Introduction

Since the discovery and successful mechanical exfoliation of graphene in 2004, there has been a surge in research on 2D materials1,2. Many 2D materials have since been predicted and experimentally synthesized, such as transition metal dichalcogenides3,4,5,6,7, borophene8,9,10,11,12,13,14, phosphorene15,16, and MA₂Z₄ family materials17,18, revealing novel physical and chemical properties as well as promising applications. Compared to bulk materials, 2D materials exhibit unique structures that confer new electronic and optoelectronic properties. Their thermoelectric transport is confined to a single plane, which makes them suitable for applications in electronics, optoelectronics, spintronics, sensors, and field-effect transistors. Studies have found that 2D semiconductor materials possess many excellent functional properties, including rectification effects, gate behavior, and spin filtering effects, which can be utilized in developing next-generation nanoelectronic devices19,20,21.

In semiconductor device applications, corrugated 2D materials22 such as arsenene, monolayer SnSb, and monolayer GeSe have multiple advantages over planar 2D materials: wide adjustable bandgaps23, in-plane anisotropy24, high carrier mobility23, and high rectification ratios22,23,25. Additionally, 4-atomic-layer corrugated 2D materials exhibit higher stability and more diverse phases than their 2-atomic-layer counterparts26. For example, IV-V group X₂Y₂ type 2D materials have lower formation energies than XY type materials27,28,29,30,31,32. Barreteau et al. successfully constructed layered SiP, SiAs, GeP, and GeAs with monoclinic crystal structures in the C2/m space group33,34,35. Based on this, Lou et al. proposed stable Ge₂Y₂ structures and studied their thermal transport and thermoelectric properties, indicating promising applications in thermoelectric devices27. Despite these advancements, the effects and electronic transport properties in different device architectures remain to be explored. Key aspects to be thoroughly investigated include: (a) the degree of anisotropy in electronic transport, (b) their potential to function as field-effect transistors (FETs), and (c) their optoelectronic properties.

IV-V group Ge₂Y₂ materials exhibit excellent kinetic stability with two stable structural phases, α-Ge₂Y₂ and β-Ge₂Y₂, both showing semiconductor properties27. In this paper, we design and study conceptual nano-device structures based on monolayers of α- and β-Ge₂Y₂ (Y = As, P, N), such as p–n junction diodes, p-i-n junction field-effect transistors, and phototransistors. We calculate their electronic and optoelectronic transport properties using first-principles methods, revealing their functional characteristics and potential applications in next-generation high-performance electronic devices.

Computational results and discussion

Electronic structure of Ge₂Y₂ monolayers

The ground-state crystal structures of fully relaxed Ge₂Y₂ (Y = As, P, N) monolayers are shown in Fig. 1a and b. Each Ge₂Y₂ unit cell contains 4 atoms: 2 Ge atoms and 2 Y atoms. The x and y directions refer to the zigzag and armchair directions, respectively. The α- and β-Ge₂Y₂ monolayers consist of two Ge-Y layers connected by Ge–Ge bonds, forming AA and AB stacking. Ge and Y atoms are arranged in hexagonal honeycomb lattices, forming two structures: α-Ge₂Y₂ monolayers(see Fig. 1a) correspond to the 2D crystal space groups [\(P\overline{6}m2\)](No. 187), while β-Ge₂Y₂ monolayers(see Fig. 1b) correspond to the [\(P\overline{3}m1\)](No. 164). Based on total energy calculations from spin-polarized and non-polarized approaches, α-Ge₂Y₂ and β-Ge₂Y₂ monolayers exhibit non-magnetic ground states, consistent with recent reports26,30. Table 1 lists the lattice parameters of Ge₂Y₂ (Y = As, P, N) monolayers and previously reported values. The lattice constants of α- and β-Ge₂As₂ monolayers are a = 3.800 Å and a = 3.815 Å, respectively, the band gap data also exhibit excellent agreement, consistent with prior data27. Furthermore, we conducted additional calculations of the band gap for the Ge₂Y₂ monolayers using the hybrid functional (HSE06) method. The results indicated that the band gap of the α-Ge₂As₂ monolayer is 1.42 eV, while that of the β-Ge₂As₂ monolayer is 1.79 eV. These findings confirm that the Ge₂Y₂ monolayers possesses semiconducting properties. Additional results are provided in Supplementary Table 1. Our results also indicate that the lattice constants of Ge₂Y₂ (Y = As, P, N) monolayers decrease as the atomic radius of element Y decreases from As to N.

