Abstract
It has been proposed that adding disorder to a topologically trivial mercury telluride/cadmium telluride (HgTe/CdTe) quantum well can induce a transition to a topologically nontrivial state. The resulting state was termed topological Anderson insulator and was found in computer simulations of the BernevigHughesZhang model. Here, we show that the topological Anderson insulator is a more universal phenomenon and also appears in the KaneMele model of topological insulators on a honeycomb lattice. We numerically investigate the interplay of the relevant parameters and establish the parameter range in which the topological Anderson insulator exists. A staggered sublattice potential turns out to be a necessary condition for the transition to the topological Anderson insulator. For weak enough disorder, a calculation based on the lowestorder Born approximation reproduces quantitatively the numerical data. Our results thus considerably increase the number of candidate materials for the topological Anderson insulator phase.
Similar content being viewed by others
Introduction
Topological insulators (TIs) are novel materials which have raised a great deal of interest over the past decade^{1,2}. One of their distinguishing features is the existence of conducting boundary states together with an insulating bulk. The boundary states are protected by timereversal symmetry (TRS) and exist both in twodimensional (2D) and threedimensional (3D) TIs. In 2D TIs, the boundary states lead to an edge conductance of one conductance quantum per edge for chemical potentials inside the bulk band gap^{3,4,5}.
It is a challenging task to find candidate materials for TIs. So far, only a limited number of materials are known. The most prominent 2D TIs are HgTe/(Hg, Cd)Te quantum wells (HgTeQWs)^{6} and InAs/GaSb heterostructures^{7,8}, whereas 3D TIs were found for instance in Bi_{1−x}Sb_{x}^{9}. The fact that their metallic surface states emerge due to a topological property of the bulk band structure means that they are robust to weak disorder. However, one expects that a large amount of disorder should ultimately localize the surface states and render them insulating.
All the more surprising, it was predicted that the opposite transition can happen in certain parameter ranges: adding strong disorder can convert a trivial insulator without edge states into a topological insulator with perfectly conducting edge states. Materials that exhibit this new state have been termed topological Anderson insulators (TAIs).
This effect was first theoretically predicted based on the lattice version of the BernevigHughesZhang (BHZ) model for HgTeQWs in the presence of Anderson disorder^{10,11}. For Anderson disorder, a random onsite potential, uniformly distributed in an energy window of width 2W, is assigned to each lattice site of a tightbinding model. From Anderson’s theory of localization^{12} one expects that a system with finite conductance without disorder undergoes a transition to a system with localized states and suppressed conductance as the disorder is increased beyond a certain threshold value. The behavior of TAIs instead is quite different. A TAI is an ordinary band insulator in the clean limit. Above a critical disorder strength W, an interesting topological state appears, in which the material features a quantized conductance. For even stronger W, above the disorder strength at which the states of the conduction and valence band localize, it was proposed that tunneling across the bulk becomes possible^{13}, probably enabled by percolating states^{14} and the conductance is again suppressed.
The disorderinduced transition can be understood by a renormalization of the model parameters. The BHZ model with disorder and band mass m can be approximated by an effective model of a clean system and renormalized mass . Using an effectivemedium theory and the selfconsistent Born approximation (SCBA), it was shown that for certain model parameters, can become negative even if the bare mass m is positive^{15}. As a consequence, the effective model becomes that of a TI and features edge states with a quantized conductance of G_{0} = e^{2}/h^{16}.
Furthermore, TAIs have been predicted in several related systems, for instance in a honeycomb lattice described by the timereversalsymmetry breaking Haldane model^{17}, a modified Dirac model^{17}, the BHZ model with s_{z} nonconserving spinorbit coupling^{18}, as well as in 3D topological insulators^{19}. Moreover, similar transitions from a topologically trivial to a topologically nontrivial phase have been found to be generated by periodically varying potentials^{20} or phonons^{21}. In contrast to onsite Anderson disorder, certain kinds of bond disorder cannot produce a TAI as they lead only to a positive correction to m^{22,23}. So far, however, the TAI phase was not found in the KaneMele model on a honeycomb lattice, describing for example graphene or proposed TIs such as silicene, germanene and stanene^{24,25,26,27}. First indications to this phase were already found, showing that the KaneMele model without a staggered sublattice potential hosts extended bulk states even for large disorder strengths^{28}.
