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Electric Field Controlled Indirect-Direct-Indirect Band Gap Transition in Monolayer InSe


Electronic structures of monolayer InSe with a perpendicular electric field are investigated. Indirect-direct-indirect band gap transition is found in monolayer InSe as the electric field strength is increased continuously. Meanwhile, the global band gap is suppressed gradually to zero, indicating that semiconductor-metal transformation happens. The underlying mechanisms are revealed by analyzing both the orbital contributions to energy band and evolution of band edges. These findings may not only facilitate our further understanding of electronic characteristics of layered group III-VI semiconductors, but also provide useful guidance for designing optoelectronic devices.


Since the pioneering work on the experimental realization of a single-layer graphite, namely graphene [1, 2], atomically thin two-dimensional (2D) materials have been paid lots of attentions [3, 4]. Various monolayer 2D materials have been theoretically predicted or experimentally discovered including silicene [57], germanane [8], black phosphorus [9, 10], transition metal dichalcogenides (TMDs) [1113], and hexagonal boron nitride [1416]. Although these atomically thin 2D materials have the similar honeycomb lattice structures, their electronic structures and conductivity properties are quite different including metal [1, 2, 58], semiconductor [913], and insulator [1416]. Therefore, according to their electronic characters, these single layer 2D materials may find applications in the design of multifunctional electronic and optical devices [3, 4]. For example, tunable optical devices with high-quality factor based on Si-graphene metamaterials [17], Cu-graphene metamaterials [18], and MoS2-SiO2-Si waveguide structures [19] are proposed. Perfect valley or/and spin polarization devices based on the ferromagnetic graphene [20], strained graphene with Rashba spin-orbit coupling and magnetic barrier [21], and strained silicene with an electric field are suggested [22, 23]. Moreover, the interaction effects between the decomposition components of SF6 and different materials including N-doped single-wall carbon nanotubes [24], Pt3-TiO2(1 0 1) surface [25], Ni-doped MoS2 monolayer [26], and Pd (1 1 1) surface [27] are investigated by using the density functional theory (DFT).

Group III–VI compounds MXs (M = Ga, In and X = S, Se, Te) are another family of layered 2D materials. Due to their unique electrical characters, these materials have drawn many researchers’ attentions [28]. DFT [2933] and tight-binding model [34] calculations show that energy band gap of layered MXs is thickness dependent, increasing from 1.3 to 3.0 eV as the number of layers is decreased. At the same time, direct-indirect band gap transition is observed, which is opposite to the behaviors of layered black phosphorus [9, 10] and TMDs [1113]. This sizable energy band gap modulation of layered MXs may be used to design optoelectronic devices [35, 36]. In addition, the stability of InSe doped with oxygen defects is investigated and found that it is more stable than black phosphorus in the air [37]. The magnetism of InSe monolayer can be tuned by adsorbing As [38], C, and F [39]. Huge spin-charge conversion effect is found in bilayer InSe due to the broken mirror symmetry [40]. Moreover, the electronic structure and the current-voltage characteristics of monolayer InSe nanoribbons strongly depend on the edge states [41]. On the other hand, experimental researches verify the layer-dependent electronic structures of MXs and they can responds to the light spanning the visible and near-infrared regions [4245]. Also, the carrier mobilities of MXs are found to be high, enabling that they may be used to design field effect transistors. For bulk GaS and GaSe, the carrier mobilities are about 80 and 215 cm 2 V −1 S −1 [46], respectively. For the monolayer InSe, the carrier mobility is even up to almost 10 3 cm 2 V −1 S −1 [47]. Moreover, band gap of layered InSe can be manipulated by uniaxial tensile strain, which is identified by the photoluminescence spectra [48].

From the viewpoint of the optoelectronic device designment, the efficiency of the devices based on direct band gap semiconductors are better than those based on indirect band gap ones. Therefore, transforming indirect band gap few-layer MXs to direct band gap type is a challenge for scientific community. Very recently, band gap manipulation and indirect-direct band gap transition are found in monolayer InSe by uniaxial strain [49]. Also, direct band gap semiconductors have been obtained by stacking 2D n-InSe and p-GeSe(SnS). And the band gap values and band offset of these van der Waals heterojunctions can be tuned by the interlayer coupling and external electric field [50]. In addition, the possible stacking configurations of bilayer InSe and the influence of the perpendicular electric field on their electronic structures are studied. Indirect band gap bilayer InSe can be transformed to the metallic type by varying the electric field strength [51]. Similarly, in other buckled 2D materials like silicene [52], germanene [53], transition metal dichalcogenides [54, 55], and black phosphorus [56], a perpendicular electric field is also proposed to tune their band gap and electronic characteristics. In light of these previous studies, a natural question may be inquired what are the electric field effects on the electronic structures of the monolayer InSe.

In this letter, the effects of a perpendicular electric field on the electronic structures of the monolayer InSe are investigated by using the tight-binding model Hamiltonian. Indirect-direct-indirect band gap transition can be achieved in the considered system with increasing electric field strength. At the same time, band gap of the monolayer InSe is decreased gradually, eventually rendering it metallic. The underlying physics mechanisms of these effects are unraveled by analyzing the orbital decomposition for the energy band and the electric field-modulated energy position shift of the band edges. Our studies may benefit to fundamentally understand the electronic properties of few-layer InSe as well as provide theoretical bases for 2D optoelectronic devices.


