Thickness-Dependent Electronic and Optical Properties of Bernal

Article pubs.acs.org/JPCC

Thickness-Dependent Electronic and Optical Properties of Bernal-Stacked Few-Layer Germanane Huabing Shu, Yunhai Li, Shudong Wang, and Jinlan Wang* Department of Physics, Southeast University, Nanjing 211189, China ABSTRACT: Few-layer germanane has been successfully fabricated experimentally very recently and shown great potential applications in electronic and optoelectronic devices. In this work, we investigate thickness-dependent electronic and optical properties of Bernal-stacked few-layer germanane within the framework of many-body perturbation theory. Few-layer and bulk germanane are all direct band gap semiconductors, and the band gaps are tunable in a broad range. Strong excitonic effects are observed in few-layer germanane, leading to a distinct red shift in optical absorption spectra, and the exciton binding energy in monolayer germanane can be as large as 750 meV, which is 15 times larger than that of bulk germanane. More importantly, the quasi-particle band gaps, optical gaps, and exciton binding energies rapidly decrease with the increase of the layer and follow a power law of A + B/Nβ (0 < β < 2) with the stacking layer.

1. INTRODUCTION van der Waals (vdW) layer materials have received extensive attention in the past few years because of their unique structures, rich physical and chemical properties, and possible applications in electronics,1−3 optoelectronics,4−6 energy conversion,7−9 and sensing.10,11 These materials generally exhibit thickness-dependent band gaps and even undergo indirect− direct band gap transition when exfoliated into a monolayer.12,13 For example, the band gaps of hydrogenated fewlayer graphene monotonically decrease with the increasing thickness.14 Bulk and few-layer MoS2 are all indirect band gap semiconductors, while monolayer MoS2 is a direct band gap semiconductor, and their band gap nonlinearly decreases with the increase of the stacking layer.15,16 Strong thicknessdependent photoluminescence was found in MoS2 as well; the photoluminescence intensity and luminescence quantum efficiency significantly increase with the decrease of the layer.12 Moreover, layer-controlled band gap and anisotropic excitons were observed in few-layer black phosphorus.13 Very recently, few-layer germanane, an analogue of graphane, has been successfully synthesized by topochemical deintercalation of CaGe2, and it can be mechanically exfoliated into a monolayer, which does not require a substrate to be stabilized.17,18 More importantly, monolayer germanane owns a direct band gap of ∼1.4 eV19,20 and a mobility of 18195 cm2/(V·s), which is five times higher than that of crystalline germanium.17 In addition, monolayer germanane exhibits strong excitonic effects, and the exciton binding energy can be as large as 600 meV,21 which make germanane hold great potential application in new optoelectronics devices. In this work, we systematically study the electronic and optical absorption properties of few-layer and bulk germanane by employing density functional theory (DFT) combined with © XXXX American Chemical Society

the many-body Green’s function (GW) and Bethe−Salpeter equation (BSE). The GW band gaps, optical gaps, and exciton binding energies of few-layer germanane are thickness-dependent and can be fitted as a power law as Eg(N) = 1.91 + 0.9/N0.42, Ee(N) = 1.81 + 0.25/N1.28, and Eb(N) = 0.04 + 0.71/N0.22, respectively, with the stacking layers. Moreover, strong excitonic effects are observed in few-layer germanane, and the exciton binding energy in the monolayer can be as large as 750 meV, which is 15 times larger than bulk germanane.