Fig. 1
figure 1

The top and side views of (a) α-Ge₂Y₂ and (b) β-Ge₂Y₂ monolayers. (c) shows the band structure and density of states of the α-Ge₂As₂ monolayer. (d) shows the band structure and density of states of the β-Ge₂As₂ monolayer.

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Table 1 Crystal Structure Parameters and Band Gaps of α- and β-Ge2Y2 Monolayers.
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Figure 1c and d are the electronic band structure diagram, density of states of the Ge2As2 monolayers, from which it can be directly observed that the Ge2As2 monolayers are indirect bandgap semiconductors. For the α-Ge2As2 monolayer, the conduction band minimum is located near the M point, and the valence band maximum is at the Γ point. The effective mass of holes at the valence band maximum along the x- and y-axis is 2.413 (2.412) me, and the effective mass of electrons at the conduction band minimum along the x- and y-axis is 0.2446 (0.5609) me. The Fermi velocities at the M point of the conduction band minimum along the x- and y-axis are 1.87 × 105 and 7.81 × 104 m/s, respectively. The β-Ge2As2 monolayer exhibits similar characteristics; the effective mass of electrons at the conduction band minimum (and holes at the valence band maximum) is 0.275 (2.185) me, with Fermi velocities at the M point of the conduction band minimum along the x- and y-axis being 2.10 × 105 and 4.01 × 104 m/s, respectively, demonstrating significant anisotropy. However, the α phase of the Ge2As2 monolayer displays more pronounced anisotropy compared to the β phase. The contribution of Ge and As atoms near the Fermi level (EF) of α- and β-Ge2As2 monolayers conduction band is similar, and the contribution of As atoms near the Fermi level (EF) of valence band is mainly caused by p orbitals of As atoms.

Figure 2a and b represent the band structure diagrams, density of states for the α-Ge2P2 and β-Ge2P2 monolayers, Fig. 2c, d corresponds to α-Ge2N2 and β-Ge2N2 monolayers respectively. Both the Ge2P2 and Ge2N2 monolayers are indirect bandgap semiconductors. Like the Ge2As2 monolayers, the Ge2P2 monolayers have its conduction band minimum at the M point and its valence band maximum at the Γ point. However, the Ge2N2 monolayers has its conduction band minimum at the Γ point, with its valence band maximum located between the Γ point and the M point. The electron states near the Fermi level (EF) of Ge2P2 monolayers conduction band are equally contributed by Ge atoms and P atoms, and the electronic states near the Fermi level (EF) of valence band are mainly contributed by the p orbitals of P atoms. The α-Ge2N2 and β-Ge2N2 monolayers are different from the Ge2As2 and Ge2P2 monolayers in that the electron states near the Fermi level in the conduction band are contributed by Ge atoms, while the electron states near the Fermi level in the valence band are mainly contributed by the p orbitals of N atoms. Additionally, energy dispersion exhibits anisotropy away from the Γ point, like many two-dimensional materials, showing anisotropic transport properties in both the x and y directions36. The specific Fermi velocities of other materials in the Ge2Y2 family are listed in Table 2, among which the Fermi velocity of the Ge2P2 monolayers is close to that of silicene.