The top view of InSe monolayer is sketched in Fig. 1a, where the big purple spheres represent indium ions while the small green ones depict selenium ions. This two types of ions form graphene-like hexagonal structure in the xy plane with lattice constant a, the distance between the nearest In or Se ions. Figure 1b shows the schematic of side view of InSe monolayer. Differing from graphene, two sublayers with mirror symmetry in the xz plane are observed. The vertical distance between In (Se) ions of different sublayers is set at d (D). Therefore, a unit cell of monolayer InSe consists of four ions Se1, In1, Se2, and In2, as shown by the red ellipse in Fig. 1b, in which number 1 (2) indicates the sublayer index.

Fig. 1
figure 1

(Color online) Top (a) and side (b) view of the monolayer InSe in the xy and xz planes, respectively. The lattice constant between the nearest In or Se ions in the xy plane is a, and the distance between the nearest In (Se) ions in different sublayers is d (D). A perpendicular electric field along z-axis Ez is applied to the monolayer InSe. c Energy band of monolayer InSe

The tight-binding Hamiltonian up to second-nearest neighbor interactions including all possible hoppings between the s and p orbitals of In and Se ions reads [34]

$$ H=\sum\limits_{l} H_{0l}+H_{ll}+H_{ll'}, $$

in which the sum runs over the sublayers l=1 and 2, and l=2(1) as l=1(2). H0l, Hll, and \(\phantom {\dot {i}\!}H_{ll^{\prime }}\) consist of terms coming from the on-site energies, hopping energies within and between the two sublayers, respectively. And the explicit expressions of them are given as [34]

$$\begin{array}{@{}rcl@{}} H_{0l}=\sum\limits_{i}[\varepsilon_{\text{In}_{s}}a_{lis}^{\dag}a_{lis}+ \sum\limits_{\alpha}\varepsilon_{\text{In}_{p_{\alpha}}}a_{{lip}_{\alpha}}^{\dag}a_{{lip}_{\alpha}}+ \\ \varepsilon_{\text{Se}_{s}}b_{lis}^{\dag}b_{lis}+ \sum\limits_{\alpha}\varepsilon_{\text{Se}_{p_{\alpha}}}b_{{lip}_{\alpha}}^{\dag}b_{{lip}_{\alpha}}], \end{array} $$

where the sum runs over all unit cells in sublayer l. \(\phantom {\dot {i}\!}\varepsilon _{\mathrm {In(Se)}_{s}}\) is the on-site energy for the s orbital of In (Se) ions, while \(\phantom {\dot {i}\!}\varepsilon _{\mathrm {In(Se)}_{p_{\alpha }}}\) is that for orbital pα (α=x,y,z). \(a_{lis}^{\dag }\) (alis) is the creation (annihilation) operator for an electron in s orbital on In ions in unit cell i and sublayer l, but \(\phantom {\dot {i}\!}a_{{lip}_{\alpha }}^{\dag }\) (\(\phantom {\dot {i}\!}a_{{lip}_{\alpha }}\)) for an electron in pα orbital. Similarly, b (b) is the creation (annihilation) operator for an electron in the relevant orbital on Se ions.

$$\begin{array}{@{}rcl@{}} H_{ll}=H_{ll}^{(\text{In}-\text{Se})_{1}}+H_{ll}^{\text{In}-\text{In}}+H_{ll}^{\text{Se}-\text{Se}}+H_{ll}^{(\text{In}-\text{Se})_{2}}, \end{array} $$

in which [34]

$$ {{}{\begin{aligned} H_{ll}^{(\text{In}-\text{Se})_{1}}=\sum\limits_{<\text{In}_{li},\text{Se}_{lj}>}\{T_{ss}^{(\text{In}-\text{Se})_{1}}b_{ljs}^{\dag} a_{lis}+T_{sp}^{(\text{In}-\text{Se})_{1}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{lj}} \\ b_{ljp_{\alpha}}^{\dag} a_{lis}+T_{ps}^{(\text{In}-\text{Se})_{1}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{lj}}b_{ljs}^{\dag} a_{lip_{\alpha}}+\sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{(\text{In}-\text{Se})_{1}}- \\ (T_{\pi}^{(\text{In}-\text{Se})_{1}}+T_{\sigma}^{(\text{In}-\text{Se})_{1}})R_{\alpha}^{\text{In}_{li}\text{Se}_{lj}} R_{\beta}^{\text{In}_{li}\text{Se}_{lj}}]b_{ljp_{\beta}}^{\dag} a_{lip_{\alpha}}\}\}+\mathrm{H.c.}, \end{aligned}}} $$
$$ { \begin{aligned} H_{ll}^{\text{In}-\text{In}}=\sum\limits_{<\text{In}_{li},\text{In}_{lj}>}\{T_{ss}^{\text{In}-\text{In}}a_{ljs}^{\dag} a_{lis}+T_{sp}^{\text{In}-\text{In}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{In}_{lj}} a_{ljp_{\alpha}}^{\dag} a_{lis}+ \\ \sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{\text{In}-\text{In}}- (T_{\pi}^{\text{In}-\text{In}}+T_{\sigma}^{\text{In}-\text{In}})R_{\alpha}^{\text{In}_{li}\text{In}_{lj}} R_{\beta}^{\text{In}_{li}\text{In}_{lj}}]a_{ljp_{\beta}}^{\dag} a_{lip_{\alpha}}\}\} \\ +\mathrm{H.c.}, \end{aligned}} $$
$$ { \begin{aligned} H_{ll}^{\text{Se}-\text{Se}}=\sum\limits_{<\text{Se}_{li},\text{Se}_{lj}>}\{T_{ss}^{\text{Se}-\text{Se}}b_{ljs}^{\dag} b_{lis}+T_{sp}^{\text{Se}-\text{Se}}\sum\limits_{\alpha}R_{\alpha}^{\text{Se}_{li}\text{Se}_{lj}} b_{ljp_{\alpha}}^{\dag} b_{lis}+ \\ \sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{\text{Se}-\text{Se}}- (T_{\pi}^{\text{Se}-\text{Se}}+T_{\sigma}^{\text{Se}-\text{Se}})R_{\alpha}^{\text{Se}_{li}\text{Se}_{lj}} R_{\beta}^{\text{Se}_{li}\text{Se}_{lj}}]b_{ljp_{\beta}}^{\dag} b_{lip_{\alpha}}\}\} \\ +\mathrm{H.c.}, \end{aligned}} $$