2. COMPUTATIONAL DETAILS A three-step procedure was employed to determine the electronic and optical properties of few-layer germanane. First, the mean-field wave functions, energies, and the matrix elements of the exchange−correlation operator were extracted via DFT calculation within the local density approximation (LDA) along with the Perdew and Wang functional22 as implemented in the QUANTUM ESPRESSO code.23 A plane-wave kinetic energy cutoff of 60 Ry was used together with norm-conserving pseudopotentials,24 and the Brillouin zone was sampled with a 18 × 18 × 1 (18 × 18 × 3) Monkhorst−Pack grid for few-layer (bulk) structures. For the geometry optimization, all atoms were fully relaxed until the total energy was converged within 10−4 eV, and forces acting on each atom were smaller than 0.01 eV/Å. For the few-layer structures, a vacuum region of 20 Å, along the direction perpendicular to the sheet, was introduced to eliminate the mirror effect between neighboring images. Then, the quasi-particle energies EQP nk within the GW approximation25−27 were calculated using the following equation Received: April 17, 2015

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DOI: 10.1021/acs.jpcc.5b03679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. (a) Structure of the optimized bulk germanane. Magenta and red spheres refer to germanium and hydrogen atoms, respectively. (b,c) Top views of the optimized monolayer and few-layer germanane. (d−i) Band structures of bulk and 1L−5L germanane from LDA (black solid lines) and G0W0 (red solid lines) calculations. Only the uppermost valence band and lowest conduction band of G0W0 results are plotted for clarity. QP KS KS KS LDA Enk = εnk + Znk(εnk )⟨ψnk|Σ(εnk ) − VXC |ψnk ⟩

band gap with respect to the number of empty bands, the size of the dielectric matrix, and the Monkhorst−Pack grid were carefully examined, and a convergence within 0.06 eV was assured. Finally, the exciton wave functions and excitation energies were obtained via the solution of BSE30−32

(1)

LDA where εKS are the mean-field energies, wave nk , ψnk, and Vxc functions, and exchange−correlation operator obtained in the aforementioned step 1, Znk is the renormalization factor, and Σ is the frequency-dependent self-energy operator, which was the direct product of the Green’s function G and screened Coulomb interaction W in coordinate−time space. In actual calculations, both G and W were built from mean-field wave functions and energies and kept fixed, that is, single-shot G0W0 calculations were performed. The plasmon-pole model28,29 proposed by Godby and Needs were employed to treat the frequency dependence of W0. The convergence of the quasi-particle

qp qp S S (Eck − Evk )A vck + Σ k ′ v ′ c ′⟨vck|K d + 2K x|v′c′k′⟩A vS′ c ′ k ′ = ΩSA vck (2)

qp where Eqp ck and Evk are the quasi-particle energies from the second step, ⟨vck| and |v′c′k′⟩ refer to the quasi-electron and quasi-hole states, ASvck and ΩS are the coefficients of exciton wave function and excitation energies, Kd is the attractive direct term of the

B


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3. RESULTS AND DISCUSSION The bulk germanane has a lamellar structure with Bernal stacking (Figure 1a), where each germanane layer consists of two Ge atoms terminated with hydrogen atoms at either of the two sides and the individual layers are weakly coupled to the adjacent layers via vdW interaction.17 The computed equilibrium lattice constant is 3.964 Å, in good agreement with the measured value of 3.980 Å.35 When the bulk germanane is exfoliated into few layers, both the interlayer spacing (Δc) and the buckling height (Δ) decrease monotonously from the bilayer to bulk germanane from 4.581 to 4.50 and 0.721 to 0.709 Å, respectively, while the intralayer Ge−Ge bond lengths increase as the increase of the layer (Table 1). The structural stability of few-layer germanane is evaluated by formation energies according to the formula Ef = (ENL − NE1L)/N, where ENL and E1L are the total energy of N-layer (NL) and one-layer (1L) germanane in the same primitive cell, respectively, and N is the layer number. As is clear from Table 1, the formation energies of few-layer germanane are all negative and become more negative with the increase of the layer, indicating that the thicker the germanane sheet, the higher the stability. The electronic band structure of bulk germanane is presented in Figure 1d, which exhibits semiconducting nature. A direct band gap appears at the Γ point, and the self-energy correction enlarges the band gap from the DFT value of 1.06 to 1.86 eV, well reproducing the measured value of 1.70 eV.36 The topology of the band structures of 1L−5L germanane is very similar except that the direct band gap at the Γ point decreases with increasing the stacking layer (Figure 1e−i). This may stem from the fact that the conduction band minimum (CBM) and valence band maximum (VBM) of few-layer germanane have different thickness-dependent behaviors. CBM shows a relatively quick upshift, while VBM shifts down smoothly as the stacking layer decreases (Figure 2a), which eventually results in the increasing band gap in few-layer germanane. This is further reflected from the partial charge density, in which the change in CBM is prominent while that in VBM is insensitive to the thickness (Figure 2b). More interestingly, the thickness dependence of the band gaps in few-layer germanane can be fitted according to a power law with the formula Eg(N) = A + B/Nβ, where Eg(N) and A are the band gap of the NL germanane and bulk germanane, respectively, and N is the layer number. The fitting results are summarized in Table 2 and plotted in Figure 2c. The LDA band gaps of few-layer

