PHYSICAL REVIEW APPLIED 16, 024046 (2021)

Multichannel Depletion-Type Field-Effect Transistor Based on FerromagneticGermanene

Jun Zheng ,1,2,* Yang Xiang,1 Chunlei Li,3,2 Ruiyang Yuan,4 Feng Chi,5 and Yong Guo 2,†

1College of Physics Science and Technology, Bohai University, Jinzhou 121013, China

2Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University,

Beijing 100084, China3College of Elementary Education, Capital Normal University, Beijing 100048, China

4Center for Theoretical Physics, Department of Physics, Capital Normal University, Beijing 100048, China

5School of Electronic and Information Engineering, University of Electronic Science and Technology of China

Zhongshan Institute, Zhongshan 528400, China

(Received 21 January 2021; revised 18 June 2021; accepted 12 July 2021; published 25 August 2021)

Based on ferromagnetic germanene, we theoretically propose a multichannel depletion-type field-effecttransistor (FET) operated by a vertical electric field. The electron transport properties of the topologicaltransistor are investigated through the nonequilibrium Green’s function method. The results reveal thatnew edge channels can be formed by removing some Ge atoms along the x axis of the device. The pro-posed multichannel ferromagnetic germanene transistor has a twice on-state current and half subthresholdswing than that of traditional germanene FETs. Under the coaction of electric induced staggered potentialand staggered exchange field, both the 100% spin-polarized edge states and valleys can be achieved. Theferromagnetic germanene FET can be turned off just by adjusting the electric field. In the low electric fieldconditions, the switching-off mechanism is due to spin blocking caused by the edge-state mismatch, whilefor relatively high electric field cases, both the edge-state mismatch and energy-valley mismatch are uti-lized. With the moderate increase in the exchange field, the switching mechanism is not changed, but thethreshold electric field and breakdown voltage of the FET can be, respectively, reduced and enhanced sig-nificantly. All the results indicate that the proposed ferromagnetic germanene FET is a promising candidatefor ultra-low-power dissipation topological devices.

DOI: 10.1103/PhysRevApplied.16.024046

I. INTRODUCTION

Germanene is a type of two-dimensional (2D) topologi-cal insulator consisting of germanium (Ge) atoms arrangedsimilarly to graphene and silicene in a hexagonal struc-ture. In 1994, a decade before the successful fabricationof graphene, the structure of germanene was studied byTakeda and Shiraishi using density functional theory [1].However, this theoretical work did not arouse widespreadresearch interest at that time, mainly because of the com-mon belief that strictly 2D materials were thermodynami-cally unstable and could not exist at any finite temperature[2,3].

*junzheng@semi.ac.cn†guoy66@tsinghua.edu.cn

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.

The rise of graphene triggered increased interest in othergroup-IV monolayers, such as silicene and germanene[4,5]. In 2009, the electronic properties of the planar struc-ture of germanene was investigated by Lebegue and Eriks-son. Their calculations showed that the properties relatedto the Dirac cone do not appear in planar germanene [6].This conclusion was confirmed by the further theoreticalresearch of Houssa et al. [7]. In the meantime, Cahangirovet al. [8] reported a study that answered the question ofwhether a stable freestanding 2D structure can be con-structed from sp2-like hybridized germanium atoms. Thecalculation of the electronic structure showed that ger-manene consists of both sp2 and sp3 bonds, in which sp2

hybridization causes it to exhibit a Dirac electronic struc-ture similar to graphene. In addition, germanene with alow-buckled honeycomb structure is more stable than thatwith high-buckled structure. In 2011, the theoretical studyof Yao et al. [9] confirmed that germanene tends to forma low-buckled structure rather than a planar one, and Geatoms with a 2D honeycomb geometry have nontrivialtopological properties in their native structure.

