Gate-Tunable Quantum Anomalous Hall Effects in MnBi_2Te_4 Thin Films
GGate-Tunable Quantum Anomalous Hall Effects in MnBi Te Thin Films
Chao Lei and A.H. MacDonald Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA (Dated: January 19, 2021)The quantum anomalous Hall (QAH) effect has recently been realized in thin films of intrinsicmagnetic topological insulators (IMTIs) like MnBi Te . Here we point out that that the QAHgaps of these IMTIs can be optimized, and that both axion insulator/semimetal and Chern insula-tor/semimetal transitions can be driven by electrical gate fields on the ∼
10 meV/nm scale. Thiseffect is described by combining a simplified coupled-Dirac-cone model of multilayer thin films withSchr¨odinger-Poisson self-consistent-field equations.
Introduction—
Following its initial experimental re-alization in magnetically-doped topological insulators(MTI)[1], the quantum anomalous Hall (QAH) effect[2]has been widely studied[3–7]. The QAH effect is of in-terest because of its potential applications in quantummetrology[8–10] and spintronics[11, 12], and because ofits potential role as a platform for chiral topologicalsuperconductivity[13, 14], Majorana edge modes[15], andMajorana zero modes[16]. Because of strong disorder,thought to be due mainly to random magnetic dopants,the QAH effect appears only at extremely low tempera-tures in MTIs. Overcoming this disorder effect has beenrecognized as key to realizing the higher temperatureQAH effects that would bring more applications withinreach.Topological materials with spatially ordered mag-netic moments can be realized by forming heterojunc-tions between ferromagnetic insulators and topologicalinsulators[4, 17–20] or by growing intrinsic MnBi X orMnSb X magnetic topological insualtors (IMTIs) andrelated superlattices [21–55], where X=(Se,Te). TheIMTIs consist of van der Waals coupled septuple-layerbuilding blocks that have Mn local moment layers attheir centers. To date the anomalous Hall resistancesmeasured[4, 17–20] in the heterojunction systems arestill far from their ideal quantized values, mainly dueto weak exchange coupling between the surface states ofthe topological insulator and moments in the ferromag-netic insulator. On the other hand reasonably accuratelyquantized Hall resistances have been measured[37] in five-septuple layer thin films of MnBi Te (MBT) in the ab-sence of magnetic field at a temperatures exceeding 1Kand, in the presence of magnetic field ∼ R xy can deviate by as much as a factor of 3%from exact quantization and the longitudinal resistances R xx are still ∼ . − . h/e . This compares with Hallresistivity deviations smaller than 1 ppm [9, 10] at thelowest temperatures in the magnetic-dopant case. In thispaper we theoretically explore the possibility of optimiz-ing the QAH in MBT by applying electrical gate voltages to increase the QAH gap, and also address gate-tunedtransitions between insulating and semimetallic states,and in the case of ferromagnetic spin configurations be-tween insulating states with different Chern numbers. Mn Bi Te (a) (c) Semimetal
QAHIAxion InsulatorNormal Insulator (b)
MBT SL
MBT SL
MBT SLMBT SL N Electric Field [meV/nm]
0 10 20 30
FIG. 1. (a) Crystal structure of one septuple layer of MBT,which consists of seven layers with one magnetic Mn ion in thecenter and two Te-Bi-Te trilayers outside; Two Dirac cones lieat the surface. (b) Illustration of the coupling between theDirac cones at the two surfaces of antiferromagnetic MBTthin films. The Dirac-cone masses are produced by exchangecoupling to Mn local moments and are opposite in sign foreven layer-number thin films and identical for odd-number-layer thin films. (c) Phase diagram of antiferromagnetic thinfilms, with electric field on the x axis and the number of sep-tuple layers N in the film along the y axis. Different colorsrepresent different phases. The red and blue dashed lines re-spectively plot the electric fields E at which eEt N = E gs and eEt N /(cid:15) zz = E gs , where E gs ∼