Table 1. Structural Parameters and formation energies of Few-Layer and Bulk Germananea N

a (b)

1 2 3 4 5 bulk

3.938 3.949 3.954 3.956 3.957 3.964

Δc

Δ

dGe−Ge

dGe−H

Ef

4.581 4.568 4.563 4.548 4.500

0.727 0.721 0.718 0.716 0.715 0.709

2.387 2.391 2.393 2.394 2.395 2.396

1.5289 1.5299 1.5303 1.5308 1.5309 1.5312

−60.08 −80.60 −88.36 −96.42

Parameters: lattice constants a (b)/Å, interlayer spacing Δc/Å (defined as the distance between two adjacent germanane layers), buckling height Δ/Å, Ge−Ge (Ge−H) bond lengths dGe−Ge (dGe−H)/Å, and formation energy Ef/meV. a

electron−hole (e−h) interaction, and Kx is the repulsive exchange term; v, c, and k are the indices for valence and conduction states and the wave vector, respectively. The Bethe−Salpeter Hamiltonian was taken into account by using Tann−Damcoff approximation,33 in which the coupling term between the resonant and antiresonant terms of the Hamiltonian is neglected. Five valence and 10 conduction bands were included to build the e−h interaction kernel, and a denser 36 × 36 × 1 k-grid was used to converge these calculations. From the solution of eq 2, the macroscopic dielectric function can be built as 8π εM(ω) = 1 − lim 2 ∑ ∑ ρn*′ nk (q , G)· q → 0 |q| ·Ω· Nq nn ′ k mm ′ k ρm ′ mk (q , G′) ×

∑ λ

A nλ′ nk (A mλ ′ mk ′)* ω − Eλ

(3)

where Ω is the volume of unit cell, n (m) and n′ (m′) are the band indices of the conduction and valence bands in building the BSE kernel, k and k′ are k-points on a uniform mesh grid in the first Brillouin zone, ρm′mk′(q,G) = ⟨m′k′|ei(q+G)·r|mk′ − q⟩ is a convolution in reciprocal space, and Aλn′nk = ⟨n′nk|λ⟩ are the eigenvectors of Hamiltonian, respectively. In the G0W0 and BSE calculations, for few-layer structures, a box-shaped screened coulomb interaction was truncated after 36 au along the direction perpendicular to the surface to avoid spurious interaction between periodic images. The G0W0 and BSE calculations were performed by the YAMBO code.34

Figure 2. (a) VBM and CBM with respect to the vacuum level in few-layer and bulk germanane and (b) the partial charge density of VBM and CBM. (c) The LDA, G0W0 band gaps, and optical gap (i.e., first optical absorption peak) of few-layer and bulk germanane (solid). The power law fitting curves are presented by dashed lines. Hollow circles, squares, and triangles correspond to the fitting values of bulk germanane from the scaling laws. C


