# The Impact of Electron Phonon Scattering, Finite Size and Lateral Electric Field on Transport Properties of Topological Insulators: A First Principles Quantum Transport Study

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## Abstract

**:**

_{ON}/I

_{OFF}of 28 in the ballistic case. These results shed light on the opportunities and challenges in the design of topological insulator field-effect transistors.

## 1. Introduction

_{2}[19], and 1.1 V/nm for Na

_{3}Bi [20] have been reported, all of which are undesirably large. In addition, although a phase transition occurs at these fields in the monolayer, a sufficient gap may only open at an even larger electric field in the nanoribbon (NR).

_{3}Bi ribbons [16]. The authors reported an ON/OFF current ratio of 6/4 at V

_{bias}of 0.05 V/0.1 V, respectively. The simulations were performed in the ballistic case and with a channel length of 2.7 nm. Additionally, a differential voltage V

_{diff}(between the top and bottom electrodes, V

_{top}and V

_{bottom}, respectively), of around 20 V was required to switch off the TI-FET. In another work based on Na

_{3}Bi ribbons, the use of an electric field is coupled with the intrinsic defects for improving the device performance by disorder filtering [21].

## 2. Materials and Methods

^{−3}eV/Å.

## 3. Results

#### 3.1. Zigzag Nanoribbons of Stanene and Bismuthene

#### 3.1.1. Mode Space Basis

_{DD}) used for stanene will be small to ensure the transport inside the small bulk gap. The helical spin texture of the protected bands imposes a spin-momentum locking which impedes the elastic backscattering in the ribbon [35]. The bulk band gap appears to be 0.34 eV for this 4 nm wide structure. It is larger than the monolayer bulk band gap due to the confinement effects that we will further discuss in the next section.

#### 3.1.2. The Effects of Electron–Phonon Coupling

_{ch}) is 10 nm for all the structures in this work. For stanene NR, the drain bias is 50 mV. The drain bias is chosen to be small due to the small bulk gap of stanene.

_{g}, increases. At V

_{g}= 0.4 V, the entire Fermi window (the shaded area in Figure 2b) is covered by the protected edge states. Figure 2b illustrates the contribution of the left edge atoms (atoms 1 to 5), the middle atoms (atoms 6 to 17), and the right edge atoms (atoms 18 to 22) to the LDOS at V

_{g}= 0.4 V. It can be observed that the low-energy states inside the bulk gap are localized at the edges of the ribbon, while the states outside the gap, as well as high-energy sates inside the bulk gap, are delocalized.

_{AC}) are compared to represent different e-ph coupling strengths. As D

_{AC}increase, the Fermi window gets fully covered with the broadened, unprotected states at D

_{AC}= 6 eV [35]. Therefore, the backscattering is not prevented as the protected states can’t be isolated. This results in a noticeable current degradation.

_{g}= 0 V for both channel materials. At V

_{g}= 0.4 V, in both cases, the Fermi window is covered by the protected edges. As bismuthene possess a larger bulk gap, the edge localization of the protected states is preserved in bismuthene even for a D

_{AC}as large as 40 eV [35]. Hence, it is expected that bismuthene exhibits better immunity to e-ph coupling. The calculated currents for the structures based on stanene and bismuthene as well as WS

_{2}monolayer (in H-phase) are shown in Figure 2c. As the deformation potential increases from 0 to 10 eV, an important current degradation is observed for the stanene and WS

_{2}monolayers, while the transport in bismuthene is almost unaffected.

#### 3.2. Ultra-Narrow Ribbons

_{DD}= 0.05 V in this setup. It is shown that the forward moving spin up/down currents are mostly localized at the opposite edges (Figure 4a,b). However, there is a smaller portion of charge transfer taking place at the center of the ribbon as well as at the opposite edge. This indicates the contribution of the bulk and the opposite edge atoms to the spin-up/down states in the bulk gap. Figure 4c demonstrates that this contribution increases with V

_{g}and the transport happens through more delocalized states. Figure 4d shows the backward moving spin-up carriers located at the opposite edge of that of the spin-up forward moving channel. The spin-momentum locking can be deduced from these results. As e-ph scattering is included (Figure 4e,f), the transfer of forward moving electrons is reduced at the edge and mostly in the middle of the ribbon, indicating the degradation of the current.