37 meV is the surface stategap at E = 0 [21] and (cid:15) zz ∼ . Our analysis is based on the simplified coupled Dirac-cone model[21] illustrated schematically in Fig. 1 (a) and(b)) that captures most topological and electronic prop-erties, and on a self-consistent-field Sch¨odinger-Possionapproximation for the interacting carriers. We show thatgates can maximize the QAH gap either by compensat-ing for unintentional electric fields, or in the case of high-Chern-number ferromagnetic (FM) insulators, by tuning a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n the gate field to an optimal non-zero value. Gate-Field Phase Diagram—
The low-energy proper-ties of MBT thin films are accurately modeled [21, 56] by a simple Hamiltonian that includes only Dirac conesurface states on the top and bottom surface of each sep-tuple layer (as illustrated in Fig. 1 (a)), and hoppingbetween Dirac cones: H = (cid:88) k ⊥ ,ij (cid:104)(cid:16) ( − ) i ¯ hv D (ˆ z × σ ) · k ⊥ + m i σ z + V i (cid:17) δ ij + ∆ ij (1 − δ ij ) (cid:105) c † k ⊥ i c k ⊥ j . (1)Here the Dirac cone labels i and j are respectively oddand even on the top and bottom surface of each septu-ple layer, ¯ h is the reduced Planck’s constant, v D is theDirac-cone velocity and V i is the self-consistent Hartreepotential on surface i . In the following we retain onlythe largest Dirac-cone hybridization parameters, letting∆ ij → ∆ S for hopping within the same septuple layerand ∆ ij → ∆ D for hopping across the van der Waals gapbetween adjacent septuple layers. Exchange interactionsbetween Dirac-cone spins and local moments in the inte-rior of each septuple layer are captured by the mass gapparameter m i = (cid:80) α J iα M α where α is a septuple-layerlabel and M α = ± α . We include interactions between Dirac conespins and Mn local moments in the same septuple layerwith exchange constant J S and with local moments inthe closest adjacent layer with exchange constant J D . Inthis paper we set ∆ S =84 meV, ∆ D = -127 meV, J S = 36meV and J D = 29 meV, based on the fit to MnBi Te abinitio electronic structure calculations discussed in detailin Ref. 21.The Dirac-cone Hartree potentials V i in Eq. 1 are calcu-lated from a discrete Poisson equation in which positions z i are assigned to Dirac-cone states ordered sequentiallyfrom top to bottom. The position assignments are basedon microscopic charge-density-weighted average positionsdiscussed in the Supplementary Material[57]. The dis-crete Poisson equation reads˜ (cid:15) E i = ˜ (cid:15) E t + i (cid:88) j =1 δρ i = ˜ (cid:15) E i − + δρ i V i = i (cid:88) j =2 E i ( z i − z i − ) = V i − + E i ( z i − z i − ) . (2)Here E t = E is the electric field controlled by the top gateabove the top surface of the thin film, E i is the electricfield between surface i and i + 1, ˜ (cid:15) is intended to accountfor gate field screening by degrees of freedom not includedin our model, and V i and δρ i are the Hartree-potentialand net surface charge density at surface i . (We choose V = 0.) The bulk perpendicular dielectric constant (cid:15) zz of MBT has not been measured, to our knowledge, butshould be close to (cid:15) zz ∼ Te [58]. Bycalculating the imaginary part of the conductivity of the bulk limit of the Dirac cone model [57], we find that itsbulk dielectric constant (cid:15) zz ∼ .