In this paper we show the existence of TAIs in the KaneMele model by means of tightbinding calculations. The interplay between the parameters characterizing intrinsic spinorbit coupling (SOC) λ_{SO}, extrinsic Rashba SOC λ_{R} and a staggered sublattice potential λ_{ν} turns out to be crucial for the visibility of TAIs and we calculate the parameter ranges in which TAIs can be observed. We find analytically that to lowest order in W, the parameters λ_{SO} and λ_{R} are not renormalized with increasing disorder strength, in contrast to λ_{ν}. However, a new effective hopping λ_{R3} is generated due to the disorder, which is related but not identical to λ_{R}. Although λ_{R} is not a crucial ingredient for the existence of TAIs, it significantly alters the physics of topological insulators in various ways^{29,30} and, as we will show below, strongly affects the TAI state.
Even though recently first signs of a TAI phase may have been found experimentally in evanescently coupled waveguides^{31}, there has been no experimental evidence so far for the existence of the TAI phase in fermionic systems. The main difficulty is the requirement of a rather large and specific amount of disorder, which is difficult to control in the topological insulators currently investigated, where the 2D TI layer is buried inside a semiconductor structure. In contrast, producing and controlling disorder in 2D materials described by the KaneMele model could be much easier. Disorder in 2D materials with honeycomb structure can be produced by randomly placed adatoms^{32,33} or a judicious choice of substrate material^{34,35,36,37}. Moreover, a sizeable staggered sublattice potential can be generated via a suitable substrate material^{38}. Other means of engineering disorder were proposed in periodically driven systems^{39,40}. Finally, honeycomb structures with the SOC necessary to produce a topological phase have already been realized using ultracold atoms in optical lattices^{41}, in which disorder can in principle be engineered.
Results
Setup
The basis of our calculations is the KaneMele model^{5} given by the following Hamiltonian on a tightbinding honeycomb lattice
which has been supplemented by an onsite Anderson disorder term with disorder strength W and uniformly distributed random variables . The summations over the lattice sites 〈ij〉 and 〈〈i, j〉〉 include all nearest neighbors and nextnearest neighbors, respectively. The operators , are creation and annihilation operators for the site i of the lattice. The parameters t, λ_{SO} and λ_{R} are the nearestneighbor hopping strength, intrinsic SOC and Rashba SOC, respectively. If the nextnearest neighbor hopping from site j to site i corresponds to a rightturn on the honeycomb lattice, then ν_{ij} = 1, otherwise ν_{ij} = −1. Furthermore, s =(s^{x}, s^{y}, s^{z}) is the vector of Pauli matrices for the spin degree of freedom and is the unit vector between sites j and i. The Wannier states at the two basis atoms of the honeycomb lattice are separated in energy by twice the staggered sublattice potential λ_{ν}, with ξ_{i} = 1 for the A sublattice and ξ_{i} = −1 for the B sublattice. The lattice constant is a.
The band structure of this model depends strongly on the parameter set λ_{SO}, λ_{R} and λ_{ν}. In the clean limit and for λ_{R} = 0, the system will be a topological insulator for and a trivial insulator otherwise^{5}. The tightbinding lattice and examples for the band structure in the clean limit are displayed in Fig. 1. For W = λ_{ν} = 0, the system will be a topological insulator if and a metal or semimetal otherwise. For finite λ_{ν} and λ_{R} the situation is more complex and a topological transition appears for values within these two boundaries.