_{3}Bi have attracted much attention. Experiments have shown that Na

_{3}Bi monolayers have a bulk band gap of 360 meV, as well as a relatively small critical field of 1.1 V/nm [20]. TI-FETs based on ultra-short, ultra-narrow Na3Bi ribbons have been theoretically explored [16]. In a TI-FET relying on the gap opening as a switching mechanism, in addition to a small critical field, the band gap opened after the transition should be sufficiently large to efficiently obstruct the current. Previous works based on DFT have shown that for wider ribbons of Na

_{3}Bi (above 6 nm), the band gap opened after the transition is small (smaller than 0.2 eV). However, thinner Na

_{3}Bi ribbons have larger gaps (around 0.4 eV), post-topological phase transition. Therefore, it may be beneficial to use thinner Na

_{3}Bi ribbons to exploit the larger post-transition gap. Our results show that at a width of under 4 nm, the confinement effects result in a gap opening due to the overlap of states around the Fermi level. Additionally, although the bulk gap increases with a reduction in width (Figure 5b,c), the states in the bulk gap are poorly localized, akin to stanene (the grey-shaded region in Figure 5c). The opened gap and the larger contribution of the middle atoms to the states inside the bulk gap pose challenges to the use of ultra-narrow Na

_{3}Bi in a TI-FET. However, if a wider Na

_{3}Bi ribbon is used, the gap in the off state (after transition) is only around 0.2 eV, which seems challenging to have a low-leakage TI-FET operating based on electric-field-induced phase transition with Na

_{3}Bi.

#### 3.3. Electric Field-Induced Topological Phase Transition in Xenes

_{z}), a gap opens at the X point in the edge states of the stanene zigzag ribbon However, the unprotected states around the $\Gamma $ point move up into the gap as well. In Figure 6b, we see that the lateral field (E

_{y}) opens a gap at the X point as well, using an about 10× reduced field. The gap is around 0.13 eV for E

_{y}= 6 eV/nm. However, the states around $\Gamma $ remain unchanged, as they are mainly formed by the middle atoms. In bismuthene, both lateral and out-of-plane fields shift the protected and unprotected states and close the bulk gap (Figure 6c,d). Once the bulk gap is closed, the spin-polarized states cannot be isolated anymore. Therefore, the transport will be unprotected. In addition, the lateral field seems to have a more drastic influence on the energy dispersion compared to Ez in bismuthene. It can be inferred that the lateral field can tune the band structure in a different way than the out-of-plane field, and in some materials, it can be more advantageous to use the lateral field.

_{left}= 0 and V

_{left}= 4.4 V, respectively. The voltage on the other gate is V

_{right}= 0 V. Additionally, L

_{ch}= 10 nm, V

_{DD}= 0.05 V and a doping concentration of 3 $\times $ 10

^{20}cm

^{−3}is introduced to shift the Fermi level. A gap of around 0.1 eV is observed in the device with V

_{left}= 4.4 V (Figure 7c), which confirms the opening of a gap by the external lateral electric field.

_{g}= 4 V). As a result of this gap, the current decreases and an I

_{ON}/I

_{OFF}ratio of around 28 can be observed at room temperature (Figure 7e).

_{g}= 0 V there is already a strong asymmetry in the LDoS formed by the left edge atoms compared to that of the right edge atoms in the Fermi window. This difference is due to the induced electric field from the side metal gates with different work functions. Figure 7e demonstrates that having a built-in electric field effectively decreases the threshold voltage needed for a gap opening. Hence, it can be a useful approach in ultra-thin TI ribbons (TIs with large bulk gaps), as the phase transitions happen without the bulk gap closing. However, it raises the question that whether the transport properties are favorable in a TI with a built-in electric field. In other words, are the spin-momentum locking and the protected transport preserved in a specific TI with a built-in field? This question requires more investigation.

## 4. Discussion

_{3}Bi. Once, e-ph scattering is included, electrons can backscatter. This results in almost no improvement in the immunity to electron–phonon scattering.

_{y}opened a gap in the metallic states of the stanene ribbon and closed the bulk gap in bismuthene. Utilizing this gap modulation, a stanene TI-FET with a channel length of 10 nm exhibits an I

_{ON}/I

_{OFF}ratio of 28 at room temperature. The threshold voltage required to open a gap shows an improvement to the previously reported TI-FETs. The use of a built-in electric field can further reduce the threshold voltage. However, the effects of a built-in field on the transport properties of a TI warrants deeper investigation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) The electronic band dispersion of 4 nm wide zigzag stanene ribbon with fluorine edge passivation, calculated by real space (RS) Hamiltonian and mode space (MS) Hamiltonian. The green-shaded region indicates the target energy window (

**b**) same as (

**a**) for a 7 nm wide zigzag bismuthene ribbon with H edge passivation.

**Figure 2.**(

**a**) An illustration of the structure under study. (

**b**) The atom resolved LDoS for the structure based on 4 nm wide stanene zigzag ribbon in the ballistic regime. The contribution of the left edge atoms (atoms 1 to 5), middle atoms (atoms 6 to 17), and right edge atoms (atoms 18 to 22) to the LDoS are demonstrated. The shaded region denotes the Fermi window. (

**c**) The calculated current for the structures is based on the stanene and bismuthene zigzag ribbons as well as a WS

_{2}monolayer (in H-phase). The current is plotted as a function of D

_{AC}.