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quantity “Im εM” is the imaginary part of the macroscopic dielectric function εM from eq 3. Compared to the independent particle pictures, the optical absorption spectrum of few-layer germanane including the e−h interaction witnesses an evident weight redistribution of the oscillator strength and shows a significant global red shift (Figure 3a−e), while in the bulk limit of germanane, the optical absorption spectrum is nearly unchanged in the scope of visible light by the inclusion of e−h interactions (Figure 3f). In another word, excitonic effects reshape the optical absorption spectrum totally, and the main optical features are dominated by excitonic states in few-layer germanane (Figure 3a−e). This suggests that the excitonic effects are significant and play decisive roles in the optical absorption spectrum, whereas those in bulk are rather weak. The first absorption peak A1 of monolayer germanane is observed at 2.06 eV (Figure 3a), corresponding to a strongly bound excitonic state, and the exciton binding energy defined as the difference between the excitation energy and the QP energy difference is as large as 750 meV, that is, 15 times larger than bulk germanane (50 meV). The reduced dimensionality and weakened electronic screening are primary factors for fostering

Table 2. Fitted Parameters for Band Gaps Eg‑LDA (Eg‑GW), the Position of the First Optical Absorption Peak (Ee), and the Binding Energy (Eb) of the First Bound Exciton in FewLayer Germanane According to the Formula E(N) = A + B/N β (E = Eg‑LDA, Eg‑GW, Ee, or Eb) A (eV) B (eV) β

Eg‑LDA

Eg‑GW

Ee

Eb

1.08 0.30 1.06

1.91 0.90 0.42

1.81 0.25 1.28

0.04 0.71 0.22

germanane satisfy an inverse relationship with the thickness of layers as Eg‑LDA(N) = 1.08 + 0.3/N1.06. The G0W0 band gaps also follow an inversion relationship but in the form of 1.91 + 0.9/N0.42 with the stacking layer. Moreover, from the above two formulas, we can derive the LDA and G0W0 band gaps of the bulk limit, which are 1.08 and 1.91 eV, respectively, which match well with the calculation values of bulk germanane (Figure 2c). The optical absorption spectra of 1L−5L and bulk germanane are calculated with the dielectric polarization vector being applied along the x direction (Figure 1b). The plotted

Figure 3. Optical absorption spectra of 1L−5L (a−e) and bulk germanane (f) for the incident light polarized along the x direction with (red line) and without (black line) inclusion of the e−h interaction, that is, G0W0 + BSE and G0W0 + RPA, respectively. A Lorentz broadening of 0.05 eV was used in these plots. A1 denotes the first absorption peak. The vertical black dash−dot lines indicate the energies of the lowest direct QP gap. D


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Figure 4. Top view of the square of the electron wave functions of the first bound excitons for the monolayer (a), trilayer (b), five-layer (c), and bulk germanane (d), respectively, where black dots indicate the hole positions. For clarity, the atoms are shown in gray.

such enhanced excitonic effects. The absorption edge (at about 2.31 eV) of monolayer germanane has a significant blue shift of about 1.11 eV with respect to that of the bulk germanane, which is closely correlated with the thickness-dependent band gaps. Furthermore, the position of the first optical absorption peak, that is, the optical gap of few-layer germanane, also exhibits a power law with the number of layers, Ee(N) = 1.81 + 0.25/N1.28 (Figure 2c). The inverse relationship between the optical gap and the layers can be understood in light of a one-dimensional infinite square well model, where En ∼ 1/a2, with En being the energy eigenvalue of the electron and a being the well width. The optical gap and the layer number can be analogous to the energy eigenvalue (En) and well width (a) of a one-dimensional infinite square well, respectively. Nevertheless, additional factors, such as electron−electron interaction and e−h interaction, must be taken into account in few-layer germanane, which modifies the inverse square law of En ∼ 1/a2 into a lower-index law of Ee ∼ 1/Nβ (β = 1.28). Moreover, from this formula, we can derive the position of the first optical absorption peak for the bulk limit as 1.81 eV, which is exactly the same as the BSE calculation value (Figure 2c). The binding energies (Eb) of the first bound exciton with the thickness can be expressed as a power law as well, Eb(N) = 0.04 + 0.71/N0.22 (Table 2). Our calculation estimates the upper limit of the exciton bound energy as 40 meV, which also agrees with the calculated value of 50 meV of bulk germanane. The small exciton bound energy of bulk germanane is similar to the case of bulk black phosphorus (30 meV)13 but much smaller than that of bulk hexagonal BN,37 which possesses a significant exciton binding energy of 700 meV. The weak excitonic effect in bulk germanane is attributed to its stronger interlayer interactions compared with that of bulk BN, similar to the case of bulk black phosphorus.13 To gain further insight into the optical properties, we explore the excitonic wave function related to the first peak of absorption spectra. The excitonic wave function can be written as |ΨS(re, rh)⟩ =

