Startseite First-principles study of structures, elastic and optical properties of single-layer metal iodides under strain
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First-principles study of structures, elastic and optical properties of single-layer metal iodides under strain

  • Ran Ran , Cui-E Hu EMAIL logo , Yan Cheng EMAIL logo , Xiang-Rong Chen und Guang-Fu Ji
Veröffentlicht/Copyright: 7. September 2020
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Zeitschrift für Naturforschung A
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Abstract

The structure, elastic, electronic and optical properties of two-dimensional (2D) MI2 (M = Pb, Ge, Cd) under strain are systematically studied by the first-principles method. It is proved that the monolayer structure of 2D-MI2 is stable by phonon spectra. Moreover, the large ideal strain strength (40%), the large range of strain and the elastic constants of far smaller than other 2D materials indicate that the single-layer PbI2 and GeI2 possess excellent ductility and flexibility. By applying appropriate strain to the structure of 2D-MI2, the band gaps of single-layer MI2 can be effectively controlled (PbI2: 1.04 ∼ 3.03 eV, GeI2: 0.43 ∼ 2.99 eV and CdI2: 0.54 ∼ 3.36 eV). It is found that the wavelength range of light absorbed by these three metal iodides is 82–621 nm, so 2D-MI2 has great absorption intensity for ultraviolet light in a large wavelength range, and the strain of structure can effectively regulate the optical parameters.

Keywords: elastic properties; first-principles; metal iodide (MI2); optical properties

1 Introduction

For two-dimensional (2D) materials, the movement of electrons is limited in the plane, so they have some physical and chemical properties that block materials do not have. These unique properties make 2D materials develop rapidly. With the development of graphene, boron nitride (h-BN), transition metal sulfides (TMDs), monoene, transition metal carbides, transition metal oxides and other 2D materials, some drawbacks have gradually emerged [1], [2], [3], [4], [5], [6], [7], [8], [9], such as the band gap loss of graphene, the low carrier concentration of TMDs and so on. In order to meet the specific needs of production, some new 2D materials, such as metal iodide, have been explored [10], [11], [12], [13].

For single-layer metal iodides (MI2), the most representative materials are 2D lead iodide (2D-PbI2), germanium iodide (2D-GeI2) and cadmium iodide (2D-CdI2). There are many kinds of phase, among which the most stable is 2H phase, which is a sandwich structure stacked in the ABC model [14]. The layer of MI2 where the metal atom M is located is sandwiched between two I layers, and there is a van der Waals force between the layers [15]. Stable structure, band gap comparable to third-generation semiconductors, excellent physical and chemical properties are the characteristics of metal iodides. Single-layer PbI2 can be used to make radiation-proof devices because of its high atomic number and density [16], [17], [18]; single-layer GeI2 has a unique light absorption range and can be used to prepare photoelectric converter devices [19], [20]; single-layer CdI2 is a good material for making heterojunction and heterojunction arrays [21], [22]. According to previous reports, the single-layer metal iodide MI2 (M = Pb, Ge, Cd) has been successfully stripped and grown in the laboratory.

There are many methods to obtain single-layer MI2, including mechanical peeling, solution method and physical vapour deposition [23]. Most single-layer structures can be obtained by mechanical peeling, but the production efficiency of this method is relatively low. The sample prepared by this method has small transverse size and poor thickness control, so it is impossible to prepare a large area of single-layer structure. Impurities are easily introduced in the preparation of monolayer structures using solution method, thus affecting the properties. In contrast, the yield of vapour deposition method is higher and the quality of single-layer structure produced is better [24]. Monolayer structure of metal iodides has been prepared on a large scale in experiments, but the number of layers, phases, densities of defects and edge morphology of samples is uncertain due to limitations of experimental equipment. The physical properties of 2D metal iodides cannot be accurately studied. Therefore, the theoretical calculation of single-layer metal iodides is very necessary. In 2015, Zhou et al. [25] explored the structure, stability, electronic and optical properties of single-layer PbI2 and showed some significant results. In our previous work, the single-layer structure, electronic, elastic and thermal transport properties of PbI2 have been explored and some important results have been obtained [26]. However, previous studies have been limited to PbI2, which is not representative and does not take into account the properties under strain.