5. We have therefore con-cluded that most of the perpendicular screening in MBTis captured by the Dirac cone model and set ˜ (cid:15) = 1 in allthe explicit calculations we describe. The surface chargedensities used in the Poisson equation are calculated self-consistently from the electronic structure model using ρ i = − e (cid:90) d k (2 π ) N (cid:88) n =1 (cid:88) s = ↑ , ↓ | Ψ σn k ( z i ) | f ( E n k − µ ) , (3)where e is electron charge, n is a quasi-2D band label, f ( E n k − µ ) is the Fermi-Dirac distribution function, and µ is the chemical potential. The net surface charge den-sity δρ i , which appears in the Poisson equation, is de-fined as the difference between ρ i calculated from Eq. 3and the charge density calculated from the same equationwith the Fermi level in the gap and no gate field. Thisprescription is motivated by the linearity of the Pois-son equation, and by the fact that the bare bands havebeen fit to the electronic structure of neutral ungatedthin films.For a thin film with N septuple layers, the model has2 N Dirac cones located at the surfaces of each septu-ple layer and 4 N bands, 2 N of which are occupied atneutrality and temperature T = 0 – the limit consid-ered in this paper. The electric field below the bottomlayer of the thin film E b = E t at neutrality. The anti-ferromagnetic (AF) thin film phase diagrams obtainedfrom these self-consistent Hartree calculations are sum-marized in Fig. 1 (c), from which it follows that the thinfilms become semimetals when the electric field exceedsa critical value. To test the reliability of the simplifiedelectronic structure model in accounting for gate-field re-sponse, we performed corresponding DFT calculations ofthe electron structure of MBT thin film with 2-4 septuplelayers. The critical fields using the two approaches are inqualitative agreement.[57]. The critical electric field atwhich the semimetallic state is reached decreases whenthe thickness of thin film increases and asymptoticallyapproaches E gs (cid:15) zz /t N , where E gs ∼
37 meV is the sur-face state energy gap [21] and t N the thickness of an N septuple layer film. At small electric fields, the thin filmsexhibit an even-odd effect, namely that thin films withan even number of septuple layers are axion insulatorswith strong magneto-electric response properties [25, 59],whereas those with an odd layer-number ( N >
3) areQAH insulators [21]. This odd-even effect is illustratedschematically in Fig. 1 (b), where it can be seen that theDirac cones at the top and bottom surfaces have oppositemasses for even layer-number, but have identical massesfor odd layer-number, this characteristic can be modeledwith a effective toy model [57].
Gate-control of the QAH effect —
In antiferromagneticMBT thin films quantized anomalous Hall resistances areexpected when the number of septuple layers N is odd.Under these circumstances the film has residual mag-netism due to uncompensated moments, and the filmis sufficiently thick [21] that coupling between top andbottom surfaces does not induce a transition to a trivialinsulator. DFT calculations[34] predict that the QAHeffect occurs for three or more layers, and a robust QAHeffect has been measured in high-quality five-septuple-layer MBT thin films[37]. G a p [ m e V ] Chern Insulator(a) Gap with
N=3N=4N=5N=6N=7N=8N=9N=10 -40 -20 0 20 40 t [meV/nm]-40-2002040 b [ m e V / n m ] (b) Gap for N=5 G a p [ m e V ] k x [nm ]-100-50050100 E [ m e V ] sasa (c) = 0 0.5 0.0 0.5 k x [nm ]-100-50050100 btbt (d) = 18 meV/nm 0.5 0.0 0.5 k x [nm ]-100-50050100 bbtt (e) = 53 meV/nm FIG. 2. (a) Gap vs. gate electric field for N -septuplelayer charge-neutral antiferromagnetic thin films. A nega-tive sign is assigned to the gaps when the Chern number isnon-zero. The green shaded regions contain Chern insulatorphases while the grey shaded regions contain trivial insulatoror semimetal phases (see text). (b) Gap vs. gate electric fieldsfor five-septuple-layer thin films. Neutrality occurs along theyellow dashed line. (c)-(d) Self-consistent bandstructures ofneutral five-septuple-layer thin films. In the absence of anelectric field bands are labelled as layer symmetric (s) or anti-symmetric(a) and by majority spin ( ↑ or ↓ ). Bands at finiteelectric field are labelled by majority surface ( t for top or b for bottom) and majority