2331-7019/21/16(2)/024046(9) 024046-1 Published by the American Physical Society

JUN ZHENG et al. PHYS. REV. APPLIED 16, 024046 (2021)

Due to the buckled structure, the germanene latticeis decomposed into two vertically displaced sublattices.When a vertical electric field is applied, the atoms indifferent sublattices are no longer equivalent, and the inver-sion symmetry is broken. As a consequence, a tunabledirect band gap can be opened at the Dirac point [10–14]. Based on these properties, a variety of silicene andgermanene field-effect topological transistors have beenproposed based on theory [15–21]. On the experimen-tal side, a growing number of groups have successfullysynthesized germanene on conductor and semiconductorsubstrates since 2014 [22–26]. Although freestanding ger-manene has yet to be synthesized, single-layer silicene andgermanene FETs have now been fabricated successfully[27,28].

Topological insulators are characterized by a full insu-lating bulk gap and time-reversal symmetry-protected gap-less edge or surface states. The nondissipation transportproperties of the edge or surface state make the topo-logical device a promising candidate for ultra-low-powerconsumption electronics [29,30]. However, the on-statecurrent of a topological transistor, which using edge statesfor information transport is much smaller at low bias volt-ages compared with the traditional depletion-type FET.It is generally recognized that a small on-state currentincreases the delays of transistors, thus limiting large-scaleintegration applications [31].

In this paper, we propose a multichannel ferromagnetic(FM) germanene FET with a relatively high on-state cur-rent and low threshold parameter at low bias voltages. Asshown in Fig. 1, the central device is divided into tworegions, and the germanene in each region is sandwichedbetween two ferromagnetic insulators with opposite mag-netization. Some Ge atoms in the germanene nanoribbonare removed along the x axis. The dual gates are placedabove and below the dielectric layers to control the elec-tric field perpendicular to the sheet. It should be pointedout that even though its structure is similar to that of a nor-mal germanene FET, which requires only band gaps to turnthe transistor off, the fundamental switching mechanism ofthe FM germanene FET is completely different.

II. MODEL AND METHODS

The source and drain electrode and the central deviceof the FM germanene transistor can be expressed by thetight-binding Hamiltonian [32–35]:

HS = HD = −t∑

<ij >,σ

c†iσ cj σ + i

λSO

3√

3

∑

�ij �,σ σ̄

νij c†iσ sz

σ σ̄ cj σ̄ ,

(1)

where the index < ij > (� ij �) means that the sum-mation is extended over all the (next-) nearest-neighbor

(a)

(b)

yx

A sites

Left central device DML > 0 Right central device DMR < 0Magnetic

17 18

insulatorB sites

2Nx = 1

Top gate

Bottom gate

Gate dielectric layer

Gate dielectric layer

Magenetic insulator

Magenetic insulator

FIG. 1. (a) Schematic cross-section view of the ferromagneticgermanene FET, in which the central device is decomposed intoseven layers. The Ge atoms (represented by the black spheres)form a buckled germanene layer, and the unit cell of the ger-manene is marked by red spheres (A sites) and green spheres (Bsites). The number of unit cells in the x-axis direction and atomsites (black and light gray spheres) in per unit cell are Nx = 18and Ny = 24, respectively. Some Ge atoms in the germanenesheet with Nx = 18 and Ny = 4 are removed. The removed Geatoms are represented by light gray spheres, and four Ge atomsremoved along the y axis on the leftmost side are marked with rednumbers. A pair of top and bottom gates are used to provide anelectric field perpendicular to the germanene. (b) Side schematicof the germanene and magnetic insulator layers. Each magneticinsulator layer is divided into two parts, and the magnetization ofthe magnetic insulator is indicated by the red arrows in the darkregion and blue arrows in the light region.

sites. The spin-up and spin-down electrons are distin-guished with the spin index σ = −σ̄ = ±1. The first termin Eq. (1) represents the nearest-neighbor hopping betweensublattices with the transfer energy t. The second termdescribes the effective spin-orbit coupling with strengthλSO, and the site-dependent Haldane phase factor νij = +1(νij = −1) if the next-nearest-neighbor hopping is anti-clockwise (clockwise) around the positive z axis [36,37].sz is the z component of the Pauli matrix.