Numerical solution
For λ_{R} = 0, we find a TAI phase for parameters close to the topological transition at . Changing this ratio corresponds to changing the band mass in the case of the BHZ model. Figure 2 shows the conductance for different values of λ_{ν}. We find that for the system is a topological insulator. For W = 0, i.e., in the clean case, the conduction and valence bands are separated by a red region with a quantized conductance of 2G_{0}. Remarkably, with increasing disorder strength, the states in the conduction and valence bands localize, but the helical edge states that are responsible for the conductance of 2G_{0} exist for an even larger energy window. The conductances and the vanishing error bars for the two distinct energy values E_{F} = 0, E_{F} = 0.2t in the lower row of Fig. 2 show that the conductance quantization and with it the topological nature of the system, persist for the vast majority of microscopic disorder configurations. Interestingly, for λ_{ν} = 1.65t = 5.5λ_{SO}, the system is a trivial insulator at W = 0. The trivial gap closes however and at W ≈ t the system develops a topologically nontrivial gap and edge states. This can be seen from the quantized conductance. Finally, for λ_{ν} = 1.85t ≈ 6.2λ_{SO}, the closing of the trivial gap and reopening of the topological gap happens at a disorder strength which is strong enough to destabilize the emergent topological phase. Features of the conductance quantization can still be seen, but this behavior is not that robust anymore. As no averaging is done in the upper row of Fig. 2 and a new disorder configuration is taken for every data point, destabilization of the topological phase can be seen by red and white speckles in the figure.
We find that no TAI exists without staggered sublattice potential (λ_{ν} = 0). If both λ_{ν} and λ_{R} are finite, the TAI phase is in general less pronounced, see Fig. 3. The plot on the right shows the closing of a trivial gap and emergence of a topological phase at W ≈ 0.5t.
Furthermore, we observe that the simultaneous presence of intrinsic and Rashba SOC (both λ_{R} ≠ 0 and λ_{SO} ≠ 0) destroys the particlehole symmetry in the spectrum. In the absence of Rashba SOC, the symmetry operator ϒ, which acts on the lattice operators as and for the sublattices A and B, leaves the (disorderfree) Hamiltonian invariant. ϒ can be viewed as particlehole conjugation combined with spatial inversion and the inversion is needed to leave the staggered sublattice potential term invariant.
Lowestorder Born approximation
In the selfconsistent Born approximation, the selfenergy ∑ for a finite disorder strength is given by the following integral equation^{15,42}
where is the Fourier transform of H in the clean limit^{5}. The coefficient 1/3 originates from the second moment of the uniform distribution function of the disorder amplitudes and E_{F} is the chemical potential. The integration is over the full first Brillouin zone. We use the lowestorder Born approximation, which means setting ∑ = 0 on the righthand side of the equation.
After a lowenergy expansion of , the integral can be evaluated analytically^{15} for λ_{R} = 0. This requires keeping the terms up to second order in k wherever this is the leading kdependent order. The evaluation yields the renormalized staggered sublattice potential
For a certain set of parameters, the logarithm can be negative and is reduced compared to λ_{ν}. Moreover, we find that λ_{SO} is not renormalized to order W^{2}. Therefore, it is possible to obtain . The system thus makes a transition from a trivial insulator to a topological insulator with increasing W.
For a more quantitative treatment, we evaluate the integral for the full Hamiltonian numerically. The selfenergy ∑ is then written as a linear combination of several independent contributions
with and Γ^{ab} = [Γ^{a}, Γ^{b}]/(2i). Here, σ^{x}, σ^{y}, σ^{z} denote the Pauli matrices for the sublattice index. This leads to the following equations for the renormalized quantities
whereas and remain unrenormalized to lowest order in W. Surprisingly, a new coupling is created by the disorder. This coupling has the matrix structure Γ^{3}, which is similar but not identical to the one for Rashba SOC. Expressing this new term in the lattice coordinates of Eq. (1) reveals that it corresponds to a Rashbatype nearestneighbor hopping term which is asymmetric and appears only for bonds that are parallel to the unit vector (0, 1),
where 〈ij〉_{ν} stands for summations over strictly vertical bonds only. Furthermore, we find to lowest order in W that for λ_{R} = 0.