**Figure 3.**(

**a**) Electronic band dispersion for 2.5 nm and 6 nm wide zigzag stanene nanoribbons. (

**b**) LDOS in a gate-controlled 2.5 nm wide zigzag stanene nanoribbon. The contribution of atoms at the left edge (atoms 1 to 4), atoms at the center (atoms 5 to 8), and atoms at the right edge (atoms 9 to 14) are shown. (

**c**) transfer characteristics for the ballistic (mode space vs. real space) and dissipative cases with D

_{AC}of 4 eV and 10 eV.

**Figure 4.**(

**a**) The atom resolved current of forward moving spin-up electrons in the device under V

_{g}= 0.4 V and V

_{DD}= 0.05 V (the ballistic case), (

**b**) same as (

**a**) but for the spin-down electrons. (

**c**): Same as (

**a**) for V

_{g}= 1.2 V. (

**d**): Same as (

**a**) but for backward moving spin-up electrons. (

**e**,

**f**): Same as (

**a**) for D

_{AC}= 4 eV and D

_{AC}= 10 eV, respectively.

**Figure 5.**(

**a**) Spin-resolved band structure of a 6.8 nm wide zigzag Na

_{3}Bi ribbon, projected on the left edge atoms. The red and blue dots indicate spin up and spin down states, respectively. (

**b**) The contribution of the left/right edge atoms as well as the middle atoms to the LDOS of a 6.8 nm wide zigzag Na3Bi ribbon. (

**c**) Same as (

**b**) but for a 3.9 nm wide ribbon. The grey-shaded region illustrates the bulk gap.

**Figure 6.**Band structure of a 2.5 nm wide stanene zigzag ribbon under (

**a**) an out-of-plane electric field (

**b**) a transverse electric field. Band structure of a 4 nm wide zigzag bismuthene ribbon under (

**c**) an out-of-plane electric field (

**d**) a transverse electric field.

**Figure 7.**(

**a**) Illustration of the structure with side gates imposing an electric field to a 2.5 nm wide zigzag stanene nanoribbon. L

_{ch}= 10 nm and the gate oxide thickness is 1.5 nm. Current spectrum $J\left(x,E\right)$ for the structure with side gates and a 2.5 nm wide zigzag stanene nanoribbon with (

**b**) V

_{left}= 0 (

**c**) V

_{left}= 4.4 V. In both cases V

_{right}= 0 V, V

_{DD}= 0.05 V. The red lines indicate the charge neutrality points in the middle of each slab. (

**d**) The contribution of the left edge atoms (atoms 1 to 4), middle atoms (atoms 5 to 8) and the right edge atoms (atom 9 to 14) to the LDOS of the same structure. Here, the side gates are assumed to be made of metals with different work functions (work function difference (WF_dif = 3 eV). The yellow-shaded area shows the Fermi window. (

**e**) IV curve of the structure with side gates with the gate metals having the same work function (WF_diff = 0) or different work functions (WF_diff = 3 eV).

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**MDPI and ACS Style**

Akhoundi, E.; Houssa, M.; Afzalian, A.
The Impact of Electron Phonon Scattering, Finite Size and Lateral Electric Field on Transport Properties of Topological Insulators: A First Principles Quantum Transport Study. *Materials* **2023**, *16*, 1603.
https://doi.org/10.3390/ma16041603

**AMA Style**

Akhoundi E, Houssa M, Afzalian A.
The Impact of Electron Phonon Scattering, Finite Size and Lateral Electric Field on Transport Properties of Topological Insulators: A First Principles Quantum Transport Study. *Materials*. 2023; 16(4):1603.
https://doi.org/10.3390/ma16041603

**Chicago/Turabian Style**

Akhoundi, Elaheh, Michel Houssa, and Aryan Afzalian.
2023. "The Impact of Electron Phonon Scattering, Finite Size and Lateral Electric Field on Transport Properties of Topological Insulators: A First Principles Quantum Transport Study" *Materials* 16, no. 4: 1603.
https://doi.org/10.3390/ma16041603