In this work, the structure, elastic and optical properties of single-layer metal iodide MI2 (M = Pb, Ge, Cd) under strain are first investigated by using the first-principles method. Firstly, it is proved that the monolayer structure of MI2 is stable by the phonon spectra. Then, the stress-strain curve, elastic constant, Young’s modulus, layer modulus and Poisson’s ratio of 2D-MI2 are obtained. Comparing with other 2D materials, it can be concluded that 2D-MI2 has excellent flexibility and ductility, as well as good ability to adjust strain. Considering the spin–orbit coupling (SOC)  effect, the band structure and optical parameters in the strain ranging from −20 to 20% were calculated by using the hybrid functional Heyd-Scuseria-Ernzerhof (HSE06). Our results show that the strain can effectively regulate the band gap and optical absorption properties of 2D-MI2.

2 Theoretical methods and calculation details

The Vienna Ab initio Simulation Package (VASP) based on first-principles is applied to all our calculations [27], [28], [29]. The Perdew-Burke-Ernzerhof method in the generalized gradient approximation model is used in the structural optimization and elastic calculations [30], [31]. For the calculation of electronic structure, we use the hybrid functional (HSE06) approximation and consider the influence of SOC [32], [33], [34]. After the convergence tests, the plane wave cut-off energies were set to 440, 500 and 400 eV for single-layer PbI2, GeI2 and CdI2, respectively. The forces on all relaxed atoms were less than 10−4 eV/Å, and the energy convergence criterion was 10−6 eV per unit cell. The k-point sampling grids in the Brillouin zone were chosen as 15 × 15 × 1 for geometry optimization and 21 × 21 × 1 for the calculation of elastic properties.

In general, the elastic constants of materials are derived from the relationship between stress and strain. The relationship between stress and strain can be expressed as follows [35]:

(1)σij=Cijklεkl

The metal iodide studied in this paper belongs to hexagonal system, so there are only six independent elastic constants.

(2)Chexal=(C11C12C13⋅C11C13⋅⋅C33C44C44C66)

In fact, there are only five independent elastic constants because for the hexagonal structure C66 is obtained from C11 and C12(C66=(C11−C12)/2) [36]. The above elastic theory can also be applied to 2D materials, but some details are different. Different from bulk material, the strain of 2D material is in plane. We know that bulk modulus represents the resistance of bulk materials to external forces. Similarly, the ability of 2D materials to resist external forces (in-plane) can be expressed by layer modulus γ [36], which is given as follows:

(3)γ=−A∂F∂A=A∂∂A(∂E∂A)=A∂2E∂A2

In addition, some other parameters which can be used to express the elastic properties of 2D materials can also be calculated by elastic constants. Young’s modulus Y and Poisson’s ratio v can be expressed as follows [36]:

(4)Y[10]2D=C11C12−C122C22
(5)Y[01]2D=C11C12−C122C11
(6)v[10]2D=C12C22
(7)v[01]2D=C12C11

The optical properties of materials are represented by some optical parameters, such as reflectivity, refractive index, absorption coefficient, energy loss and so on. The interaction between light and the electrons and atoms it passes through produces light absorption, which reveals the optical properties of materials and also contains the information of electronic structure. To explore the optical properties of materials, we need to calculate the complex dielectric function, which is expressed as follows [37]:

(8)ε(ω)=ε1(ω)+iε2(ω)

The imaginary part of the complex dielectric function ε2 can be obtained by the sum of occupied and unoccupied wave functions, as follows [37]:

(9)εαβ(ω)=4π2e2Ωlimq→01q2∑c,v,k2wkδ(εck−εvk−ε)×〈uck+eαq|uvk〉〈uck+eβq|uvk〉∗

Among them, ex(x=α,β,γ) represents the unit vector in Cartesian direction, uk represents the periodic part of the primitive cell of pseudo wave function, c and v are the conduction band and the valence band, respectively. The real part of the complex dielectric function ε1 can be deduced by Kramer-Kronig relation on the basis of the imaginary part, as follows [38]:

(10)ε1(ω)=1+2πP∫ω′ε2(ω′)ω′2−ω2dω′

Then, the reflectivity and absorption coefficient can be obtained by the following formula [38]:

(11)R(ω)=|ε1(ω)+jε2(ω)−1ε1(ω)+jε2(ω)+1|2
(12)α=2[−ε1(ω)±ε12(ω)+ε22(ω)]

3 Results and discussion

3.1 Structure and elasticity

The most stable phase of bulk MI2 (M = Pb, Ge, Cd) is the semiconducting 2H phase, whose unit cell consists of two iodide atoms and a metal atom. Figure 1a shows the bulk structure. Recleave surface was performed to achieve a monolayer crystal structure, that is, the bulk phase of MI2 was cleaved along (0 0 1) plane. After convergence test, we found that the vacuum layer of 8 Å is enough when eliminating the interaction between layers from Figure 1b. In addition, the lattice constants, bond lengths and bond angles of the optimized monolayer MI2 are shown in Table 1. Our previous work has proved the stability of single-layer PbI2 [26]. It can be seen in Figure 2 that all phonon modes in monolayer MI2 have positive frequency indicating the stability of this structure.

Figure 1: (Colour online) (a) Unit cell of 2H-metal iodide. Grey and red balls represent metal atoms and I atoms, respectively; (b) The energy dependences of the height of the vacuum layer.
Figure 1:

(Colour online) (a) Unit cell of 2H-metal iodide. Grey and red balls represent metal atoms and I atoms, respectively; (b) The energy dependences of the height of the vacuum layer.

Table 1:

The lattice constants, I–M bond lengths and M–I–M bond angles of monolayer MI2 (M = Pb, Ge, Cd).

Single-layer PbI2Single-layer GeI2Single-layer CdI2
Lattice constants (Å)4.664.304.34
I–M bond lengths (Å)3.233.043.04
M–I–M bond angles89.84°90.31°91.10°
Figure 2: (Colour online) The phonon spectra of (a) 2D-PbI2 [26], (b) 2D-GeI2 and (c) 2D-CdI2.
Figure 2:

(Colour online) The phonon spectra of (a) 2D-PbI2 [26], (b) 2D-GeI2 and (c) 2D-CdI2.

After the stable single-layer structure is obtained, the strain-stress curve is calculated to explore the ability of 2D-MI2 to resist external strain. Figure 3 depicts the stress per unit length (N⋅m−1) as a function of strain (%). It can be seen from Figure 3 that the strain-stress curves of single-layer PbI2 and GeI2 are relatively similar. The maximum stresses are 10.6 and 12.11 N⋅m−1, respectively, and the corresponding ideal strains are all around 40%, indicating that their structures can stretch to 40% of the original. The stress-strain curve of CdI2 is different from those of PbI2 and GeI2. Its maximum stress is 18.81 N⋅m−1, and the corresponding ideal strain is 24%, which is obviously lower than the former two. Compared with the ideal strain strength of other 2D materials, such as MoS2 (20%) and graphene (24%) [39], [40], [41], [42], the ideal strain strength of single-layer PbI2 and GeI2 as high as 40% makes them show excellent strain capacity and flexibility. On the one hand, this is due to their fold structures. For example, the surfaces of graphene and h-BN are flat, and there are no folds in z direction; some 2D materials have fold structures but close to plane structures. These 2D materials have no space to coordinate the strain when they are stressed, either they have high hardness and are not easy to deform or they have general hardness and their structures are easily damaged. What is more is that this phenomenon can also be explained by elasticity theory.