$$ { \begin{aligned} H_{ll}^{(\text{In}-\text{Se})_{2}}=\sum\limits_{<\text{In}_{li},\text{Se}_{lj'}>}\{T_{ss}^{(\text{In}-\text{Se})_{2}}b_{lj's}^{\dag} a_{lis}+T_{sp}^{(\text{In}-\text{Se})_{2}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{lj'}} \\ b_{lj'p_{\alpha}}^{\dag} a_{lis}+T_{ps}^{(\text{In}-\text{Se})_{2}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{lj'}}b_{lj's}^{\dag} a_{lip_{\alpha}}+\sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{(\text{In}-\text{Se})_{2}}- \\ (T_{\pi}^{(\text{In}-\text{Se})_{2}}+T_{\sigma}^{(\text{In}-\text{Se})_{2}})R_{\alpha}^{\text{In}_{li}\text{Se}_{lj'}} R_{\beta}^{\text{In}_{li}\text{Se}_{lj'}}]b_{lj'p_{\beta}}^{\dag} a_{lip_{\alpha}}\}\}+\mathrm{H.c.} \end{aligned}} $$

include the hopping terms between the nearest-neighbor In-Se, In-In, Se-Se, and next-nearest In-Se pairs within the same sublayer l, respectively. \(T_{ss/sp/ps}^{\mathrm {X}}\) is the hopping integral for the ss/sp/ps orbitals between the corresponding pair X, while \(T_{\pi (\sigma)}^{\mathrm {X}}\) is that for the parallel p and p orbitals perpendicular to (lying along) the hopping vector \(R_{\alpha }^{\mathrm {X}}\) [57]. For example

$$\begin{array}{@{}rcl@{}} R_{\alpha}^{(\text{In}-\text{Se})_{1}}=\frac{\mathrm{\mathbf{R}}_{\text{Se}_{lj}}-\mathrm{\mathbf{R}}_{\text{In}_{li}}} {|\mathrm{\mathbf{R}}_{\text{Se}_{lj}}-\mathrm{\mathbf{R}}_{\text{In}_{li}}|}\cdot \hat{\alpha}, \end{array} $$

where \(\phantom {\dot {i}\!}\mathrm {\mathbf {R}}_{{\text {In}_{li}}/{\text {Se}_{lj}}}\) is the position vector for Inli/Selj, \(\hat {\mathbf {\alpha }}\) is a unit vector along α.

$$\begin{array}{@{}rcl@{}} H_{ll'}=H_{ll'}^{(\text{In}-\text{In})_{1}}+H_{ll'}^{\text{In}-\text{Se}}+H_{ll'}^{(\text{In}-\text{In})_{2}}, \end{array} $$

in which [34]

$$ { \begin{aligned} H_{ll'}^{({\text{In}-\text{In}})_{1}}=\sum\limits_{i}\{T_{ss}^{({\text{In}-\text{In}})_{1}}a_{l'is}^{\dag} a_{lis}+T_{sp}^{({\text{In}-\text{In}})_{1}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{In}_{l'i}} a_{l'ip_{\alpha}}^{\dag} a_{lis}+ \\ \sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{({\text{In}-\text{In}})_{1}}- (T_{\pi}^{({\text{In}-\text{In}})_{1}}+T_{\sigma}^{({\text{In}-\text{In}})_{1}})R_{\alpha}^{\text{In}_{li}\text{In}_{l'i}} R_{\beta}^{\text{In}_{li}\text{In}_{l'i}}] \\ a_{l'ip_{\beta}}^{\dag} a_{lip_{\alpha}}\}\}+\mathrm{H.c.}, \end{aligned}} $$
$$ { \begin{aligned} H_{ll'}^{\text{In}-\text{Se}}=\sum\limits_{<\text{In}_{li},\text{Se}_{l'j}>}\{T_{ss}^{\text{In}-\text{Se}}b_{l'js}^{\dag} a_{lis}+T_{sp}^{\text{In}-\text{Se}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{l'j}} \\ b_{l'jp_{\alpha}}^{\dag} a_{lis}+T_{ps}^{\text{In}-\text{Se}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{Se}_{l'j}}b_{l'js}^{\dag} a_{lip_{\alpha}}+\sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{\text{In}-\text{Se}}- \\ (T_{\pi}^{\text{In}-\text{Se}}+T_{\sigma}^{\text{In}-\text{Se}})R_{\alpha}^{\text{In}_{li}\text{Se}_{l'j}} R_{\beta}^{\text{In}_{li}\text{Se}_{l'j}}]b_{l'jp_{\beta}}^{\dag} a_{lip_{\alpha}}\}\}+\mathrm{H.c.}, \end{aligned}} $$