Fig. 2
figure 2

(a) shows the band structure and density of states of the α-Ge₂P₂ monolayer. (b) shows the band structure and density of states of the β-Ge₂P₂ monolayer. (c) shows the band structure and density of states of the α-Ge₂N₂ monolayer. (d) shows the band structure and density of states of the β-Ge₂N₂ monolayer.

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Table 2 Fermi Velocities of α- and β-Ge2Y2 Monolayers.
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The structural stability of Ge2Y2 (Y = As, P, N) monolayers has been previously studied and calculated. Results indicate that the phonon spectra of all six Ge2Y2 structures exhibit no imaginary frequencies throughout the entire Brillouin zone, demonstrating that Ge2Y2 monolayers are dynamically stable in equilibrium27. To provide a more intuitive assessment of the semiconductor properties of the Ge2Y2 monolayers, take the α-Ge2As2 monolayer as an illustrative example, the carrier mobility is further investigated using the deformation potential theory37,38:

$${\mu }_{2D}=\frac{e{\hslash }^{3}{C}_{2D}}{{k}_{B}T\left|{m}^{*}\right|{m}_{d}{E}_{1}^{2}}$$
(1)

where m* represents the effective mass in the transport direction and kB is the Boltzmann constant. E1 indicates the deformation potential constant along the transport direction for holes at the VBM or electrons at the CBM, as determined by \({E}_{1}=\frac{\Delta E}{\left(\Delta l/{l}_{0}\right)}\). md is the average effective mass of the carriers, as defined by \({m}_{d}=\sqrt{{m}_{x}^{*}{m}_{y}^{*}}\). C2D is the elastic modulus of the homogeneously deformed crystal, and for a 2D material, the elastic modulus can be calculated by \({C}_{2D}=\left[{\partial }^{2}E/\partial {\left(\Delta l/{l}_{0}\right)}^{2}\right]/{S}_{0}\).

The results of the calculations are presented in Table 3. The effective mass of electrons along the zigzag (armchair) direction is \(\left|{m}_{e}\right|\) = 0.245 \({m}_{e}\)(0.561 \({m}_{e}\)), while that of holes is \(\left|{m}_{h}\right|\) = 2.413 \({m}_{e}\)(2.412 \({m}_{e}\)). These results indicate that the effective mass of electrons is significantly lower than that of holes, suggesting that electron-dominated carriers exhibit higher mobility compared to hole-dominated carriers. The elastic constant \({C}_{2D}\) in the zigzag (armchair) direction is 73.496 (85.165) \(N/m\), indicating that the armchair direction exhibits greater resistance to applied strain relative to the zigzag direction. The deformation potential constants for electrons and holes along the zigzag (armchair) direction are \({E}_{1}\) = 4.786 (2.360) eV and \({E}_{1}\) = 2.753 (2.685) eV, respectively. A smaller deformation potential constant implies minimal changes in carrier energy after phonon scattering, which facilitates efficient carrier transport. Based on the obtained values of \(\left|{m}^{*}\right|\), \({C}_{2D}\), and \({E}_{1}\), and using Formula (1), the carrier mobility of α-Ge2As2 monolayers at room temperature (T = 300 K) was calculated. Specifically, the electron mobility \({\mu }_{e}\) = 678.862 (1410.761) \({cm}^{2}{V}^{-1}{S}^{-1}\) and hole mobility \({\mu }_{h}\) = 31.954 (38.955) \({cm}^{2}{V}^{-1}{S}^{-1}\) were observed along the zigzag (armchair) direction. Compared to MoS2 monolayers39,40, α-Ge2As2 monolayers exhibit superior electron mobility. This enhanced carrier mobility suggests that carriers within the material can be easily mobilized under an applied electric field, a critical factor for achieving high-performance, low-power nanoelectronic devices.