Figure 3: (Colour online) The stress-strain curves for single-layer MI2 (M = Pb, Ge, Cd).
Figure 3:

(Colour online) The stress-strain curves for single-layer MI2 (M = Pb, Ge, Cd).

As it is known, another parameter that can indicate the resistance to compression is the elastic constant. The value of elastic constant (C11), Young’s modulus (Y2D) and shear modulus (G2D) is listed in Table 2. Compared with other 2D materials, the Young’s modulus of 2D-MI2 (M = Pb, Ge, Cd) is far lower than that of h-BN (289 N⋅m−1), 2D TMDs like MoS2 (199 N⋅m−1), WS2 (272 N⋅m−1) and TcS2 (93 N⋅m−1) but similar to that of phosphorene (armchair) (44 N⋅m−1) [36], [43], [44], [45]. Therefore, the 2D-MI2, which is more flexible, also has potential applications in flexible optoelectronics. Nevertheless, although both single-layer PbI2 and GeI2 are metal iodides, the Young’s modulus of PbI2 is nearly 40% smaller than that of GeI2. To further explore why single-layer PbI2 is more flexible, the Bader charge analysis has be done. The I–M bond is a transitional form between the covalent bond and ionic bond. In general, the stronger the ionicity, the weaker the resistance of the chemical bond to an external force and therefore the smaller the elastic constant. Each I atom gets 0.45 e transferred from the Pb atom when Pb atom is bonding with I atom, but instead, each I atom gets 0.36 e transferred from the Ge atom when Ge atom is bonding with I atom. So single-layer PbI2 is more ionic, which leads to a smaller Young’s modulus. What is more is that it can be seen from Table 2 that the ratio of layer modulus and sheer modulus (γ/G2D) of GeI2 is less than 1.75, indicating its brittleness [46], whereas that of MI2 (M = Pb, Cd) is greater than 1.75, showing good ductility.

Table 2:

Calculated elastic constants (C11 and C12 in N⋅m−1), sheer modulus (G2D in N⋅m−1), Young’s modulus (Y2D in N⋅m−1) and layer modulus (γ in N⋅m−1).

C11C12G2DY2Dγγ/G2D
Monolayer PbI214.693.955.3013.639.351.80
Monolayer GeI223.765.369.2022.5514.561.58
Monolayer CdI222.858.267.2919.8615.562.13

By summarizing the above calculation, it can be concluded that 2D-MI2 has low hardness, excellent flexibility and the ability to coordinate strain, so it can be utilized to synthesize some semiconductor devices that require extensive strain.

3.2 Electronic structure

Materials will be deformed due to the action of external forces in the process of practical application, so it is meaningful to explore the change of electronic structure in the case of strain. The electronic structure of monolayer PbI2 and GeI2 under strain is systematically studied in this section. Firstly, the strain structure is optimized, and then, the electronic structure is calculated. All kinds of lattice parameters along lattice vector of a/b direction are chosen. A biaxial strain is defined as ε = ∆a/a0, where ∆a is the difference between the frozen and optimized lattice constant along lattice vector a/b direction and a0 is the equilibrium lattice constant.

The electronic structure of single-layer MI2 (M = Pb, Ge, Cd) under different strain is shown in Figure 4a–c. In the calculation process, the top of valence band is Fermi surface. It can be clearly seen that the conduction band of single-layer PbI2 decreases continuously and the valence band increases continuously with the stretching of the structure. In this process, the position of the bottom of the conduction band is always at the Γ-point, but the position of the top of the valence band changes from the position between Γ-point and M-point in equilibrium state to the position of M-point (the ε is more than 15%). For single-layer GeI2, the variation trend of its valence band and conduction band is basically the same as that of single-layer PbI2. The only difference is the position of the conduction band bottom when ε = −10%. The change of electronic structure of CdI2 under strain is different from that of PbI2 and GeI2. With the compression of the structure, the bottom of the conduction band decreases gradually. In this process, the position of the bottom of the conduction band and the top of the valence band has not changed. When the structure is stretched, the conduction band rises slowly to ε = 5% and then drops sharply. At the same time, the position of the bottom of the conduction band changes from M-point (ε = 0 ∼ 5%) to Γ-point (ε > 10%), and the position of the top of the valence band changes from Γ-point to the position between Γ-point and M-point. The band gap of single-layer MI2 is always indirect, no matter in tension or compression.