$$ { \begin{aligned} H_{ll'}^{({\text{In}-\text{In}})_{2}}=\sum\limits_{i}\{T_{ss}^{({\text{In}-\text{In}})_{2}}a_{l'js}^{\dag} a_{lis}+T_{sp}^{({\text{In}-\text{In}})_{2}}\sum\limits_{\alpha}R_{\alpha}^{\text{In}_{li}\text{In}_{l'j}} a_{l'jp_{\alpha}}^{\dag} a_{lis}+\\ \sum\limits_{\alpha,\beta}\{[\delta_{\alpha\beta}T_{\pi}^{({\text{In}-\text{In}})_{2}}- (T_{\pi}^{({\text{In}-\text{In}})_{2}}+T_{\sigma}^{({\text{In}-\text{In}})_{2}})R_{\alpha}^{\text{In}_{li}\text{In}_{l'j}} R_{\beta}^{\text{In}_{li}\text{In}_{l'j}}] \\ a_{l'jp_{\beta}}^{\dag} a_{lip_{\alpha}}\}\}+\mathrm{H.c.} \end{aligned}} $$

include the hopping terms between the nearest-neighbor In-In, In-Se, and next-nearest In-In pairs between sublayers l and l, respectively. If a perpendicular electric field along z-axis is applied to the monolayer InSe, its effects can be introduced by a modification of the on-site orbtial energies of In and Se ions, that is,

$$\begin{array}{@{}rcl@{}} \varepsilon'=\varepsilon+eE_{z}z, \end{array} $$

where e is the electron charge and Ez is the strength of the perpendicular electric field. The perpendicular electric field can be achieved by adding top and bottom gates to the monolayer InSe. Moreover, two insulating layers are inserted between the monolayer InSe and gates to eliminate the electric current along z-axis. As a result, the electric field strength can be tuned by varying the gating voltage.

By transforming the tight-binding Hamiltonian in Eq. (1) into the k space and then diagonalizing it, energy bands E(k) of monolayer InSe without or with a perpendicular electric field can be obtained conveniently, where k is wave vector. At the same time, the coefficient of eigenvector Cnk(o) at band n, orbital o, and wave vector k can also be achieved.

Numerical Results and Discussions

The lattice parameters of monolayer InSe in Fig. 1a and b are taken as a=3.953 Å, d=2.741 Å, and D=5.298 Å, which are obtained by the local density approximation [30]. The on-site and hopping energies in the tight-binding Hamiltonian Eq. (1) are given in Table 1, which are fitted by the density functional theory data with scissor correction [34]. Although only the numerical results of the monolayer InSe are given here, qualitatively similar results have also been found in the bilayer InSe and the bulk InSe. For conciseness, they are not presented in this letter.

Table 1 Parameters (eV) of the tight-binding Hamiltonian in Eq. (1)

Figure 1c shows the energy band of the monolayer InSe. The conduction bands around point Γ display parabola-like energy dispersion, which are similar to that of other normal semiconductors. However, the band structure along Γ−K is slightly asymmetrical with that along Γ−M. And the lowest two conduction bands crossing each other along both these two directions, as indicated by the red cycles. In contrast to the conduction bands, the highest valence band is flat but slightly inverted around point Γ, forming an interesting Mexican hat-like structure. Therefore, monolayer InSe is an indirect band gap semiconductor, which is quite different from that of bulk InSe since it is a direct band gap semiconductor. The energy gap of monolayer InSe can be obtained by \(E_{\mathrm {g}}^{\text {id}}=E_{\mathrm {C}}-E_{\mathrm {A}}=2.715\) eV, which is much enlarged by comparing with that of bulk InSe \(E_{\mathrm {g}}^{\mathrm {d}}=1.27\) eV [34]. However, the other valence bands show normal parabola-like energy dispersion.

In order to comprehend the energy band of monolayer InSe shown in Fig. 1c, the orbital decomposition |Cnk(o)|2 for the energy band is given in Fig. 2. As the two sublayers of the monolayer InSe is symmetrical along z-axis, the ions in different sublayers have the same orbital contributions to the energy band. Here, In and Se ions in sublayer 2, as shown in Fig. 1b, are taken as examples. The upper panels indicate orbital contributions from In ions while the down panels represent those of Se ions. The thickness of lines is proportional to normalized orbital contribution. It can be seen that the lowest conduction band around point Γ is contributed firstly from pz orbital of Se ion and then s orbital of In ion. The second conduction band around K point dominantly originates from px orbital of In ion and then pz orbital of Se ion. However, the highest valence band is principally contributed from pz orbital of Se ion. The other valence bands result from both px and py orbitals of Se ion. These results are consistent with those results obtained by the DFT calculations [34].

Fig. 2
figure 2

(Color online) Orbital decompositions for the energy band of monolayer InSe. Thicker lines indicate a more dominant contribution. Only In and Se ions in sublayer 2 are selected as examples since the two sublayers of the monolayer InSe with mirror symmetry along z-axis (ah)