Table 3 For the α- Ge2As2 monolayers electrons and holes, the effective mass (m*), deformation potential constant (E1), elastic modulus (C2D), and carrier mobility (μ) along the zigzag (x) and armchair (y) directions.
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Constructing Ge2Y2 monolayers p–n junction diodes

The p–n junction diodes of α-Ge2Y2 and β-Ge2Y2 monolayers (Fig. 3a) achieve p-type and n-type doping using the virtual crystal approximation method41,42,43,44,45. The chosen doping method is electrostatic doping by atomic compensation charge, widely used for simulating various nanodevices46. Each diode consists of drain (D) and source (S) probes and a central scattering p–n junction. The D/S probes are simulated by semi-infinite p and n-doped supercells along the transport direction. When a forward D-S bias Vb is applied, a positive current flows from D to S, and vice versa. The current through the α-Ge2Y2 and β-Ge2Y2 monolayers p–n junction diodes is determined as:

$$I\left( {V_{b} } \right) = \frac{2e}{h}\int {_{ - \infty }^{\infty } T\left( {E,V_{b} } \right)} \left[ {f_{D} \left( {E - \mu_{D} } \right) - f_{S} \left( {E - \mu_{S} } \right)} \right]dE$$
(2)
Fig. 3
figure 3

Electron transport properties of p–n junction diode in α-Ge2Y2 monolayers. (a) Schematic of the p–n junction diodes. I-V curves and rectification ratio curves based on p–n junction diodes of α-Ge2As2 monolayer (b, c) α-Ge2P2 monolayer (d, e) and α-Ge2N2 monolayer (f, g).

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The Fermi–Dirac distribution function of the D(S) electrode is:

$$f_{D\left( S \right)} = \left\{ {1 + exp\left[ {{{\left( {E - \mu_{D\left( S \right)} } \right)} \mathord{\left/ {\vphantom {{\left( {E - \mu_{D\left( S \right)} } \right)} {k_{B} T_{D\left( S \right)} }}} \right. \kern-0pt} {k_{B} T_{D\left( S \right)} }}} \right]} \right\}^{ - 1}$$
(3)

Here, \(\mu_{D\left( S \right)}\) and \(T_{D\left( S \right)}\) represent the chemical potential and electron temperature, respectively. In this work, the bias voltage ranges from -0.5 to 0.5 V with a sampling interval of 0.1 V. The paper primarily discusses the β-Ge2Y2 monolayers p–n junction diodes, with the results for the β-Ge2Y2 monolayers provided in the supplementary materials (FIG. S1).

The current–voltage (I-V) curves of the α-Ge2As2 monolayer p–n junction diode (Fig. 3b) demonstrate a strong rectification effect in both Z-type and A-type configurations. Under a reverse bias of -0.5 V, the total current of the Z-type and A-type p–n junction diodes of the α-Ge2As2 monolayer are 205.85 and 324.16 mA/mm, respectively (Fig. 3b), showing electrical anisotropy with a ratio of 0.64. Figure 3c demonstrates the strong rectification effect of the p–n junction diodes along the zigzag and armchair directions: both have a rectification ratio RR = I(-Vb)/I(Vb) as high as 105.

Other monolayers p–n junction diodes in the Ge2Y2 family exhibit similar behavior. For α-Ge2P2, the total currents for Z-type and A-type p–n junction diodes at a reverse bias of -0.5 V are 1.45 and 1.91 mA/mm, respectively (Fig. 3d), with an electrical anisotropy ratio of 0.75. Similarly, for α-Ge2N2, the total currents for Z-type and A-type p–n junction diodes at a reverse bias of -0.5 V are 225.64 and 187.41 μA/mm, with an electrical anisotropy ratio of 1.20(Fig. 3f). As illustrated in Fig. 3e and g, the monolayers of α-Ge2P2 and α-Ge2N2 both exhibit good rectification effects. For the α-Ge2P2 monolayers, the A-type device shows better rectification than the Z-type. Similarly to the α-Ge2As2 monolayers, the Z-type device of the α-Ge2N2 monolayer has a better rectification effect. These properties suggest that Ge2Y2 monolayers p–n junction diodes have potential applications in rectifiers.