HC = HS + λE

∑

i,σ

μic†iσ ciσ

+ M∑

i∈L,σ

μic†iσ sz

σσ ciσ − M∑

i∈R,σ

μic†iσ sz

σσ ciσ , (2)

where the second term represents the staggered poten-tial induced by the electric field with strength λE andμi = +1 (−1) for the A (B) sites [11,38]. The strengthof λE can be controlled by applying a z-directed elec-tric field Ez(x, y) with λE = �Ez(x, y), in which � =0.33Å is the sublattice separation distance [9]. The ver-tical electric field applied to the germanene can be writ-ten as Ez(x, y) = (Vt − Vb)/[d0 − 2(dD + dM ) + 2(dD +dM )/(εD + εM )] [15], where Vt and Vb are top gate andbottom gate voltages, d0 is the distance between the twogates, dD and dM are the thickness of the dielectric and

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magnetic insulator regions, εD and εM are the dielectricconstants of the dielectric layer and magnetic insulatorlayer, respectively. The corresponding total gate voltage isVg = Vt + Vb.

The last two terms describe the exchange fields of dif-ferent directions with strength M operating on the Geatoms at A (μi = +1) and B (μi = −1) sublattices inthe left (i ∈ L) and right (i ∈ R) central device regions.As proposed for graphene and silicene, a large exchangefield could be induced in germanene by the magneticproximity effect with a magnetic insulator such as EuO[39–41]. In the simulation calculation, the magnetizationof adjacent ferromagnets is set to be opposite, the stag-gered exchange field applied to the left (right) centraldevice region is �ML = M A

L − M BL = 2M (�MR = M A

R −M B

R = −2M ) [42]. In addition, four Ge atoms aroundNy/2 in the device are removed along the x direction.Hence, in the following calculation, i(j ) /∈ {Ny/2, Ny/2 ±1, Ny/2 + 2}. Due to the edge states are protected fromthe effects of weak electron interactions [43,44], andwe focus on the influence of electric and exchangefields in this paper, the electron-electron interaction isneglected.

Within the Landauer-Büttiker formalism, the currentflowing through the drain electrode can be expressed as[45,46]

ID = eh

∑

σ

∫Tσ (ε) [fS(ε) − fD(ε)]dε, (3)

where Tσ (ε) is the spin-dependent transmission coef-ficient. Without local external fields, the transmissioncoefficient satisfies Tσ = TSDσ = TDSσ , where TSDσ =Tr[�Sσ Gr

σ�Dσ Gaσ ] is related to the electron transport from

the source to the drain electrode with the linewidthfunction �S(D)σ = i(r

S(D)σ − aS(D)σ ). The advanced and

retarded self-energy function of the α lead can beobtained from a(r)

ασ = HCαga(r)ασ HαC, and the Hamilto-

nian matrix HCα describes the coupling between thecentral device and α lead. The surface Green’s func-tion gr(a)

ασ can be calculated by the Green’s functionmethod or transfer matrix [47,48]. The Green’s func-tion of the entire system is given by Gr

σ (ε) = [εI − H0 −r

Sσ − rDσ ]−1 = [Ga

σ (ε)]†, in which H0 is the Hamilto-nian matrix of the central device. The Fermi functionfα(ε) = [e(ε−μα)/kBT + 1]−1, where T is the system equi-librium temperature, which is set to 4.2 K. μα is theelectrochemical potential of the α electrode. For a givenbias voltage VDS, the electrochemical potentials of the elec-trodes are fixed at μS = eVS = eVDS/2 and μD = eVD =−eVDS/2.

For the general tight-binding Hamiltonian H = ∑iσ ,j σ̄

c†iσ Hiσ ,j σ̄ cj σ̄ , the local current flowing from a site i with

spin σ under a dc bias can be expressed as [49–54]

Jiσ =∑

j σ̄

Jiσ ,j σ̄ = eh

∑

j σ̄

∫Re[Hiσ ,j σ̄ G<

j σ̄ ,iσ (ε)]dε. (4)

When the device is under a small bias and at zerotemperature, by applying the Keldysh equation G<(ε) =Gr(ε)[i�SfS(ε) + i�DfD(ε)]Ga(ε) and the electron cor-relation function Gn(ε) = Gr(ε)�SGa(ε) [55], the spin-dependent nonequilibrium local current can be writtenas