Energy band of the monolayer InSe with a perpendicular electric field along z-axis is shown in Fig. 3a. The electric field strength is taken as Ez=2.0 V/nm. By comparing with the energy band in Fig. 1c, each conduction and valence band is lifted to the higher energy region as a whole. However, the energy shift of each band is different since its orbital decomposition from the pz orbital of In and Se ions is different. Position of the maximum value of the highest valence band is changed to point Γ while that of the minimum value of conduction band keeps unchanged. Therefore, the monolayer InSe is transformed into a direct band gap semiconductor. And the energy gap is decreased to \(E_{\mathrm {g}}^{\mathrm {d}}=2.61\) eV. Furthermore, the crossings along both Γ−K and Γ−M directions are opened so that energy gaps are generated, as displayed by the red cycles, since the symmetry along z-axis is broken by the perpendicular electric field. When the electric field strength is increased to Ez=6.0 V/nm, the energy gap at point Γ is decreased but those at the crossings is increased further, as shown in Fig. 3b. Interestingly, position of the minimum value of conduction band is altered from point Γ to that around point K, while that of the maximum value of the highest valence band stay at point Γ. This phenomenon means that the monolayer InSe is transited into indirect band gap semiconductor again and the indirect energy gap of the whole band \(E_{\mathrm {g}}^{\text {id}}=1.30\) eV. Similarly, the band gap of monolayer InSe can be controlled by biaxial strain. The band gap ranges from 1.466 to 1.040 eV when the strain is varied from 1 to 4%. In addition, indirect-direct band gap transition is also observed when the monolayer InSe is under uniaxial strain [49]. For the bilayer InSe with a perpendicular electric field, its band gap decreases as the electric field strength increases and it will be closed when the electric field strength is increased to 2.9 V/nm [51].

Fig. 3
figure 3

(Color online) Energy bands of the perpendicular electric field-modulated monolayer InSe at different strengths Ez=2.0 V/nm(a) and 6.0 V/nm (b), respectively. Red circles in a and b mean the opened energy gaps around the crossing points shown in Fig. 1c. c Energies at points A (the black solid line), B (the magenta dashed line), C (the blue dotted line), and D (the green dash-dotted line) shown in Fig. 1c as a function of the electric field strength. d Global band gap as a function of the strength of the electric field. The yellow line means the direct band gap while the red and blue lines indicate the indirect band gaps

For the sake of understanding the changing process of electronic structure of monolayer InSe in the presence of a perpendicular electric field more clearly, energies at the wave vectors corresponding to points A, B, C, and D at the band edges shown in Fig. 1c as a function of the strength of electric field are depicted in Fig. 3c. Energies with respect to all these points move upward as the increasing electric field strength, confirming the evolution of the energy bands in Fig. 3a and b. When the electric field strength Ez<1.6 V/nm, energy at point A in the valence band is higher than that of point B while the bottom of conduction band locates at point C. Therefore, the electric field-modulated monolayer InSe within this strength range is an indirect band gap semiconductor, as shown by the red area. However, energies with respect to points A and B will cross at TP1, and then energy at point B will be higher than that of point A as the electric field strength is increased further. Simultaneously, the bottom of conduction band keeps unchange until the electric field strength is increased to 4.0 V/nm. As a result, the electric field-modulated monolayer InSe within this strength range is a direct band gap semiconductor, as shown by the yellow area. Similar to the energy crossover between points A and B in the valence band, transit point is also observed in the energies at points C and D in the conduction bands, as indicated by TP2. Energy at point D is lower than that of point C while the top of valence band still stay at point B if only the electric field strength is smaller than 9.23 V/nm. Consequently, the electric field-modulated monolayer InSe is turned into an indirect band gap semiconductor again, as shown by the blue area. Interestingly, energies at point B in the highest valence band and point D in the lowest conduction band will cross at TP3 too, which means that the energy band gap is closed. Moreover, energy at point B will be higher than that of point D when the electric field strength is larger than 9.23 V/nm. Therefore, the lowest conduction band and highest valence band will overlap so that the electric field-modulated monolayer InSe becomes a metal in this case, as shown by the cyan area. The global band gap corresponding to different colored areas in Fig. 3c is plotted in Fig. 3d. The band gap corresponding to the red area is almost independent of the varied electric field strength, as shown by the red line. However, the band gap of the yellow area is decreased linearly with increasing electric field strength. Similar band gap behavior is also found in the blue area but with a larger slope. The band gap is decreased to zero as long as the electric field strength is larger than that at point TP3, as shown by the cyan line. The electric field-modulated band gap behaviors indicate that layered III–VI semiconductors have potential applications in designing novel optical detector and absorbers. Moreover, the spectral response frequency of these devices ranges continuously from the violet light (ν≈6.57×1014 Hz as Ez=1.6 V/nm) to the infrared light (ν<3.97×1014 Hz as Ez>5.18 V/nm).

As well known, electronic characteristics of materials are mainly determined by energy band edges. According to the orbital decomposition for the energy band in Fig. 2, both the conduction and valence band edges of monolayer InSe are dominantly contributed from pz orbital of Se ion. Therefore, only pz orbital decompositions of Se ion in sublayer 2 for energy bands shown in Fig. 3a and b are displayed in Fig. 4a and b, respectively. By comparing with Fig. 2h, pz orbital contribution to the conduction bands is slightly changed. Therefore, the shape of these band structures undergoes little affection. However, the pz orbital contribution to the valence bands is strongly modified, resulting in the change of shape of these band structures. Moreover, according to the pz orbital decomposition for the energy band of monolayer InSe with a perpendicular electric field, the relative position of each conduction band keeps unchanged although gaps are opened at band crossings, as indicated by the red cycles. On the contrary, the relative position of each valence band is changed. The energies of the lower valence bands around Γ point increase and surpass those of the highest valence band finally, leading to indirect-direct band gap transition.