S *(rh) ψck(re)ψvk ∑ Acvk cvk

where re and rh are the real space electron and hole coordinates, respectively. The Ψ is the quasi-particle wave function. The coefficients AScvk are obtained by diagonalizing the Hamiltonian of BSE. To represent the six-coordinate function, the fixed hole position (black dot) is shown in Figure 4, and the modulus square of the real-space quasi-particle wave function (|ΨS[re,rh = (0,0,0)]|2) is projected onto the x−y plate. As clearly shown in Figure 4a−d, electron probability distributions |ΨS[re,rh]|2 of the first bound excitons in 1L, 3L, 5L, and bulk germanane show an obvious decaying nature, which is an indication of the binding between the quasi-electron and quasi-hole governed by the excitonic effects. Moreover, the electron and the hole are confined within a narrower area with the decrease of thickness, reflecting the enhanced binding between excited e−h pairs. The electron distribution range is found to be in close relation with the binding energy. This can be understood in light of a simple hydrogen atom model whose excitation energy E can be expressed as E = Eg − [1/(4πεε0)2]·[e2/2r], where Eg is the band gap, −(e2/2r) is the energy eigenvalue of the hydrogen atom ground state, and r is Bohr radius. Obviously, the larger the Bohr radius, the smaller the (Eg − E). As the electron distribution range and the binding energy can be analogous to the Bohr radius r and the (Eg − E) of the hydrogen atom model, we can easily draw a similar conclusion that the narrower area of the electron distribution, the larger the binding energy. Further, the electron and hole distributions also exhibit huge overlaps between excited e−h wave functions in the monolayer, leading to a reduced electronic screening that governs the strong binding of the exciton.

4. CONCLUSION In summary, we have performed DFT combined GW and BSE calculations to explore the electronic and optical properties of few-layer and bulk germanane. A close relationship between the thickness and the electronic and optical properties has been addressed. Few-layer and bulk germanane are all direct band gap semiconductors, and the band gap can be effectively engineered in the broad range of 2.81−1.86 eV by controlling the stacking layer. The e−h interactions induce a significant red shift of the whole absorption spectrum of few-layer germanane

(4) E


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and a strong exciton binding energy as large as 750 meV in monolayer germanane. More interestingly, the layer-dependent band gaps, optical gaps, and exciton binding energies can be formulated as certain scaling laws with the layer number as E(N) = A + B/Nβ with 0 < β < 2. The widely tunable band gap and the large exciton binding energy in few-layer germanane may provide additional possibilities in multijunction solar cells to effectively harvest solar energy over a large energy range.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +86-25-52090600-8210. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work is supported by the NBRP (2011CB302004), NSF (21173040, 21373045), and Jiangsu (BK20130016) and SRFDP (20130092110029) in China. The authors thank the computational resources at the SEU.

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REFERENCES

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Thickness-Dependent Electronic and Optical Properties of Bernal

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