spin. In Fig.2 (a) we plot the charge gap of several MBTthin films vs. gate electric field. For even N all in-sulators are trival, and gaps decrease with gate fields.The parameters we have chosen for the coupled Dirac- cone model place the N=3 thin film on the trivial side ofthe topological phase transition, but an increase of themagnetic exchange coupling parameters by as little asseveral meV would drive the system from a trivial insu-lator state to a Chern insulator state. If the ideal un-gated three-setptuple layer antiferromagnetic MBT thinfilm is indeed a trivial insulator, our calculations showthat a gate field would not be able to drive the systeminto a Chern insulator state since the condition to be aChern insulator is m > √ ∆ + V [57] in the presenceof electric field. Here m ,∆, V are the parameters the ofmass, hybridization of top and bottom surface states,and Hartree potential induce by the electric field in thetoy model illustrated in the supplemental material. Onthe other hand a small electric field due to asymmetricunintended doping would close the gap of the Chern in-sulator state [57], even if it were stable in the ideal case.In contrast odd N films with N ≥ N than for odd N because the approaching valence and conduction bandextrema states have the same dominant spin in the for-mer case, strengthening level repulsion effects. In thelimit of thick films the critical electric field E requiredto close the gap approaches the value E gs (cid:15) zz /et N where E gs ∼
37 meV is the energy gap of isolated surface states.The energy gap in the quasi-2D band structure of the N = 5 thin film is plotted vs. top and bottom gate fieldsin Fig. 2 (b), where the yellow dashed line marks the neu-trality line. We see here that as the carrier densities ofthe films thicken, the overall gap is controlled more andmore by independent screening of external electric fieldsby carriers near either surface. When carriers are present,the screened electric field drops toward near the middle ofthick films and larger electric fields are generally requiredto close the gap. Of course, when the Fermi level does notlie in the gap and away from the neutrality line, the re-sulting states are magnetically ordered two-dimensionalFermi liquids with large momentum-space Berry curva-tures [60], not Chern insulators. Because these itinerantelectron ferromagnets are strongly gate tunable, they arepotentially interesting for spintronics.The band structure evolution with gate field for N = 5neutral antiferromagnets is illustrated in Fig. 2 (c)-(e).In these plots we have labeled the subbands based on theprojection of their densities-of-states to individual spins,and to Dirac cones associated with particular septuplelayers. For N = 5 antiferromagnets the Hamiltonian inthe absence of a gate field (Fig. 2 (c)) possesses a z → − z mirror symmmetry which leads to band eigenstates thatare either symmetric ( s ) or antisymmetric ( a ) under thissymmetry operation. At finite gate fields the four sub-bands around the Fermi level reside mainly on the top orbottom surfaces ( t for top and b for bottom) of the thinfilm and that they are strongly spin-polarized ( ↑ or ↓ ).Either t ↑ and b ↓ or t ↓ and b ↑ subbands lie close to theFermi level depending on the direction of the electric fieldand the spin-configuration. Fig. 2 (d) shows the bandsnear the critical value at which band touching first oc-curs, near 18 meV/nm for (cid:15) zz ∼ .
5. At larger fields (Fig.2 (e)), the b ↓ and t ↑ subbands are inverted. The inver-sion changes the polarizations of the subband just belowFermi level, and drives the thin film from a state withChern number C = 1 to semimetal state with C = 0.In the semimetal state beyond the critical electric field,a small gap reopens in our simplified electron-structuremodel due to weak-coupling between top and bottom sur-faces. These small gaps are not expected to survive theanisotropic band dispersion of more fully realistic models. High-Chern-number QAH Systems—
