typical example of N-doped Sb2Te3 as displayed in Fig. 2(a). Strong coupling between 2$p$-orbital of N and Te 5$p$, Sb 5$s$/5$p$ orbitals results in above orbitals near the Fermi level to split along spin directions. The spin-up bands are nearly fully occupied while the spin-down bands are partially filled, leading to a magnetic moment of 0.58 $\mu_{B}$ per unit cell. The spin density distribution in Fig. 2(b) further reveals that the ferromagnetism arises from a synergistic effect of the N-dopant and its first-, second-nearest neighboring Sb and Te atoms in one QL. The neighboring Sb and Te atoms are ferromagnetically coupled to the N atom. No $d$ state is observed near the Fermi level, confirming the pure sp-electron ferromagnetism. In addition, little amount of spin polarization along the other direction is observed in one QL. Thus, one can say that the whole QL can be spin-polarized (long-range ferromagnetism) by substituting only one Te atom by one N atom in Sb2Te3. In fact, the magnetic ground state for N-doped Sb2Te3 have been calculated with the inclusion of spin-orbit coupling, and the results indicate that out-of-plane spin polarization is less preferred, which is about 0.5 meV (per unit-cell with one N atom) lower than that of the horizontal axis.

One can also find that no local magnetic moment is induced for N-substitution at the Te1 site of Sb2Te3. As shown in Fig. 2(d), it can be attributed to the strong bonding interaction (the bond length is 1.997 Å) existing between substituted N atom and the nearest Sb atom, which decreases the interaction of the Sb atom and its surrounding Te (Te2 site) atoms in the same QL. While, in the case of N-substitution at Te2 site, the bonding interaction (the bond length is 2.098 Å) between substituted N atom and its nearest Sb atom is weaker than that in the case of substituting at Te1 site. From Fig. 2(c), one can see that this weaker bonding interaction does not decrease the interaction between the substituted N atom and its surrounding Sb atoms, which is in agreement with above observation in Fig. 2(a) that ferromagnetism in the case of N-substitution at Te2 arises from a synergistic effect of the N dopant and its first-, second-nearest neighboring Sb and Te atoms within one QL.

Aside from the ferromagnetic ordering of N- and C-doped TIs, two other prerequisites  he2018topological for the observation of QAHE include: (1) The bulk gap exists in N- and C-doped TIs; (2) The surface bandgap can be opened due to the ferromagnetism. To this point, let us continue to investigate the variance of band structures due to the C- and N-doping. First, we focused on the bulk band structures of N- or C-doped Bi2Se3, Bi2Te3, and Sb2Te3 TIs. Figure 3(a) displays the bulk band structure of 11% N-doped Sb2Te3 at Te2 site, opening a small bulk gap about 9 meV. In fact, the bulk gap of N-doped Sb2Te3 can be dramatically increased to 39 meV [See Fig. 3(b)] when N-doping concentration increases from 11% to 22%. In 11% N-doped Sb2Te3, further analysis of the characters of different elements [see Figs. 3(a-1)-3(a-3)] reveals that the valence band (conduction band) close to the band gap is dominated by Te (Sb) atoms, while N contributes mainly to the lower part of the valence bands. These results directly reflect the strong bonding interaction between N and surrounding Sb atoms due to the larger electronegativity of N. If doping concentration increases to 22%, the contribution from Sb (Te) atoms to conduction (valence band) band close to the gap obviously decreases [see Figs. 3(b-1)-3(b-3)]. This is because the more N dopants substitute Te atoms, the stronger bonding interaction between N and surrounding Sb atoms arises, and the resulting large energy difference between bonding and anti-bonding orbitals helps enlarge the bulk gap. In addition, our results presented in Figs. 3(c)-3(f) confirm that appropriate in-plane tensile strains can not only greatly enlarge the bulk gap from 9 mev to 46 (1% tensile strain), 74 meV (2% tensile strain), and 85 meV (3% tensile strain), but also shift the Fermi level closer to the bulk band gap. The strain effect on bulk band gap and Fermi level will be beneficial for realization of high-temperature QAHE in N-doped Sb2Te3. Calculated results for other doping configurations can be found in Supplemental Materials. Unfortunately, no global bulk band gap appears in these systems. Thus, our results prove that only N-doped Sb2Te3 is an ideal candidate for high-temperature QAHE and its corresponding bulk gap can be modulated via doping concentration or applying in-plane strain.