Figure 4: (Colour online) The electronic structure of single-layer MI2 (M = Pb, Ge, Cd) under different strains.
Figure 4:

(Colour online) The electronic structure of single-layer MI2 (M = Pb, Ge, Cd) under different strains.

The curve of band gap calculated by the HSE06 + SOC method with strain is shown in Figure 5. It can be seen that the band gaps of single-layer PbI2 and single-layer GeI2 continue to increase with the compression strain of the structures. When they increase to 3.03 and 2.89 eV (compression strain to −10%), respectively, the band gap drops sharply. As the tensile strain of the structure goes on, the band gap decreases. The curve of single-layer CdI2 is different from the former two. With the compression of the structure, the band gap of the single-layer CdI2 decreases gradually. With the structure stretching, the band gap increases slowly at first and suddenly drops sharply when it increases to 3.46 eV (tensile strain is around 5%).

Figure 5: (Colour online) The curves of band gaps of single-layer MI2 (M = Pb, Ge, Cd) calculated by the HSE06 + SOC method.
Figure 5:

(Colour online) The curves of band gaps of single-layer MI2 (M = Pb, Ge, Cd) calculated by the HSE06 + SOC method.

Through the above analysis, we can conclude that compared with compression strain, stretching can regulate the band gap of single-layer CdI2 more efficiently, while single-layer PbI2 and GeI2 are the opposite, because compression has a more significant role in regulating the band gap of both. In addition, in the process of strain changing from −20 to 20%, the effect of strain on the band gap of single-layer MI2 is very obvious (CdI2: 0.54 ∼ 3.46 eV, PbI2: 1.04 ∼ 3.03 eV, GeI2: 0.43 ∼ 2.99 eV). Therefore, the structural strain can effectively control the band gap of single-layer MI2. In the application, the appropriate strain can be applied to the single-layer MI2 to obtain the desired band gap so that it can be used in the photoelectric devices that need specific band gap.

3.3 Optical properties

Now, we calculate the optical properties of the monolayer metal iodide MI2 (M = Pb, Ge, Cd). We first calculate the real part ε1 and the imaginary part ε2 of the dielectric function. Then, we use Eqs. (8) and (9) to get the absorption curve and reflection curve. The real part ε1 and imaginary part ε2 of dielectric function, absorption coefficient, reflectivity of the single-layer MI2 as a function of photon energy are shown in Figure 6. It can be seen from Figure 6a and b that the real part and imaginary part of 2D-MI2 have a maximum value in the low energy region (infrared, visible and near ultraviolet). Clearly, we can see from Figure 6c that PbI2 and GeI2 monolayers have two similar absorption bands. The absorption curve of single-layer CdI2 is different from that of single-layer PbI2 and GeI2, which is due to the fact that Cd atom is not in the same group as Pb and Ge atoms. For single-layer PbI2, GeI2 and CdI2, the absorptions begin when the photon energy is zero, and the energies at the absorption edge are basically consistent with the corresponding band gaps. However, it can be found that the energy at the absorption edge is slightly lower than that in the band gap because the energy required for free carrier absorption is lower than that for the intrinsic absorption, and free carrier absorption occurs when the energy is lower than the forbidden band energy. With the increase of photon energy, the ε2 of MI2 rises rapidly in the visible light region and then the peak appears, indicating the absorption part of the visible light. The first peak appeared in the absorption spectrum of MI2 near 4 eV, and the light absorption of single-layer PbI2 and GeI2 reached 790,000 cm−1 and 750,000 cm−1, respectively, indicating that the single-layer PbI2 and GeI2 have a strong absorption capacity for ultraviolet light, while CdI2 has relatively weak absorption capacity for UV light. However, the absorption spectrum of single-layer MI2 in the range of UV light appears several peaks continuously, and the peak value is also very large. To sum up, we can conclude that the monolayer MI2 has a strong absorption capacity for ultraviolet light in a wide range of wavelengths. It can be seen from Figure 6d that the first peak appears in the visible light region for the reflection, and then many peaks appear in the ultraviolet light region, with the highest reflectivity around 50, 47 and 34% for single-layer PbI2, GeI2 and CdI2, respectively. In general, there are many peaks in the reflection curve of single-layer MI2, which are concentrated in the UV region, and the peak value is small, showing that the main reflective parts of the 2D-MI2 are visible light and ultraviolet light, but the reflectivity is not high. However, they have strong absorption capacity for UV light and can be used to make UV detection devices or anti-UV radiation devices.