Because their an-tiferromagnetic interlayer exchange interactions are ex-ceptionally weak, the Mn local moment spins in MBTare aligned by external magnetic fields larger than ∼ ∼ . N > vs. gate electric fields for FM thinfilms with from with N from 7 to 10, crossing the thick-ness at which the Chern number jumps from 1 to 2 atzero electric field. Overall the gaps tend to decrease withgate field as in the antiferromagnetic case. An excep-tion occurs for the N = 9 and N = 10 films, for whichthe gaps initially increase as the system moves furtherfrom the C = 2 to C = 1 boundary with gate field. InFig. 3 (b), where we assign a positive sign to the gapof odd-Chern-number(C=1) states and a negative signfor even-Chern-number(C=0 or 2) states, we illustratethe thickness dependence for a series of gate fields. Gateelectric fields can induce transitions between insulatorswith different Chern numbers as illustrated in Fig. 3 (c).The green and magenta curves for N = 9 and N =10 in Fig. 3 (a) show that the maximum QAH gapsare reached at around 10-15 meV/nm. At around 25meV/nm, a topological phase transition occurs at whichthe Chern numbers change from 2 to 1, and furtherchange to 0 as the electric fields increase (shown as in G a p [ m e V ] Odd Chern NumberEven Chern Number(a) Gap with
N=7N=8N=9N=10 G a p [ m e V ] C=2C=1(b) Gap with thickness =0=5=10=15=20 C h e r n N u m b e r s (c) Chern Numbers N=7N=8N=9N=10 k x [nm ]0.060.040.020.000.020.040.06 E ( e V ) tbtb (d) 15 meV/nm 0.2 0.0 0.2 k x [nm ]0.060.040.020.000.020.040.06 E ( e V ) tb (e) 25 meV/nm FIG. 3. (a) Gaps vs. gate electric fields for ferromagneticthin films from 7 to 10 septuple layers. For the N = 7 thin filmno gap closing happens occurs below 30 meV/nm, whereasfor the N = 8 thin film, the QAH gap closes at E ∼ N =9/10 thin films, the QAH gap closes at E ∼
25 meV/nm; (b) QAH gaps vs. thickness for severalelectric fields from E = 0 to E = 20 meV/nm; (c) Chernnumbers vs. electric field for several film thicknesses from N = 7 to N = 10; (d) and (e) are bandstructure of N = 9ferromagnetic thin film in different electric fields. Fig. 3 (c)). These transitions are illustrated further inFig. 3 (d) and (e), where we show the bandstructures ofFM thin films with N = 9 in Fig. 3 (d) and (e), with zeroelectric field (red dashed curve in (d)), with electric field E = 15 meV/nm (blue curve in (d)), and with electricfield E = 25 meV/nm (in (e)). Unlike the bandstruc-tures of the AF thin film with N = 5 in Fig.2, the bandsclosest to the Fermi level are not located primarily in the t ↑ and b ↓ septuple layers and are instead spread acrossthe thin film. As the electric field increases, the t ↑ sub-band is pushed down and the b ↓ subbands is pulled up.Before these two subbands touch up at E ∼
25 meV/nm(shown in Fig. 3(e)), hybridization between the electronand hole subbands spread across the entire thin film in-creases and thus increase the QAH gap. When the t ↑ and b ↓ subbands touch at E ∼
25 meV/nm, a topolog-ical phase transition occurs at which the Chern numberchanges from 2 to 1. Similar band inversions occur forother two subbands which are not shown in the figure,and eventually change the Chern number from 1 to 0.
Discussion—
In this paper we have focused on neu-tral MBT thin films. Gate-tuning will be phenomeno-logically richer in the case of electrostatic doping, wherethe ground states are expected to be extremely tunablemagnetically ordered two-dimensional metals. The Halleffect will remain quantized in doped samples, providedthat the added charges are localized. Since magnetizationtextures are charged in Chern insulators, we anticipatethe possibility of engineering Skyrmion lattice groundstates[61–64] at finite doping. Separately, gate fields canbe used to engineer strong spin-orbit coupling[65], whichis ubiquitous, in Fermi liquid states and to control the in-terplay between the itinerant electron and Mn local mo-ment contributions to the magnetization.In summary, we have studied gate tuning effects inMBT thin films using self-consistent Schr¨odinger-Poissonequations, demonstrating that gates can optimize theQAH gap either by compensating for unintentional elec-tric fields, or in the case of high-Chern-number ferro-magnetic (FM) insulators, by tuning the gate field to anoptimal non-zero value. Our theory provide an explana-tion for the absence of the QAH effect in AF thin filmwith three septuple layers, and sheds light on strategiesto optimize the QAH effect in both antiferromagneticand ferromagnetic MBT thin films using gates. The gateelectric fields that induce large changes are on the scaleof 10 meV/nm, which is easily realized experimentally.