Jiσ ,j σ̄ = 2e2

hIm[Hiσ ,j σ̄ Gn

j σ̄ ,iσ (ε)](VS − VD). (5)

III. RESULTS AND DISCUSSION

In the numerical results presented below, the size of thecentral device region is fixed as Nx = 80 and Ny = 84. Thenearest-neighbor hopping energy t and effective spin-orbitcoupling strength λSO of germanene are taken as t = 1.3eV and λSO = 43 meV [56]. The strength of the exchangefield induced by the magnetic proximity effect is fixed asM = λSO/2, the corresponding staggered exchange fieldsacting on the left and right sides of the central deviceare �ML = 2M = λSO and �MR = −2M = −λSO. In thesimulation calculation of the drain current ID (local currentJiσ ,j σ̄ ), the system equilibrium temperature is set to 4.2 K(0 K). The conversion between the current units e/h andμA is 1e/h = 38.74μA.

The drain currents ID of the multichannel ferromagneticFET as a function of the electric-field-induced staggeredpotential λE for bias voltages of VDS = 10, 30, and 50 mVare plotted in Fig. 2(a). As a comparison, under the samebias voltage, Figs. 2(b) and 2(c) show the drain current ofthe dual-channel and multichannel traditional germaneneFET. It can be seen from the figures that multichannelcan effectively increase the on-state current, but traditionaltransistors still need a strong electric field to cut off thecurrent.

As shown in Fig. 2(a), the multichannel ferromagneticgermanene FET has two advantages over the traditionaldual-channel FET: (i) a larger on-state current and lowersubthreshold swing and (ii) a lower threshold staggeredpotential.

When λE = 0, the on-state current of the multichannelFET [see Fig. 2(a)] can be up to twice that of the dual-channel germanene FET [see Fig. 2(b)]. This is because,for the traditional dual-channel FET with the local cur-rent density distribution shown in the inset of Fig. 2(b),only two boundary channels are involved in the currenttransport, in which the spin-down (spin-up) current trans-fers from the source to the drain electrode through theupper (lower) edge. However, as shown in Fig. 2(d), ifsome Ge atoms located at hollow gray circles are removed,

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(a)

(c) (b)

(d)

(e)

(f) (g)

FIG. 2. (a) Drain current ID of the multichannel ferromagneticgermanene FET as a function of electric-field-induced staggeredpotential λE with different values of the bias voltage VDS . (b)Drain current ID of the dual-channel normal germanene FET ver-sus λE with bias voltages of VDS = 10, 30, and 50 mV. Insetshows the corresponding on-state local current density distribu-tion with λE = 0 and the energy ε = 10 meV. (c) Drain currentID of the multichannel normal germanene FET as a function ofλE with bias voltages of VDS = 10, 30, and 50 mV. (d) Localcurrent density distribution of multichannel germanene transis-tor in the on state with λE = 0 and ε = 10 meV, and with centralregion size Nx = 40 and Ny = 24. The thick red (thin blue) arrowcorresponds to the spin-up (spin-down) current. The length anddirection of the arrow represent the magnitude and direction ofthe current density vector at each lattice site. The black solid cir-cle represents the position of the Ge atom, and the gray hollowcircle corresponds to the location of the removed Ge atom. (e)Local current density distribution of multichannel transistor inthe off state with λE = λSO/2, and the other parameters are thesame as those in Fig. 2(d). Logarithmic and linear current of thedual-channel normal germanene FET (f) and the multichannelferromagnetic germanene FET (g) versus λE with bias voltageVDS = 50 mV.