Figure 6: (Colour online) (a) The real part ε1 and (b) imaginary part ε2 of dielectric function, (c) absorption coefficient, (d) reflectivity of single-layer MI2 (M = Pb, Ge, Cd) as a function of photon energy.
Figure 6:

(Colour online) (a) The real part ε1 and (b) imaginary part ε2 of dielectric function, (c) absorption coefficient, (d) reflectivity of single-layer MI2 (M = Pb, Ge, Cd) as a function of photon energy.

In view of the strong ability of 2D-MI2 to absorb UV light in the equilibrium state, the optical absorption properties of single-layer MI2 under strains are further calculated, and the results are shown in Figure 7. Among them, Figure 7 (a), (c) and (e) show the absorption spectra of singe-layer PbI2, GeI2 and CdI2 under compressions (ε = −20 ∼ 0%), respectively, while (b), (d) and (f) show the absorption spectra of the three under tensions (ε = 0 ∼ 20%). It can be seen that the light absorptions still start at 0 eV even though the structures are strained. When the structure is compressed, the absorption edge value moves to the left, while the absorption edge value moves to the right when the structure is stretched. This is due to the change of band gap. The absorption band of 2D-MI2 becomes narrower and lower with the tensile strain. On the other hand, the compressive strain can improve the optical absorption of 2D-MI2 expanding the range of absorbing ultraviolet. Therefore, the strain can effectively regulate the light absorption ability of single-layer MI2 so that it can be used in different photoelectric detection devices.

Figure 7: (Colour online) Absorption spectra of 2D-PbI2 (a, b), 2D-GeI2 (c, d) and 2D-CdI2 (e, f) under compressions (−20 ∼ 0%) and tensions (0 ∼ 20%).
Figure 7:

(Colour online) Absorption spectra of 2D-PbI2 (a, b), 2D-GeI2 (c, d) and 2D-CdI2 (e, f) under compressions (−20 ∼ 0%) and tensions (0 ∼ 20%).

4 Conclusions

The structural, elastic and optical properties of 2D metal iodide MI2 (M = Pb, Ge, Cd) in equilibrium and strain are systematically studied by using the first-principles method. It is proved that the monolayer structure of MI2 (M = Pb, Ge, Cd) is stable, and it presents a fold structure similar to “sandwich”. Because of this fold structure, single-layer PbI2 and GeI2 have large ideal strain strength (40%), indicating that they have a good strain capacity and excellent flexibility. In addition, the layer modulus γ, Young’s modulus Y2D, Poisson’s ratio v and shear modulus G2D are calculated. The results show that the layer modulus of single-layer MI2 is much smaller than other two-dimensional materials, showing the characteristics of small hardness and good flexibility. The excellent flexibility, ductility and strong ability of coordinating strain all make the 2D-MI2 be used to manufacture photoelectric devices with large strain range.