ACKNOWLEDGEMENTS
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CONTENTS
Acknowledgements 5References 5List of Figures 2DFT calculations 3Discretization of Poisson Equation 3Calculation of bulk dielectric constant 3Trilayer MBT thin film 3Charge distribution 3Gap vs.
Electric Field 4Toy model for even-odd effect in AF thin films 4Phase transition vs. electric field 6
LIST OF FIGURES x axis and the number of septuple layers N inthe film along the y axis. Different colors represent different phases. The red and blue dashed linesrespectively plot the electric fields E at which eEt N = E gs and eEt N /(cid:15) zz = E gs , where E gs ∼
37 meVis the surface state gap at E = 0 [S21] and (cid:15) zz ∼ . vs. gate electric field for N -septuple layer charge-neutral antiferromagnetic thin films. Anegative sign is assigned to the gaps when the Chern number is non-zero. The green shaded regionscontain Chern insulator phases while the grey shaded regions contain trivial insulator or semimetalphases (see text). (b) Gap vs. gate electric fields for five-septuple-layer thin films. Neutrality occursalong the yellow dashed line. (c)-(d) Self-consistent bandstructures of neutral five-septuple-layer thinfilms. In the absence of an electric field bands are labelled as layer symmetric (s) or anti-symmetric(a)and by majority spin ( ↑ or ↓ ). Bands at finite electric field are labelled by majority surface ( t for topor b for bottom) and majority spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (a) Gaps vs. gate electric fields for ferromagnetic thin films from 7 to 10 septuple layers. For the N = 7thin film no gap closing happens occurs below 30 meV/nm, whereas for the N = 8 thin film, the QAHgap closes at E ∼
17 meV/nm, For N =9/10 thin films, the QAH gap closes at E ∼
25 meV/nm; (b)QAH gaps vs. thickness for several electric fields from E = 0 to E = 20 meV/nm; (c) Chern numbers vs. electric field for several film thicknesses from N = 7 to N = 10; (d) and (e) are bandstructure of N = 9 ferromagnetic thin film in different electric fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4S1 Projected density of states of antiferromagnetic MnBi Te thin films with thickness of 3-6 septuplelayers. The green solid vertical lines are labeling the outer Te atoms of each septuple layers, while themagenta dashed lines are the weighted location of each Dirac cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4S2 Gap vs. electric field of trilayer(N=3) MnBi Te thin film with different value of exchange interaction J S , which is varied from 36 meV to 41 meV. The topoloigcal phase transition happens at J S ≈ meV in the absence of electric field, while in the range of Chern insulator phase, a small electric field woulddrive the thin film into a trivial insulator phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5S3 Charge distribution at the neutrality points for antiferromagnetic MBT thin films, (a)-(d) are corre-sponding to the case of N = 3-6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5S4 Electric field distribution of acrossing the antiferromagnetic MBT thin films with different externalfield. In these plots the electric fields are ploted with the unit of the external electric field E t , the fieldinside the thin films decrease due to the screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5S5 Effective dielectric constant vs. electric fields for MBT thin film with thickness of N = 3-10. . . . . . . . . 5S6 Dependence of gap of thin films vs. external electric field, (a) and (b) are the case from DFT calculationswhile (c) and (d) are got from the couple Dirac cone model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 DFT calculations