Ge2Y2 Monolayers p-i-n Junction Field-Effect Transistors

A vertical electric field can also enhance device transport performance. Therefore, the field-effect characteristics of p-i-n junction field-effect transistors (FETs) based on monolayers Ge2Y2 (Y = As, P, N) were studied, as shown in Fig. 4a. The FET consists of two side electrodes and a middle intrinsic scattering region. The side electrodes are made of p-doped and n-doped Ge2Y2 monolayers. The approximately 3 nm middle intrinsic region (i) is composed of intrinsic Ge2Y2 monolayers, serving as the FET channel, and is covered by top and bottom gates spanning the entire intrinsic region. The current through the p-i-n junction FET is determined using the Eq. 47:

$$I(V_{b} ,V_{g} ) = \frac{2e}{h}\int_{ - \infty }^{\infty } T(E,V_{b} ,V_{g} )\left[ {f_{D} (E - \mu_{D} ) - f_{S} (E - \mu_{S} )} \right]dE$$
(4)
Fig. 4
figure 4

(a) Schematic of p-i-n junction field-effect transistor in α-Ge2As2 monolayer. I-V curves and rectification ratio curves at gate voltages of 0 (b), -5 (c), and 5 V (d).

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Figure 4b shows the I-V curves of Z-type and A-type p-i-n junction FETs based on the α-Ge2As2 monolayer at zero gate voltage (Vg) within a bias range of -0.5 to 0.5 V. It is evident that FETs based on α-Ge2As2 monolayer exhibit excellent rectifying effects, similar transport mechanisms, and strong electrical anisotropy as their p–n junction diodes. However, due to the semiconductor characteristics of the middle intrinsic region, the current density is much lower than that of the p–n junction diodes. When a gate voltage of 5 or -5 V is applied, as shown in Fig. 4c and d,the current density increases sharply, and the RR curves of both Z-type and A-type FETs also increase. Additionally, at gate voltages of 5 and -5 V, the transport behavior is almost identical.

The p-i-n junction FETs based on Ge2P2 and Ge2N2 monolayers also exhibit similar device characteristics to their corresponding p–n junction diodes and show equally excellent field-effect behaviors as the p-i-n junction FETs based on α-Ge2As2 monolayers, as shown in Figs. 5a and 6a. It is noted that, when a positive gate voltage is applied, the threshold voltage of the Z-type p-i-n junction FETs based on Ge2Y2 is significantly lower than that of the A-type. As the gate voltage increases, the current density also increases significantly, demonstrating field-effect behavior. Figures 5b and 6b show the rectification ratio curves of the p-i-n junction FETs based on Ge2P2 and Ge2N2 monolayers at different gate voltages. As the gate voltage increases, the rectification ratio increases by at least two orders of magnitude, and in some cases, by up to five orders of magnitude, particularly for Z-type Ge2P2 and A-type Ge2N2 p-i-n junction FETs at -0.5 V, where the rectification ratio reaches 109. The β-Ge2Y2 monolayers p-i-n field-effect transistors also exhibit similarly good gate control behavior, with the specific results shown in Supplementary FIG. S2, S3, and S4. These highly tunable transistor characteristics of monolayer Ge2Y2 monolayers p-i-n junctions make them not only ideal materials for efficient current modulation but also promising candidates for the development of next-generation nanoscale field-effect transistor devices.

Fig. 5
figure 5

Transport characteristics of p-i-n junction field-effect transistor (FET) based on α-Ge2P2 monolayer at different gate voltages. I-V curves (a) and rectification ratio curves (b) of α-Ge2P2 monolayer p-i-n junction FET at gate voltages of 0, 5, and 10 V.

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Fig. 6
figure 6

Transport characteristics of p-i-n junction field-effect transistor (FET) based on α-Ge2N2 monolayer at different gate voltages. I-V curves (a) and rectification ratio curves (b) of α-Ge2N2 monolayer p-i-n junction FET at gate voltages of 0, 5, and 10 V.