new edge channels with opposite spin will appear at thecritical interface. Therefore, under a small forward bias,the spin-up and down-spin currents can flow to the drainelectrode mainly through four edge channels. Figures 2(f)

and 2(g) depict the logarithmic and linear current of thedual-channel normal germanene FET and the multichannelferromagnetic germanene FET versus the electric-field-induced staggered potential. By comparing two log(ID)plots, the subthreshold slope (the reciprocal value of thesubthreshold swing) of the multichannel ferromagneticFET in the relatively high-electric-field region is nearlydoubled than that of dual-channel normal germanene FET[see the orange solid lines in Figs. 2(f) and 2(g)]. There aretwo steep slopes in the log(ID)-λE plot as shown by the redcircles in the gray area of Fig. 2(g), the larger and smallersloped regions correspond to different current cutoff mech-anisms. In the larger slope region (λE around 2λSO), boththe energy-band mismatch and two valley gaps are usedto cut off the drain current. However, in the smaller sloperegion, only the energy-band mismatch is used. In addition,the current cutoff of the dual-channel normal germaneneFET utilizes only the band gap opened by the electric field,and hence the log(ID)-λE plot in the gray area of Fig. 2(f)only has one steep slope.

As shown in Fig. 2(b), for a dual-channel normal ger-manene FET, the output current ID can reach about zeroonly when the staggered potential satisfies λE > λSO +VDS/2. However, with the aid of the magnetic proxim-ity effect, the FM germanene FET can be turned off withλE > λSO − M + VDS/2. A much lower threshold stag-gered potential means not only lower power consump-tion but also a different switch-off mechanism. When theFM germanene transistor is in the off state, there arestill a small number of spin-up electrons that can prop-agate through the edge into the device region near thesource lead [see Fig. 2(e)]. This indicates that the ger-manene under a weak electric field is not transformedinto a topological trivial insulator. Hence, the FM ger-manene FET does not utilize the band gap to block thecurrent.

To understand the switching mechanism and transportproperties, the energy band and corresponding conduc-tance diagrams of the normal and FM germanene are givenin Figs. 3 and 4. As presented in Figs. 3(a) and 3(d), with-out considering the effect of the external field, the energylevels of the edge states degenerate and cross at the �

(kxa = π ) point. Moreover, because some of the Ge atomsare removed along the x axis in the real space in the cal-culation, a pair of new edge states with opposite spins isformed in the bulk band-gap region, and the degeneracy ofthe states is 4. When a relatively low staggered exchangefield is applied (|�ML(R)| = 2M < 2λSO), the width ofthe bulk band gap in the K valley (K ′ valley) becomessmaller (larger), but the edge state is not destroyed, andits degeneracy is still 4 [see Figs. 3(b) and 3(c)]. Under thecondition of a small positive bias and considering the spin-momentum locking, electrons can flow from the left to theright electrode through two spin-up and two spin-downedge states [see Fig. 2(d)].

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Top gate

Bottom gate

Gate dielectric layer

Gate dielectric layer

Magenetic insulator

Magenetic insulator

(a) (b) (c) (d)

lE = 0, DML = 0 lE = 0, DMR = 0lE = 0, DML > 0 lE = 0, DML < 0

FIG. 3. Energy-band diagrams of (a) the source electrode with λE = 0 and �ML = 0; (b) the left region of the central device withλE = 0 and �ML = λSO; (c) the right region of the central device λE = 0 and �MR = −λSO; (d) the drain electrode with λE = 0 and�MR = 0. The purple lines represent spin-degenerate energy states.

It should be pointed out that from the comparison ofFigs. 3(a) and 3(b), Figs. 3(b) and 3(c), Figs. 3(c) and3(d), the slopes of edge states are different, which meansthat the velocities of electrons in the electrodes and theleft and right central device regions are different. There-fore, as shown in Fig. 2(d), when the current flows fromthe source to the drain electrode, there will be internal cur-rent due to interface scattering. However, due to the degreeof edge states mismatch is low, electrons transport to the

drain electrode mainly through the edge channels, and theinternal current is much smaller than the edge current.