The density functional theory (DFT) calculations were performed using the Vienna Ab initio simulation package(VASP) [S66–S69] using Generalized Gradient Approximation PBE [S70, S71] pseudopotentials. For the plane waveexpansion of the DFT calculation we used a cutoff energy of 600 eV, a total electronic energy convergence thresholdof 10 − eV per unit cell, a 9 × × Discretization of Poisson Equation
The Dirac-cone Hartree potentials V i in main text are assigned with positions z i which are based on microscopiccharge-density-weighted average positions: z i = (cid:88) j w j z j (S1)with j = Bi , Te and w j = p j / (cid:80) j p j , where p j is the projected density of states for Bi and Te atoms besides Mnion(shown in Fig. S1 for 3-6 septuple layers MBT) got from density-functional-theory. Calculation of bulk dielectric constant
The bulk dielectric constant is calculated based on the optical conductivity: (cid:15) zz = lim ω → (1 + 4 iπ ∂σ zz ∂ω ) , (S2)with the optical conductivity as: σ zz ( ω ) = ie ¯ h (cid:90) d k (2 π ) (cid:88) nm f n k − f m k E n k − E m k (cid:104) ψ m k | ∂ z H k | ψ n k (cid:105) (cid:104) ψ n k | ∂ z H k | ψ m k (cid:105) E n k − E m k − (¯ hω + iη ) , (S3)where ω is the frequency and f n k is the Fermi-Dirac func-tion. Trilayer MBT thin film
A topological phase transition happens at J S ≈ meV for the N=3 thin film in the coupled Dirac-conemodel, the parameters we have chosen place the thin filmon the trivial side, in Fig. S2 we see that the electric fieldrequired to drive the N=3 thin film depends on the valueof J S , and a small electric field would destroy the Cherninsulator state when the film is a Chern insulator. Charge distribution
Fig. ?? are the charge distribution at the neutralitypoints for antiferromagnetic MBT thin films, with the total charge density to be 0 and the two surfaces of thinfilms are electrically polarized.The electric field distributions across the antiferromag-netic MBT thin films are shown in Fig. S4, (a)-(d) areshowing the cases with thickness from 3 to 6 septuples.Due to the screening, the electric fields inside the thinfilms decrease. In Fig. S5 we estimate the effective di-electric constants with the methods as follow: (cid:15) eff = E t × d × NV b − V t , (S4)where E t is the eleectric field at the top surface, V t/b isthe Hartree potential at the top/bottom surface, and dis the thickness of the thin films. P D O S [ a r b . un i t s ] (a) N=3 0 1 2 3 4 5z [nm]0.10.20.30.40.50.6 P D O S [ a r b . un i t s ] (b) N=40 1 2 3 4 5 6z [nm]0.10.20.30.40.50.6 P D O S [ a r b . un i t s ] (c) N=5 0 1 2 3 4 5 6 7z [nm]0.10.20.30.40.50.6 P D O S [ a r b . un i t s ] (d) N=6 FIG. S1. Projected density of states of antiferromagnetic MnBi Te thin films with thickness of 3-6 septuple layers. The greensolid vertical lines are labeling the outer Te atoms of each septuple layers, while the magenta dashed lines are the weightedlocation of each Dirac cones. Gap vs.
Electric Field
The gaps at Γ points from both DFT calculations andthe couple Dirac cone model are shown in Fig. S6, (a)and (b) are the case from DFT calculations while (c) and(d) are got from the couple Dirac cone model. In theseplots we assign a minus sign for the thin films with Chernnumber of 1.
Toy model for even-odd effect in AF thin films
The antiferromagnetic MBT thin films may be mod-eled by a toy model, thin films with odd and even numberof layers may be modeled by a topological insulator thinfilm with the same mass and opposite mass term withinthe Dirac cone at the two surface.For MBT thin films with odd number of layers, thetwo surface Dirac cones have the same mass term, the G a p [ m e V ] Chern Insulator J S = 36 meV/nm J S = 37 meV/nm J S = 38 meV/nm J S = 39 meV/nm J S = 40 meV/nm J S = 41 meV/nm FIG. S2. Gap vs. electric field of trilayer(N=3) MnBi Te thin film with different value of exchange