The effect of strain on the band of 2D-MI2 was investigated using the HSE06 + SOC method. It is found that the effect of structural compression on the band gap of single-layer PbI2 and GeI2 is more significant, but structural stretching can change the band gap of single-layer CdI2 more efficiently. From ε = −20% to ε = 20%, the change of the band gap of single-layer MI2 is very obvious (CdI2: 0.54 ∼ 3.46 eV; PbI2: 1.04 ∼ 3.03 eV; GeI2:0.43 ∼ 2.99 eV). Therefore, the strain can effectively control the band gap of single-layer MI2. By exploring the optical properties, it is found that the single-layer MI2 has strong absorption ability for ultraviolet light over a large wavelength range. Moreover, strain can not only effectively change the band gap but also effectively control the light absorption ability.

Corresponding authors: Cui-E Hu, College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing400047, China, E-mail: ; and Yan Cheng, College of Physics, Sichuan University, Chengdu610064, China, E-mail:

Funding source: NSAF Joint Fund

Funding source: Chinese Academy of Engineering Physics

Award Identifier / Grant number: U1830101

Funding source: Science Challenge Project

Award Identifier / Grant number: TZ2016001

Acknowledgments

This work was supported by the NSAF Joint Fund Jointly setup by the National Natural Science Foundation of China and the Chinese Academy of Engineering Physics (Grant No. U1830101) and the Science Challenge Project (Grant No. TZ2016001).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by the NSAF Joint Fund Jointly setup by the National Natural Science Foundation of China and the Chinese Academy of Engineering Physics (Grant No. U1830101) and the Science Challenge Project (Grant No. TZ2016001).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2020-06-13
Accepted: 2020-08-05
Published Online: 2020-09-07
Published in Print: 2020-10-25

© 2020 Ran Ran et al., published by De Gruyter, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. General
  3. Graphene in curved Snyder space
  4. Dynamical Systems & Nonlinear Phenomena
  5. Resonant interactions between the fundamental and higher harmonic of positron acoustic waves in quantum plasma
  6. Hydrodynamics
  7. Analysis of Carreau fluid flow by convectively heated disk with viscous dissipation effects
  8. Quantum Theory
  9. From quantum foundations to spontaneous quantum gravity – An overview of the new theory
  10. Variational principle for time-periodic quantum systems
  11. Solid State Physics & Materials Science
  12. Design and analysis of 6CB nematic liquid crystal–based rectangular patch antenna for S-band and C-band applications
  13. First-principles study of structures, elastic and optical properties of single-layer metal iodides under strain
  14. Influence of deposition time and annealing treatments on the properties of chemically deposited Sn2Sb2S5 thin films and photovoltaic behavior of Sn2Sb2S5-based solar cells
https://doi.org/10.1515/zna-2020-0157
Schlagwörter für diesen Artikel
elastic properties; first-principles; metal iodide (MI2); optical properties

Artikel in diesem Heft

  1. Frontmatter
  2. General
  3. Graphene in curved Snyder space
  4. Dynamical Systems & Nonlinear Phenomena
  5. Resonant interactions between the fundamental and higher harmonic of positron acoustic waves in quantum plasma
  6. Hydrodynamics
  7. Analysis of Carreau fluid flow by convectively heated disk with viscous dissipation effects
  8. Quantum Theory
  9. From quantum foundations to spontaneous quantum gravity – An overview of the new theory
  10. Variational principle for time-periodic quantum systems
  11. Solid State Physics & Materials Science
  12. Design and analysis of 6CB nematic liquid crystal–based rectangular patch antenna for S-band and C-band applications
  13. First-principles study of structures, elastic and optical properties of single-layer metal iodides under strain
  14. Influence of deposition time and annealing treatments on the properties of chemically deposited Sn2Sb2S5 thin films and photovoltaic behavior of Sn2Sb2S5-based solar cells
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