The staggered electric potential and the staggeredexchange field can break the time-reversal symmetry andthe spin rotation symmetry. Under the combined actionof staggered electric potential and staggered exchangefield, and if λE + M > λSO, the edge states can trans-form from a quantum spin Hall state to the spin-polarizedquantum anomalous Hall state, and the K and K ′ valleys

(a) (b) (c)

(d) (e) (f)

lE = 3lSO/4, DML > 0

lE = 3lSO/2, DML > 0

lE = 3lSO/4, DMR < 0

lE = 3lSO/2, DMR < 0

FIG. 4. Energy-band diagrams of germanene with different values of the electric-field-induced potential and the staggered exchangefield: (a) λE = 3λSO/4, �ML = λSO; (b) λE = 3λSO/4, �MR = −λSO; (d) λE = 3λSO/2, �ML = λSO; (e) λE = 3λSO/2, �MR =−λSO. The red and blue lines represent spin-up and spin-down energy states, respectively. (c) Conductance of ferromagnetic (FM) ger-manene as a function of the energy ε with λE = 3λSO/4 and �ML(R) = ±λSO. (f) Conductance G versus the energy ε with λE = 3λSO/2and �ML(R) = ±λSO.

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are +100% or −100% spin polarized in different energyranges. As illustrated in Fig. 4(a) [Fig. 4(b)] for �ML > 0(�MR < 0), spin-down (spin-up) edge states are destroyedand lead to a direct band gap of 2|λSO − |λE| − M | atthe K (K ′) point. The spin directions of the undestroyededge states in Figs. 4(a) and 4(b) are opposite, and theedge-state mismatch between the left and right parts ofthe device gives rise to zero conductance within the cor-responding energy window [see Fig. 4(c)]. The strength ofelectric-field-induced staggered potential in Figs. 4(a)–4(c)is λE = 3λSO/4, the corresponding x-axis coordinate λE =3λSO/4 = 32.25 meV displayed in Fig. 2(a) is marked witha red dot. It can be seen from Fig. 2(a) that under the con-dition of a weak electric field, only using the edge-statemismatch cannot cutoff the current generated by relativelylarge bias.

Figures 4(d) and 4(e) show the energy bands of FM ger-manene with the same intensity exchange fields as in Figs.4(a) and 4(b), but a higher electric-field-induced staggeredpotential λE = 3λSO/2 = 64.5 meV, which is marked witha blue dot in the x-axis coordinate of Fig. 2(a). When thestaggered exchange field �ML > 0 (�MR < 0), the bandgap of the spin-down (spin-up) edge states in the K-valley(K ′-valley) region increases with the strength of the elec-tric field. At the same time, the spin-up (spin-down) K ′valley (K valley) moves toward the Fermi level, and theenergy valley shows 100% spin polarization, as shownby the red (blue) thin solid line and thick dashed line inFig. 4(d) [Fig. 4(e)]. Because of the mismatch between theenergy bands of the edge states and the energy valleys inthe �ML > 0 and �MR < 0 regions, the conductive pathscan be blocked in a relatively larger energy range [seeFig. 4(f)]. It is worth note that if the magnetic proximityeffect is not considered, a larger electric field λE = 2λSOis needed to make germanene produce a large enoughband gap to achieve the same degree of conductance sup-pression, which is shown in Fig. 4(f). In addition, thesubthreshold slope of the dual-channel normal germaneneFET and the multichannel ferromagnetic germanene FETis different, due to the switching-off mechanisms of thetwo transistors are completely different. The dual-channelnormal germanene FET requires a large electric field toopen a band gap, while the multichannel ferromagneticgermanene FET mainly uses energy-band mismatch. Whenthe energy-valley mismatch and the K-valley and K ′-valley gaps are used to turn off the transistor, the subthresh-old slope of the multichannel ferromagnetic germaneneFET can be nearly double that of the dual-channel normalgermanene FET.

As the magnetic proximity effects of different ferromag-netic materials have been studied in previous theoreticalwork, we next discuss the influence of the exchange fieldstrength on the off -state threshold parameter and break-down voltage of the device. Figures 5(a)–5(c) depict thedrain current ID as a function of the electric-field-induced

so so

DS DS DS

so

Ez

(c)(b)(a)

Off Off Off

FIG. 5. Drain current ID of the multichannel FM germanenetransistor for different values of the electric-field-induced stag-gered potential λE and bias voltage VDS for (a) M = λSO/4, (b)M = λSO/2, and (c) M = 3λSO/4.

staggered potential λE and bias voltage VDS for differentvalues of the exchange field M . It can be seen from thefigure that whether the intensity of the exchange field isweak or strong, the FM germanene transistor can be turnedon and off by adjusting the z-axis electric field. In the onstate with λE = 0, because the current in the low bias-voltage regime is transported only through the edge state,the corresponding current values are the same under dif-ferent M . However, in the case of a high bias voltage, thecurrent intensity increases with the increase in M due to thebulk energy bands participating in the electron transport[see the black regions in Figs. 5(a)–5(c)]. In addition, thenumerical simulation results show that with the increasein the exchange field, the threshold electric field strengthof the off state will decrease, and the breakdown volt-age the device can withstand will increase significantlyunder the same electric field [see the white regions in Figs.5(a)–5(c)].