For W = λ_{R} = 0, the upper and lower edge of the gap are at the energies . This is the case for both topological and trivial insulators. Extrapolation of these equations to finite W leads to the conditions . The solid black lines in Fig. 2 are the two solutions to these equations and describe the closing and reopening of the gap qualitatively for small W.
For finite λ_{R} and therefore finite λ_{R3}, there is no analytical expression of the gap energy. In this case, we read off the positions of the gap edges from band structure calculations for several values of λ_{R} and λ_{R3}. An interpolation leads to two functions h_{U,L}(λ_{ν}, λ_{R}, λ_{R3}) for the upper and lower band edge in the clean system. Replacing the unperturbed by the renormalized parameters yields two equations
The solutions of these equations are indicated by the solid black lines in Fig. 3. Hence, these results agree with the numerical data for small W without any fitting parameter. Deviations appear for larger W, when the lowestorder Born approximation is not applicable.
Phase diagram
Figure 4 shows a phase diagram as a function of λ_{ν} and λ_{R} based on the tightbinding simulations. The dark color marks the regions for which a critical disorder strength W_{c} exists above which the system is a TAI (blue for λ_{SO} = 0.3t, red for λ_{SO} = 0.15t). The TAI phase is located along the boundary separating trivial from topological insulators in the clean case. Towards larger λ_{R}, the TAI region becomes narrower and eventually vanishes above a critical λ_{R}. Figure 5 shows the critical disorder strength W_{c} as a function of λ_{ν} for a fixed value of λ_{R}.
In Figs 2 and 3 rather large values of the parameters λ_{SO}, λ_{ν} and λ_{R} were chosen to better visualize the effect. The TAI phenomenon scales down also to smaller values of the parameters, as the red region in the Fig. 4 indicates, but the TAI phase becomes less pronounced in the conductance plots and is harder to identify. Material parameters for stanene for example are t = 1.3 eV, λ_{SO} = 0.1 eV^{43} and λ_{R} = 10 meV^{44}. We suspect that disorder, e.g., originating from missing or dislocated atoms, can reach disorder strengths in the eV range.
Alternative disorder models
Anderson disorder is a special model for disorder which is not necessarily representative for all TI materials. To better understand the effect of the disorder model, we briefly remark on the following disorder Hamiltonian
In contrast to the Anderson disorder model, where a random potential is assigned to every lattice site, here the distribution function for η_{i} is such that only a fraction 0 < ρ ≤ 1 of the sites are affected by disorder. Denoting the total number of sites by N, we assume η_{i} = 1 on ρN/2 sites, η_{i} = −1 on ρN/2 sites and η_{i} = 0 on the remaining sites. The disorder amplitude W is constant. Because , the normalization factor in Eq. (8) ensures that the mean squared disorder strength is equal to the Anderson disorder case for ρ = 1.
For general ρ, the prefactor in Eq. (2) is thus replaced by ρW^{2}/3. The lowestorder Born approximation for the disorder model (8) therefore predicts that a reduced disorder density ρ can be exactly compensated by an increased amplitude W. For large enough ρ, this is indeed confirmed in the tightbinding simulations.
However, because a single impurity (ρ = 1/N) cannot destroy the topological phase, it is clear that the TAI phase should eventually vanish for ρ → 0 at arbitrary W. Nevertheless, we find numerical evidence for the TAI phase at surprisingly low impurity densities. A TAI region remains pronounced for densities as low as ρ = 0.1.
Discussion
In conclusion, we have shown that the topological Anderson insulator is a significantly more universal phenomenon than previously thought. Using a combination of an analytical approach and tightbinding simulations, we have established that the topological Anderson insulator appears in the KaneMele model that describes potential topological insulators such as silicene, germanene and stanene and that can also be realized in optical lattices. We have observed a transition from a trivially insulating phase to a topological phase at a finite disorder strength and have mapped out the phase diagram as a function of the staggered sublattice potential (~λ_{ν}) and the Rashba spinorbit coupling (~λ_{R}). The new Anderson insulator exists at the boundary between trivial and topological insulators for small λ_{R} and finite λ_{ν}, but not at the boundary between a semimetal and a topological insulator for small λ_{ν} and finite λ_{R}. Since the KaneMele model on a honeycomb lattice describes a wide class of candidate materials for topological insulators, we hope that our work will trigger experimental efforts to confirm the existence of the topological Anderson insulator.