Fig. 4
figure 4

(Color online) a and b show pz orbital decomposition of the Se ion in sublayer 2 for the energy bands of the monolayer InSe with a perpendicular electric field shown in Fig. 3a and b, respectively. Thicker lines represent a more important contribution


Electronic structures of monolayer InSe under the modulation of a perpendicular electric field are investigated. Indirect-direct-indirect band gap transition is found for the monolayer InSe by tuning the electric field strength. Simultaneously, global band gap of this system is decreased monotonously to zero with increasing electric field strength, which means that semiconductor-metal transition is achieved. The evolution of energy band of monolayer InSe in the presence of the perpendicular electric field is clarified by analyzing the energy change of band edge and orbital decomposition for energy band. These results may be helpful in further understanding of the electronic structures of monolayer InSe as well as the designment of monolayer-InSe-based photoelectric devices responding from violet to far-infrared light.

Availability of Data and Materials

The datasets supporting the conclusions of this article are included within the article.





Density functional theory


Transition metal dichalcogenides


  1. Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA (2004) Electric field effect in atomically thin carbon films. Science 306:666–669.

    CAS  Article  Google Scholar 

  2. Zhang YB, Tan YW, Stormer HL, Kim P (2005) Experimental observation of quantum Hall effect and Berry’s phase in graphene. Nature 438:201–204.

    CAS  Article  Google Scholar 

  3. Novoselov KS, Fal’ko VI, Colombo L, Gellert PR, Schwab MG, Kim K (2012) A roadmap for graphene. Nature 490:192–200.

    Article  CAS  Google Scholar 

  4. Geim AK, Grigorieva IV (2013) Van der Waals heterostructures. Nature 499:419–425.

    CAS  Google Scholar 

  5. Aufray B, Kara A, Vizzini S, Oughaddou H, Léandri C, Ealet B, Lay GL (2010) Graphene-like silicon nanoribbons on Ag(110): a possible formation of silicene. Appl Phys Lett 96:183102.

    Article  CAS  Google Scholar 

  6. Liu CC, Feng WX, Yao YG (2011) Quantum spin Hall effect in silicene and two-dimensional germanium. Phys Rev Lett 107:076802.

    Article  CAS  Google Scholar 

  7. Feng BJ, Ding ZJ, Meng S, Yao YG, He XY, Cheng P, Chen L, Wu KH (2012) Evidence of silicene in honeycomb structures of silicon on Ag(111). Nano Lett 12:3507–3511.

    Article  CAS  Google Scholar 

  8. Bianco E, Butler S, Jiang S, Restrepo OD, Windl W, Goldberger JE (2013) Stability and exfoliation of germanane: a germanium graphane analogue. ACS Nano 7:4414–4421.

    Article  CAS  Google Scholar 

  9. Li LK, Yu YJ, Ye GJ, Ge QQ, Ou XD, Wu H, Feng DL, Chen XH, Zhang YB (2014) Black phosphorus field-effect transistors. Nat Nanotechnol 9:372–377.

    Article  CAS  Google Scholar 

  10. Liu H, Neal AT, Zhu Z, Luo Z, Xu XF, Tománek D, Ye PD (2014) Phosphorene: an unexplored 2D semiconductor with a high hole mobility. ACS Nano 8:4033–4041.

    Article  CAS  Google Scholar 

  11. Mak KF, Lee C, Hone J, Shan J, Heinz TF (2010) Atomically thin MoS 2: a new direct-gap semiconductor. Phys Rev Lett 105:136805.

    Article  CAS  Google Scholar 

  12. Coleman JN, Lotya M, O’Neill A, Bergin SD, King PJ, Khan U, Young K, Gaucher A, De S, Smith RJ, Shvets IV, Arora SK, Stanton G, Kim HY, Lee K, Kim GT, Duesberg GS, Hallam T, Boland JJ, Wang JJ, Donegan JF, Grunlan JC, Moriarty G, Shmeliov A, Nicholls RJ, Perkins JM, Grieveson EM, Theuwissen K, McComb DW, Nellist PD, Nicolosi V (2011) Two-dimensional nanosheets produced by liquid exfoliation of layered materials. Science 331:568–571.

    Article  CAS  Google Scholar 

  13. Debbichi L, Eriksson O, Lebégue S (2014) Electronic structure of two-dimensional transition metal dichalcogenide bilayers from ab initio theory. Phys Rev B 89:205311.

    Article  CAS  Google Scholar 

  14. Kubota Y, Watanabe K, Tsuda O, Taniguchi T (2007) Deep ultraviolet light-emitting hexagonal boron nitride synthesized at atmospheric pressure. Science 317:932–934.

    Article  CAS  Google Scholar 

  15. Gorbachev RV, Riaz I, Nair RR, Jalil R, Britnell L, Belle BD, Hill EW, Novoselov KS, Watanabe K, Taniguchi T, Geim AK, Blake P (2011) Hunting for monolayer boron nitride: optical and Raman signatures. Small 7:465–468.

    Article  CAS  Google Scholar 

  16. Achilli S, Cavaliere E, Nguyen TH, Cattelan M, Agnoli S (2018) Growth and electronic structure of 2D hexagonal nanosheets on a corrugated rectangular substrate. Nanotechnology 29:485201.

    Article  CAS  Google Scholar 

  17. He XY, Liu F, Lin FT, Shi WZ (2019) Investigation of terahertz all-dielectric metamaterials. Opt Express 27:13831–13844.

    Article  Google Scholar 

  18. Shi CYY, He XY, Peng J, Xiao GN, Liu F, Lin FT, Zhang H (2019) Tunable terahertz hybrid graphene-metal patterns metamaterials. Opt Laser Technol 114:28–34.