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Study on the optoelectronic properties of Ge2Y2 monolayers

We further explored the potential applications of α-Ge2Y2 and β-Ge2Y2 monolayers in optoelectronics, Z-type and A-type p-i-n junction phototransistors based on Ge2Y2 monolayers were designed and investigated their photocurrent transport properties under illumination, (see Fig. 8a) and investigated their photocurrent transport properties under light to evaluate their potential applications as optoelectronic devices. As shown in Fig. 7, the optical absorption spectra of α- and β-phase Ge2As2 and Ge2P2 monolayers show isotropy in the x- and y-axis directions while the Ge2N2 monolayers show a weak anisotropy trend. Light absorption occurs when the photon energy exceeds the bandgap of the material. With the increase of photon energy, the absorption intensity gradually increases, and the maximum absorption coefficient in the purple light region can reach 106 cm−1. This remarkable ability to absorb light makes the Ge2Y2 monolayer a potential candidate for use in photovoltaic solar cells and other optoelectronic devices.

Fig. 7
figure 7

Optoelectronic properties of Ge2Y2 monolayers. Light absorption coefficients of (a) α- and (b) β-Ge2As2 monolayers. Light absorption coefficients of (c) α- and (d) β-Ge2P2 monolayers. Light absorption coefficients of (a) α- and (b) β-Ge2N2 monolayers.

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In the simulated photoelectric transport process presented in this paper, the incident light is linearly polarized, with the photon energy range set from 1.6 to 3.2 eV, which corresponds to the visible light range. Specifically, the photocurrent injected into the electrode α can be determined by48,49:

$${J}_{\text{D}}^{(ph)}=\frac{ie}{h}\int Tr\left\{{\Gamma }_{m}\left[{G}_{ph}^{<}+{f}_{m}(\varepsilon )({G}_{ph}^{>}-{G}_{ph}^{<})\right]\right\}d\varepsilon$$
(5)

where \(m (\text{D},\text{S})\) denotes the electrode, \({\Gamma }_{m}=i\left({\sum }_{m}^{r}-{\sum }_{m}^{a}\right)\) denotes the coupling of the central area with the electrode \(m\). \({G}_{ph}^{</>}={G}_{0}^{r}{\sum }_{ph}^{</>}{G}_{0}^{a}\) denotes the lesser (greater) Green’s function of the central area; \({\sum }_{ph}^{</>}\) is the self-energy of electrodes with the electron–photon interaction50, which includes the information about the polarization of the polarized light of a complex vector \(e\); \({f}_{m}\left(\varepsilon \right)\) represents the Fermi–Dirac distribution function to the electrode \(m\). The normalized photocurrent can be denoted by the following photoresponse function51,52:

$$I={J}_{\text{D}}^{(ph)}/\text{e}{I}_{\upomega }$$
(6)

where \({I}_{\upomega }\) is the photon flux. Under zero bias (no power supply), We then give, in Fig. 8b, c and d, the photocurrent (I) for different photon energies in the α-Ge2As2, α-Ge2P2 and α-Ge2N2 monolayers p-i-n junction phototransistors. This indicates that in the higher photon energy range of 2.6 to 3.2 eV, the photocurrent (I) in the Z-type for the α-Ge2As2 and α-Ge2P2 monolayers is significantly greater than that in the A-type. Additionally, the Ge2Y2 monolayer p-i-n junction phototransistor exhibits weaker anisotropic optoelectronic characteristics. The photocurrent of the α-Ge2As2 and α-Ge2P2 monolayers p-i-n junction phototransistors reach the maximum of 3.49 and 3.12 respectively at 3.2 eV, suggests that they have potential applications in optoelectronic devices within the visible light range, particularly in the violet light spectrum. The α-Ge2N2 monolayer p-i-n junction phototransistor exhibits a stronger photoelectric response to yellow-green light in the visible spectrum, with its Z-type device achieving a maximum photocurrent of 4.63 at 2.2 eV. The experimental results of β-Ge2Y2 monolayer are shown in Figure S5 of supplementary materials. These values correspond to different regions of visible light, suggesting their potential applications in photodetectors for various visible light spectra. Additionally, the β-Ge2Y2 monolayers p-i-n junction phototransistors also exhibit good optoelectronic response and anisotropy, with specific results shown in the supplementary materials. These excellent optoelectronic properties make Ge2Y2 monolayers promising materials for photovoltaic devices and photodetectors.