interaction J S ,which is varied from 36 meV to 41 meV. The topoloigcalphase transition happens at J S ≈ meV in the absenceof electric field, while in the range of Chern insulator phase,a small electric field would drive the thin film into a trivialinsulator phase. C a rr i e r D e n s i t y [ c m ] N=3 t =20 meV/nm t =50 meV/nm t =100 meV/nm 0 1 2 3 4z [nm]3210123 C a rr i e r D e n s i t y [ c m ] N=4 t =20 meV/nm t =50 meV/nm t =100 meV/nm0 1 2 3 4 5 6z [nm]3210123 C a rr i e r D e n s i t y [ c m ] N=5 t =20 meV/nm t =50 meV/nm t =100 meV/nm 0 1 2 3 4 5 6 7z [nm]432101234 C a rr i e r D e n s i t y [ c m ] N=6 t =20 meV/nm t =50 meV/nm t =100 meV/nm FIG. S3. Charge distribution at the neutrality points for an-tiferromagnetic MBT thin films, (a)-(d) are corresponding tothe case of N = 3-6. effective Hamiltonian in the absence of electric fields is: H odd = m v D k + ∆ 0 v D k − − m m − v D k + − v D k − − m , (S5)where k ± = k x ± ik y , v D is the Dirac velocity, ∆ is thehybridization of the two Dirac-cone at the surface and m is the mass. ∆ is the effective hybridization which maybe got from the parameters ∆ S and ∆ D in the main text,and m may be approximately only dependent on J S . Theeigenvalues are: E = ± (cid:113) v D | k | + ( m ± ∆) , (S6) [ t ] N=3 t =20 meV/nm t =50 meV/nm t =100 meV/nm 0 1 2 3 4 5z [nm]0.30.40.50.60.70.80.91.0 [ t ] N=4 t =20 meV/nm t =50 meV/nm t =100 meV/nm0 1 2 3 4 5 6z [nm]0.20.40.60.81.0 [ t ] N=5 t =20 meV/nm t =50 meV/nm t =100 meV/nm 0 1 2 3 4 5 6 7z [nm]0.20.40.60.81.0 [ t ] N=6 t =20 meV/nm t =50 meV/nm t =100 meV/nm FIG. S4. Electric field distribution of acrossing the antifer-romagnetic MBT thin films with different external field. Inthese plots the electric fields are ploted with the unit of theexternal electric field E t , the field inside the thin films decreasedue to the screening. e ff Displacement Potential with N=3N=4N=5N=6N=7N=8N=9N=10
FIG. S5. Effective dielectric constant vs. electric fields forMBT thin film with thickness of N = 3-10. G a p [ m e V ] (a) FM Thin Films (DFT)N = 2N = 3N = 4 0 20 40 60 80 100Electric Field [meV/nm]020406080100 G a p [ m e V ] (b) AF Thin Films (DFT)N = 2N = 3N = 40 20 40 60 80 100Electric Field [meV/nm]-60-30030 G a p [ m e V ] (c) FM Thin Films (Dirac Cone Model)N=2N=3N=4 0 20 40 60 80 100Electric Field [meV/nm]020406080100 G a p [ m e V ] (d) AF Thin Films (Dirac Cone Model)N=2N=3N=4 FIG. S6. Dependence of gap of thin films vs. external electricfield, (a) and (b) are the case from DFT calculations while(c) and (d) are got from the couple Dirac cone model. where | k | = (cid:113) k x + k y . In the presence of electric field, aperturbation Hamiltonian may be modeled as: V H = V V − V
00 0 0 − V , (S7)with ± V the Hartree potential due to the presence ofelectric field.For MBT with even number of layers, the effectiveHamiltonian in the absence of electric field is: H even = m v D k + ∆ 0 v D k − − m − m − v D k + − v D k − m , (S8)in which the two Dirac cones at the two surfaces haveoppisite mass. The eigenvalues are: E = ± (cid:113) v D | k | + m + ∆ , (S9) Phase transition vs. electric field
The topological phase transition point may be got viathe band energy at Γ point, for MBT thin films with oddnumber of layers, the eigenvalues at Γ point are: E = ± m ± (cid:112) ∆ + V , (S10)where V is the Hartree potential caused by the electricfield. In the absent of electric field, the thin films arein topological phase when m > ∆. In the presence ofelectric field, this condition becomes as m > √ ∆ + V ,which means the exchange interaction needed to inducea topological phase transition is larger and increase withthe increase of electric field. While for MBT thin filmswith even number of layers, the eigenvalues at Γ pointare: E = ± (cid:112) ( m ± V ) + ∆ ,,