    Article  CAS  Google Scholar 

  19. He XY, Liu F, Lin FT, Xiao GN, Shi WZ (2019) Tunable MoS2 modified hybrid surface plasmon waveguides. Nanotechnology 30:125201.

    Article  CAS  Google Scholar 

  20. Wu QP, Liu ZF, Chen AX, Xiao XB, Liu ZM (2014) Generation of full polarization in ferromagnetic graphene with spin energy gap. Appl Phys Lett 105:252402.

    Article  CAS  Google Scholar 

  21. Wu QP, Liu ZF, Chen AX, Xiao XB, Liu ZM (2016) Full valley and spin polarizations in strained graphene with Rashba spin orbit coupling and magnetic barrier. Sci Rep 6:21590.

    Article  CAS  Google Scholar 

  22. Li Y, Zhu HB, Wang GQ, Peng YZ, Xu JR, Qian ZH, Bai R, Zhou GH, Yesilyurt C, Siu ZB, Jalil MBA (2018) Strain-controlled valley and spin separation in silicene heterojunctions. Phys Rev B 97:085427.

    Article  CAS  Google Scholar 

  23. Li Y, Jiang WQ, Ding GY, Peng YZ, Wen ZC, Wang GQ, Bai R, Qian ZH, Xiao XB, Zhou GH (2019) Electrically tunable valley-dependent transport in strained silicene constrictions. J Appl Phys 125:244304.

    Article  CAS  Google Scholar 

  24. Gui Y, Tang C, Zhou Q, Xu L, Zhao Z, Zhang X (2018) The sensing mechanism of N-doped SWCNTs toward SF6 decomposition products: a first-principle study. Appl Surf Sci 440:846–852.

    Article  CAS  Google Scholar 

  25. Wang Y, Gui Y, Ji C, Tang C, Zhou Q, Li J, Zhang X (2018) Adsorption of SF6 decomposition components on Pt3 -TiO2 (1 0 1) surface: a DFT study. Appl Surf Sci 459:242–248.

    Article  CAS  Google Scholar 

  26. Wei H, Gui Y, Kang J, Wang W, Tang C (2018) A DFT study on the adsorption of H2S and SO2 on Ni doped MoS2 monolayer. Nanomaterials 8:646.

    Article  CAS  Google Scholar 

  27. Liu D, Gui Y, Ji C, Tang C, Zhou Q, Li J, Zhang X (2019) Adsorption of SF6 decomposition components over Pd (1 1 1): a density functional theory study. Appl Surf Sci 465:172–179.

    Article  CAS  Google Scholar 

  28. Huang W, Gan L, Li H, Ma Y, Zhai T (2016) 2D layered group IIIA metal chalcogenides: synthesis, properties and applications in electronics and optoelectronics. CrystEngComm 18:3968–3984.

    Article  CAS  Google Scholar 

  29. Zólyomi V, Drummond ND, Fal’ko VI (2013) Band structure and optical transitions in atomic layers of hexagonal gallium chalcogenides. Phys Rev B 87:19540.

    Article  CAS  Google Scholar 

  30. Zólyomi V, Drummond ND, Fal’ko VI (2014) Electrons and phonons in single layers of hexagonal indium chalcogenides from ab initio calculations. Phys Rev B 89:205416.

    Article  CAS  Google Scholar 

  31. Do DT, Mahanti SD, Lai CW (2015) Spin splitting in 2D monochalcogenide semiconductors. Sci Rep 5:17044.

    Article  CAS  Google Scholar 

  32. Sun HZ, Wang Z, Wang Y (2017) Band alignment of two-dimensional metal monochalcogenides MXs (M=Ga,In; X=S,Se,Te). AIP Adv 7:095120.

    Article  CAS  Google Scholar 

  33. Ayadi T, Debbichi L, Said M, Lebègue S (2017) An ab initio study of the electronic structure of indium and gallium chalcogenide bilayers, journal=J Chem Phys 147:114701.

  34. Magorrian SJ, Zólyomi V, Fal’ko VI (2016) Electronic and optical properties of two-dimensional InSe from a DFT-parametrized tight-binding model. Phys Rev B 94:245431.

    Article  Google Scholar 

  35. Magorrian SJ, Zólyomi V, Fal’ko VI (2017) Spin-orbit coupling, optical transitions, and spin pumping in monolayer and few-layer InSe. Phys Rev B 96:195428.

    Article  Google Scholar 

  36. Magorrian SJ, Ceferino A, Zólyomi V, Fal’ko VI (2018) Hybrid k · p tight-binding model for intersubband optics in atomically thin InSe films. Phys Rev B 97:165304.

    Article  CAS  Google Scholar 

  37. Xiao KJ, Carvalho A, Neto Castro AH (2017) Defects and oxidation resilience in InSe. Phys Rev B 96:054112.

    Article  Google Scholar 

  38. Sun YN, Wang XF, Zhai MX, Yao AL (2017) Tunable magnetism and metallicity in As-doped InSe quadruple layers. J Phys D Appl Phys 50:215003.

    Article  CAS  Google Scholar 

  39. Fu Z, Yang B, Zhang N, Ma D, Yang Z (2017) First-principles study of adsorption-induced magnetic properties of InSe monolayers. Appl Surf Sci 436:419–423.

    Article  CAS  Google Scholar 

  40. Zhou M, Zhang D, Yu S, Huang Z, Chen Y, Yang W, Chang K (2019) Spin-charge conversion in InSe bilayers. Phys Rev B 99:155402.

    Article  CAS  Google Scholar 

  41. Yao AL, Wang XF, Liu YS, Sun YN (2018) Electronic structure and I-V characteristics of InSe nanoribbons. Nanoscale Res Lett 13:107.