Methods
The numerical simulations were done with the tightbinding Hamiltonian (1) on a honeycomb lattice with rectangular shape of width w = 93a and length l = 150a using the Kwant code^{45}. A smaller version of the sample is shown in Fig. 1. Both the upper and lower edge are taken to be of zigzag type. At the left and right edges two semiinfinite, metallic leads of width w are attached. The leads are also modeled by a honeycomb lattice with only nearestneighbor hopping and a finite onsite energy of 1.2t to bring them into the metallic regime.
Additional Information
How to cite this article: Orth, C. P. et al. The topological Anderson insulator phase in the KaneMele model. Sci. Rep. 6, 24007; doi: 10.1038/srep24007 (2016).
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).
Qi, X. & Zhang, S. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).
Bernevig, B. A., Hughes, T. L. & Zhang, S.C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757 (2006).
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766 (2007).
Knez, I., Du, R.R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2011).
Li, T. et al. Observation of a helical Luttinger liquid in InAs/GaSb quantum spin Hall edges. Phys. Rev. Lett. 115, 136804 (2015).
Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970 (2008).
Li, J., Chu, R.L., Jain, J. K. & Shen, S.Q. Topological Anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).
Jiang, H., Wang, L., Sun, Q.f. & Xie, X. C. Numerical study of the topological Anderson insulator in HgTe/CdTe quantum wells. Phys. Rev. B 80, 165316 (2009).
Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958).
Chen, L., Liu, Q., Lin, X., Zhang, X. & Jiang, X. Disorder dependence of helical edge states in HgTe/CdTe quantum wells. New J. Phys. 14, 043028 (2012).
Girschik, A., Libisch, F. & Rotter, S. Percolating states in the topological Anderson insulator. Phys. Rev. B 91, 214204 (2015).
Groth, C. W., Wimmer, M., Akhmerov, A. R., Tworzydło, J. & Beenakker, C. W. J. Theory of the topological Anderson insulator. Phys. Rev. Lett. 103, 196805 (2009).
Prodan, E. Threedimensional phase diagram of disordered HgTe/CdTe quantum spinHall wells. Phys. Rev. B 83, 195119 (2011).
Xing, Y., Zhang, L. & Wang, J. Topological Anderson insulator phenomena. Phys. Rev. B 84, 035110 (2011).
Yamakage, A., Nomura, K. & Imura, K.I., Kuramoto, Y. DisorderInduced Multiple Transition Involving Z2 Topological Insulator. J. Phys. Soc. Jpn. 80, 053703 (2011).
Guo, H.M., Rosenberg, G., Refael, G. & Franz, M. Topological Anderson insulator in three dimensions. Phys. Rev. Lett. 105, 216601 (2010).
Fu, B., Zheng, H., Li, Q., Shi, Q. & Yang, J. Topological phase transition driven by a spatially periodic potential. Phys. Rev. B 90, 214502 (2014).
Garate, I. Phononinduced topological transitions and crossovers in Dirac materials. Phys. Rev. Lett. 110, 046402 (2013).
Song, J., Liu, H., Jiang, H., Sun, Q.f. & Xie, X. C. Dependence of topological Anderson insulator on the type of disorder. Phys. Rev. B 85, 195125 (2012).
Lv, S.H., Song, J. & Li, Y.X. Topological Anderson insulator induced by intercell hopping disorder. J. Appl. Phys. 114, 183710 (2013).
Aufray, B. et al. Graphenelike silicon nanoribbons on Ag(110): A possible formation of silicene. Appl. Phys. Lett. 96, 183102 (2010).
Kara, A. et al. A review on silicene  new candidate for electronics. Surf. Sci. Rep. 67, 1 (2012).