Fig. 8
figure 8

Optoelectronic properties of α-Ge2Y2 monolayers. (a) Schematic of p-i-n junction optoelectronic transistor in α-Ge2As2 monolayer. The photocurrent of α-Ge2As2 monolayer (b), α-Ge2P2 monolayer(c) and α-Ge2N2 monolayer (d) optoelectronic transistor at zero bias voltage (no power source).

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Conclusion

In this study, we employed the first-principles calculations based on Density functional theory (DFT) to design conceptual nanodevices based on Ge2Y2 (Y = As, P, N) monolayers and investigated their electronic, transport, and optoelectronic properties. The results indicate that Ge2Y2 semiconductors with indirect bandgaps ranging from 0.92 to 1.14 eV exhibit good dynamic stability and significant mechanical anisotropy. The electronic properties of Ge2Y2 monolayers are highly sensitive to applied uniaxial and biaxial strains, with strain predicted to be an effective method for tuning bandgaps. Both Z-type and A-type Ge2Y2 monolayers p–n junction diodes demonstrate strong rectification effects, with ultra-high rectification ratios (up to 105 for Ge2As2), high current densities (up to 324.16 mA/mm for Ge2As2), significant electrical anisotropy, and high current anisotropy ratios (1.20 for Ge2N2). Field-effect transistors based on Ge2Y2 monolayers exhibit perfect rectification effects and strong electrical anisotropy similar to p–n junction diodes. Additionally, gate voltage can effectively modulate the current of field-effect transistors. Ge2Y2 monolayers and their phototransistors also show excellent photoresponse in the visible regions. The different device performances from As to N in Ge2Y2 monolayers highlight their potential applications in functional electronic and optoelectronic devices. Due to their indirect and strain-tunable bandgaps, and excellent electronic transport and optoelectronic properties, Ge2Y2 monolayers materials are promising candidates for applications in nano-devices and optoelectronic devices.

Computational methods

The electronic structure and transport properties of 2D Ge₂Y₂ monolayers were studied using density functional theory53,54,55 combined with the nonequilibrium Green’s function method56,57,58,59. Calculations were performed using the Hongzhiwei Nanodcal software60,61. Electron interactions were treated with the generalized gradient approximation62,63, and the atomic nuclei were described by optimized norm-conserving pseudopotentials64. During structural optimization, the total energy and forces on each atom were converged to less than 10−6 eV and 10−4 eV/Å, respectively, the temperature is set to room temperature 300 K. To ensure sufficient accuracy and computational efficiency, a cutoff energy of 100 Ha and a 1 × 8 × 8 Monkhorst–Pack k-points grid were used for Brillouin zone sampling and structural optimization. To avoid interactions between adjacent layers, a vacuum space of approximately 25 Å was added in the vertical direction of the 2D surface. Additionally, the hybrid functional (HSE06)65 is employed to assess the bandgap, using parameter settings identical to those of the PBE. For electronic transport calculations, the Hamiltonian and charge density were iterated until the total energy difference between steps was below 10–4 eV. The Brillouin zones of the left and right electrodes of armchair (A-type) and zigzag (Z-type) Ge₂Y₂ monolayer devices were sampled with 1 × 7 × 280 and 1 × 4 × 300 k-points grids, respectively.