    Article  CAS  Google Scholar 

  42. Mudd GW, Svatek SA, Ren T, Patanè A, Makarovsky O, Eaves L, Beton PH, Kovalyuk ZD, Lashkarev GV, Kudrynskyi ZR, Dmitriev AI (2013) Tuning the bandgap of exfoliated InSe nanosheets by quantum confinement. Adv Mater 25:5714–5718.

    Article  CAS  Google Scholar 

  43. Mudd GW, Patanè A, Kudrynskyi ZR, Fay MW, Makarovsky O, Eaves L, Kovalyuk ZD, Zólyomi V, Fal’ko V (2014) Quantum confined acceptors and donors in InSe nanosheets. Appl Phys Lett 105:221909.

    Article  CAS  Google Scholar 

  44. Tamalampudi SR, Lu YY, Kumar UR, Sankar R, Liao CD, Moorthy BK, Cheng CH, Chou FC, Chen YT (2014) High performance and bendable few-layered InSe photodetectors with broad spectral response. Nano Lett 14:2800–2806.

    Article  CAS  Google Scholar 

  45. Lei S, Ge L, Najmaei S, George A, Kappera R, Lou J, Chhowalla M, Yamaguchi H, Gupta G, Vajtai R, Mohite AD, Ajayan PM (2014) Evolution of the electronic band structure and efficient photo-detection in atomic layers of InSe. ACS Nano 8:1263–1272.

    Article  CAS  Google Scholar 

  46. Late DJ, Liu B, Luo J, Yan A, Matte HSSR, Grayson M, Rao CNR, Dravid VP (2012) GaS and GaSe ultrathin layer transistors. Adv Mater 24:3549–3554.

    Article  CAS  Google Scholar 

  47. Bandurin DA, Tyurnina AV, Yu GL, Mishchenko A, Zólyomi V, Morozov SV, Kumar RK, Gorbachev RV, Kudrynskyi ZR, Pezzini S, Kovalyuk ZD, Zeitler U, Novoselov KS, Patanè A, Eaves L, Grigorieva IV, Fal’ko VI, Geim AK, Cao Y (2017) High electron mobility, quantum Hall effect and anomalous optical response in atomically thin InSe. Nat Nanotechnol 12:223–227.

    Article  CAS  Google Scholar 

  48. Song C, Fan FR, Xuan N, Huang S, Zhang G, Wang C, Sun Z, Wu H, Yan H (2018) Largely tunable band structures of few-layer InSe by uniaxial strain. ACS Appl Mater Interfaces 10:3994–4000.

    Article  CAS  Google Scholar 

  49. Zhou M, Zhang R, Sun J, Lou WK, Zhang D, Yang W, Chang K (2017) Multiband k·p theory of monolayer XSe (X=In, Ga). Phys Rev B 96:155430.

    Article  Google Scholar 

  50. Xia CX, Du J, Huang XW, Xiao WB, Xiong WQ, Wang TX, Wei ZM, Jia Y, Shi JJ, Li JB (2018) Two-dimensional n-InSe/p-GeSe(SnS) van der Waals heterojunctions: high carrier mobility and broadband performance. Phys Rev B 97:115416.

    Article  CAS  Google Scholar 

  51. Yang X, Sa B, Zhan H, Sun Z (2017) The electric field modulated data storage in bilayer InSe. J Mater Chem C 5:12228–12234.

    Article  CAS  Google Scholar 

  52. Drummond ND, Zólyomi V, Fal’ko VI (2012) Electrically tunable band gap in silicene. Phys Rev B 85:075423.

    Article  CAS  Google Scholar 

  53. Ni ZY, Liu QH, Tang KC, Zheng JX, Zhou J, Qin R, Gao ZX, Yu DP, Lu J (2012) Tunable bandgap in silicene and germanene. Nano Lett 12:113–118.

    Article  CAS  Google Scholar 

  54. Ramasubramaniam A, Naveh D, Towe E (2011) Tunable band gaps in bilayer transition-metal dichalcogenides. Phys Rev B 84:205325.

    Article  CAS  Google Scholar 

  55. Huang D, Kaxiras E (2016) Electric field tuning of band offsets in transition metal dichalcogenides. Phys Rev B 94(R):241303.

    Article  Google Scholar 

  56. Li Y, Yang SX, Li JB (2014) Modulation of the electronic properties of ultrathin black phosphorus by strain and electrical field. J Phys Chen C 118:23970–23976.

    Article  CAS  Google Scholar 

  57. Slater JC, Koster GF (1954) Simplified LCAO method for the periodic potential problem. Phys Rev 94:1498.

    Article  CAS  Google Scholar 

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The authors thank Xiao-Ying Zhou for useful discussions. This work was supported by the NSFC (grant nos. 11664019, 11264019, 11764013, and 11864012) and the Science Foundation for Distinguished Young Scholars in Jiangxi Province of China (grant no. 20162BCB23032).

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XBX conceived the research work. QY, ZFL, and QPW carried out the computation. XBX, YL, and GPA analyzed the results and wrote the manuscript. All the authors read and approved the final manuscript.

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Correspondence to Xian-Bo Xiao.

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Xiao, XB., Ye, Q., Liu, ZF. et al. Electric Field Controlled Indirect-Direct-Indirect Band Gap Transition in Monolayer InSe. Nanoscale Res Lett 14, 322 (2019).

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  • Monolayer InSe
  • Electric field
  • Indirect-direct-indirect band gap transition