So far, the above results strongly suggest that N-doped Sb2Te3 is a highly potential candidate for exploring high-temperature QAHE. In principle, one still needs to see whether its corresponding thin-film system can also open sizable band gap or not. By analyzing the band structures, we showed that these thin-film systems can indeed support topological states (i.e., QAHE) at different doping concentrations. In the absence of dopants, the thin film of 5QLs’ Sb2Te3 hosts a massless Dirac fermion, manifested as the hallmark linear Dirac dispersion near $\Gamma$ point Sb2Te3 [see Fig. 1(c)]. When N-dopants are introduced, the Dirac fermion acquires a finite mass to open up a surface band gap, due to the presence of induced ferromagnetism. This satisfies the prerequisites to form QAHE TopologicalInsulator2 .

Figure 4 displays the band structures and Berry curvatures along high-symmetry line near $\Gamma$ point at two different N-doping concentrations of 7% (a,b) and 13% (c,d), respectively. One can find that the band gap increases with the doping concentration. It is noteworthy that our calculated band structure for 7% doping without spin opens a small gap about 7 meV due to the inter-surface hybridization. When the spin degree of freedom is invoked, the strong ferromagnetic exchange field closes the band gap and forms band crossings between spin-up and -down bands, which then naturally open up topologically non-trivial band gap when the spin-orbit coupling is considered.

Quantitatively, one can provide a unambiguous evidence for the manifestation of QAHE by integrating the Berry curvature of the occupied valence BerryCurvature1 ; BerryCurvature2 . Figures 4(b) and 4(d) display the Berry curvature distribution along high symmetry lines, which exhibits a large negative peak near $\Gamma$ point and zero elsewhere. As a consequence, the total integration of the Berry curvatures (i.e., the Hall conductance) must be nonzero. On the other hand, the region with finite Berry curvatures broadens with the increase of the surface band gap, and the curvature peak gradually decreases, implicating the unchanged quantized Hall conductance. Therefore, we showed that the N-doped Sb2Te3 thin film can realize the QAHE.

For particular doping concentrations, the Curie temperatures are respectively $T_{\rm C}=17(29)$ Kelvin at 11% (22%) N-doping concentrations, by using the magnetic coupling strengths extracted from our first-principles calculations and the mean-field theory CurieTemperature . This indicates that the Curie temperature might be raised with increasing the doping concentration. But one has to note that higher doping concentration may decrease the spin-orbit coupling of the whole system.

One merit of N-doping in TIs is that nonmagnetic doping scheme does not induce inhomogeneous distribution that occurs in magnetic dopants. Another one is that dissipative conduction channels from metallization in magnetically doped TIs cannot be formed by nonmagnetic and nonmetallic N doping. Since there were many materials by nonmagnetic dopants exhibiting robust high-temperature ferromagnetism  CZnO ; CZnO2 ; CBN ; Antimonene , our work will provide a highly desirable scheme to overcome the difficulty of observing QAHE at high temperatures ChangCuiZu ; ExperimentalQAHE1 ; ExperimentalQAHE2 ; 2015NatMat .

In conclusion, our findings demonstrate that N-doped TIs provides a versatile route to realizing high-temperature QAHE. Our proof-of-principle demonstrations revealed that nonmagnetic N-doping results in the formation of bulk gaps, long-range ferromagnetism with moderate Curie temperatures, and non-zero Berry curvatures. The estimated QAHE observation temperature is 17-29 K in 11-22% N-doped Sb2Te3 thin films, which is about two orders of magnitude higher than what was previously reported. By further increasing and tailoring the doping and TI environments, it is feasible to further increase the QAHE observation temperature.