Dvila, M. E., Xian, L., Cahangirov, S., Rubio, A. & Lay, G. L. Germanene: a novel twodimensional germanium allotrope akin to graphene and silicene. New J. Phys. 16, 095002 (2014).
Zhu, F.f. et al. Epitaxial growth of twodimensional stanene. Nat. Mater. 14, 1020 (2015).
Prodan, E. Disordered topological insulators: a noncommutative geometry perspective. J. Phys. A: Math. Theor. 44, 113001 (2011).
Orth, C. P., Strübi, G. & Schmidt, T. L. Point contacts and localization in generic helical liquids. Phys. Rev. B 88, 165315 (2013).
Rod, A., Schmidt, T. L. & Rachel, S. Spin texture of generic helical edge states. Phys. Rev. B 91, 245112 (2015).
Stützer, S. et al. Experimental realization of a topological Anderson insulator. Paper presented at CLEO: QELS Fundamental Science 2015, San Jose (CA), United States, 10–15 May 2015, 10.1364/CLEO_QELS.2015.FTh3D.2.
Weeks, C., Hu, J., Alicea, J., Franz, M. & Wu, R. Engineering a robust quantum spin Hall state in graphene via adatom deposition. Phys. Rev. X 1, 021001 (2011).
Jiang, H., Qiao, Z., Liu, H., Shi, J. & Niu, Q. Stabilizing topological phases in graphene via random adsorption. Phys. Rev. Lett. 109, 116803 (2012).
Ando, T. Screening effect and impurity scattering in monolayer graphene. J. Phys. Soc. Jpn. 75, 074716 (2006).
Ishigami, M., Chen, J. H., Cullen, W. G., Fuhrer, M. S. & Williams, E. D. Atomic structure of graphene on SiO2 . Nano Letters 7, 1643–1648 (2007).
Fratini, S. & Guinea, F. Substratelimited electron dynamics in graphene. Phys. Rev. B 77, 195415 (2008).
Varlet, A. et al. Tunable Fermi surface topology and Lifshitz transition in bilayer graphene. Synthetic Metals 210, 19 (2015) doi: 10.1016/j.synthmet.2015.07.006.
Nevius, M. S. et al. Semiconducting Graphene from Highly Ordered Substrate Interactions. Phys. Rev. Lett. 115, 136802 (2015).
Titum, P., Lindner, N. H., Rechtsman, M. C. & Refael, G. Disorderinduced Floquet topological insulators. Phys. Rev. Lett. 114, 056801 (2015).
Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
Bruus, H. & Flensberg, K. ManyBody Quantum Theory in Condensed Matter Physics (Oxford University Press, 2004).
Xu, Y. et al. Largegap quantum spin Hall insulators in tin films. Phys. Rev. Lett. 111, 136804 (2013).
Liu, C.C., Jiang, H. & Yao, Y. Lowenergy effective Hamiltonian involving spinorbit coupling in silicene and twodimensional germanium and tin. Phys. Rev. B 84, 195430 (2011).
Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X. Kwant: a software package for quantum transport. New J. Phys. 16, 063065 (2014).
Acknowledgements
C.P.O., T.S. and C.B. acknowledge financial support by the Swiss SNF and the NCCR Quantum Science and Technology. T.L.S. acknowledges support by National Research Fund, Luxembourg (ATTRACT 7556175).
Author information
Authors and Affiliations
Contributions
C.P.O. and T.S. performed the numerical simulations with input from C.B. and T.L.S. All authors contributed to writing the manuscript.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Orth, C., Sekera, T., Bruder, C. et al. The topological Anderson insulator phase in the KaneMele model. Sci Rep 6, 24007 (2016). https://doi.org/10.1038/srep24007
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep24007
This article is cited by

Fredholm Homotopies for StronglyDisordered 2D Insulators
Communications in Mathematical Physics (2023)

Observation of spinpolarized Anderson state around charge neutral point in graphene with Feclusters
Scientific Reports (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.