Finally, we briefly discuss the effect of edge hydrogena-tion and random impurities on the numerical simulationresults. In a realistic germanene nanoribbon, there will bea dangling bond at the edge. Usually, the Ge atoms in theboundary are passivated by hydrogen to improve the sta-bility of zigzag germanene nanoribbons. When both thezigzag edges are fully Ge—H bonded, the characteristicedge states of germanene are completely preserved, the H-edge-hydrogenated zigzag germanene nanoribbon behaveslike a bare freestanding germanene nanoribbon. When theedges of germanene are Ge—H2 bonded, however, theedge states are destroyed and an energy gap is openedat the Fermi level, the H2-edge-hydrogenated zigzag ger-manene nanoribbon behaves like a 2D semiconductor[57,58]. Since the on state of the topological transistor,which we study in the paper, mainly uses the edge state

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to conduct the current, and the transistor in the off state ismainly utilize the edge-state mismatch. Therefore, whenthe Ge atoms at the edge are passivated by one hydro-gen atom, all results in the paper can also be obtained.In addition, real materials always have random impurities.The time-reversal symmetry-protected edge states could bedestroyed when magnetic impurities are introduced intothe system. Due to there being more edge states carryingthe current in multichannel FET than in dual-channel FET,the magnetic impurities have a more obvious influence onthe multichannel FET. However, since nonmagnetic impu-rities do not break the time-reversal symmetry, the helicalproperty of the edge states is topologically protected, boththe multichannel FET and dual-channel FET are robust inthe presence of nonmagnetic impurities.

IV. CONCLUSIONS

In summary, we theoretically study the electron trans-port properties of a multichannel germanene transistor, inwhich the FM-sandwiched germanene is subjected to az-directed electric field. It is found that the transistor con-sidered can be switched between on and off states bytuning only the strength of the electric field. Comparedwith the normal germanene FET, the multichannel FM ger-manene FET has the following main features. (1) When Geatoms are removed along the x axis, two new edge channelswith opposite spin are created at the boundary, the on statecurrent of the multichannel germanene FET at λE = 0 canbe up to double. (2) When the staggered potential satisfiesλE > λSO − M + VDS/2, the edge state transformed fromthe quantum spin Hall state to the spin-polarized quantumanomalous Hall state, and the K and K ′ valleys are +100%or −100% spin polarized in different energy ranges. Con-sequently, the FM germanene FET can be turned off. (3)The switching-off mechanism of the FM germanene FET isrelated to the electric field strength. For λE < λSO, the tran-sistor in the off state is mainly due to spin blocking causedby the edge-state mismatch, for λE > λSO, both the edge-state mismatch and the energy-valley mismatch are used.For λE > 2λSO, besides the energy-valley mismatch, theK and K ′ valley gaps are also used, and the subthresholdswing can be reduced to half that of the dual-channel nor-mal germanene FET. (4) The strength of the exchange fieldin the range of 0 < M < λSO does not change the switch-ing mechanism. However, the stronger the exchange fieldis, the lower the threshold electric field required for theoff state is, and the higher the breakdown voltage of thetransistor becomes.

ACKNOWLEDGMENTS

This work is supported by the National Natural Sci-ence Foundation of China (Grants No. 12174038 andNo. 12074209), the LiaoNing Revitalization Talents Pro-gram (XLYC2007141), the Open Project of State Key

Laboratory of Low-Dimensional Quantum Physics (GrantNo. KF201910), and the Science Technology Founda-tion from Education Commission of Beijing (Grant No.KM201810028022).

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