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Cite this:DOI: 10.1039/c9cp03128f

Six new silicon phases with direct band gaps

Qun Wei,*a Wen Tong,a Bing Wei,a Meiguang Zhang*b and Xihong Peng *c

Six new silicon phases with direct band gaps were found through silicon atomic substitution of carbon

in the known carbon structures via high-throughput calculations. The six newly discovered Si phases are

in the space groups of Im %3m, C2/c, I4/mcm, I4/mmm, P21/m, and P4/mbm, respectively. Their crystal

structures, stabilities, mechanical properties, elastic anisotropy, and electronic and optical properties

were systematically studied using first-principles density functional theory calculations. All the new

phases were proved to be thermodynamically and mechanically stable at ambient pressure. The direct

band gap values in the range of 0.658–1.470 eV and the excellent optoelectronic properties of these six

Si allotropes suggest that they are promising photovoltaic materials compared to diamond silicon.

Introduction

With increasing concerns regarding energy resource depletion,interest has arisen in the development of environmentally cleanand safe processes that are capable of expanding energy infra-structure. Solar energy is one of the most abundant alternativeclean energies available for conversion. However, a full deploymentof solar cells requires tremendous amount of natural resourcesincluding raw materials. Earth-abundant materials with a highconversion rate are apparent target photovoltaic materials as activelayers for next-generation solar cells. The leading semiconductorsilicon draws great research attention due to its abundance,stability and wide usage. However, its indirect band gap with lowconversion efficiency limits its potential in photovoltaic applica-tions. Its direct band gap of 3.4 eV is too high and eliminates asignificant portion of low-energy photon absorption in the solarspectrum. Some metastable silicon allotropes such as T12-Si,1 R8,2

I%4,3 and lonsdaleite structures4 were reported in the literature.However, all these materials do not perform well in solar absorp-tion. Therefore, there is a need to search for silicon allotropes withan appropriate direct band gap for high efficiency solar energyabsorption.

Aligned with this demand, the material study of new siliconphases with a direct band gap has attracted research attentionin the past few years and many silicon allotropes have beenproposed. For instance, Botti et al.5 predicted numerous low-energy silicon allotropes with quasi-direct band gaps between

1.0 and 1.5 eV which show better optical properties thandiamond silicon. Similarly, Guo et al.6 reported a silicon allo-trope structure named h-Si6 (the space group: P63mmc) withdirect band gaps of 0.61 eV and prominent optical properties.Fan et al.3 found a stable silicon allotrope (Amm2) with a quasi-direct band gap of 0.742 eV. In addition, Lee et al.7 optimizedvarious crystal structures containing multiple (10 to 20) siliconatoms per unit cell via conformational space annealing andfound out that these silicon structures have a direct band gapand good photovoltaic properties using first principles calcula-tions. Oh et al.8 reported an ultra-stable pure silicon super-lattice structure with dipole-allowed direct band gaps less than1.12 eV. Fan et al.9 proposed two new silicon phases: Cm%32 andP21/m (different from the P21/m phase in this work) and foundthat the absorption spectra of the two phases significantlyoverlap with the solar spectrum and have direct band gapvalues of 1.85 and 0.83 eV, respectively. He et al.10 suggestedfive new silicon phases with direct or quasi-direct band gapsranging from 1.2 to 1.6 eV, which are suitable for applicationsin thin-film solar cells. Moreover, Bai et al.11 recently describeda new silicon allotrope in the monoclinic P2/m space group witha band gap of 1.51 eV and the first-principles calculationsindicate the stability of the structure.

In the present work, we continue the effort to search forsilicon allotropes with direct band gaps by replacing the carbonatoms with silicon in the known carbon phases. After high-throughput calculations, we obtained six new silicon allotropeswith direct band gaps in the Im%3m, C2/c, I4/mcm, I4/mmm,P21/m, and P4/mbm space groups, respectively. The stabilityand the mechanical, electronic, and optical properties of thesesix new silicon phases are systematically studied. These newphases have direct band gaps in the range of 0.658–1.470 eV,which is superior for solar spectrum absorption and demon-strates great potential as photovoltaic materials.

a School of Physics and Optoelectronic Engineering, Xidian University,

Xi’an 710071, P. R. China. E-mail: qunwei@xidian.edu.cnb College of Physics and Optoelectronic Technology, Baoji University of Arts and

Sciences, 721016 Baoji, P. R. China. E-mail: zhmgbj@163.comc College of Integrative Sciences and Arts, Arizona State University, Mesa,

Arizona 85212, USA. E-mail: xihong.peng@asu.edu

Received 3rd June 2019,Accepted 16th August 2019

DOI: 10.1039/c9cp03128f

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Calculation details

The first-principles density functional theory (DFT) calculations withthe projector augmented wave (PAW) potentials12 were performedusing the Vienna ab initio simulation package (VASP).13,14 ThePerdew–Burke–Ernzerhof (PBE) version of the generalized gradientapproximation (GGA)15,16 was utilized as the exchange and correla-tion functional. A plane wave basis set cut off energy of 400 eV wasused for expanding the wave functions. The convergence criteria forstructural optimizations were set to be no more than 10�5 eV in thetotal energy of the two adjacent steps and less than 0.05 eV Å�1 inthe force per atom. The Brillouin zone was represented by theMonkhorst–Pack k-point mesh 7 � 7 � 5 (P4/mbm), 5 � 5 � 5(P21/m), 4� 4� 4 (Im%3m), 9� 6� 11 (C2/c), 4� 4� 5 (I4/mcm), and5 � 5 � 4 (I4/mmm), respectively. The electronic band structureswere determined using both the hybrid HSE06 functional17,18 andPBE methods. The phonon spectra were calculated using thePHONOPY code.19

Results and discussion

We have predicted six new silicon allotropes using the abovementioned methods. A schematic representation of the sixcrystal structures in a unit cell is displayed in Fig. 1 and theWyckoff positions are listed in Table 1. There are 8, 6, 8, 16, 14and 16 atoms in the primitive cell of Im%3m, C2/c, I4/mcm,I4/mmm, P21/m, and P4/mbm phases, respectively. Their crystalstructure parameters are listed in Table 2. Our calculated lattice

constant of diamond silicon is in good agreement with theexperimental value20–22 indicating that the calculation methodand the parameters were reasonably chosen. The enthalpies ofthese new phases relative to diamond silicon were calculatedusing the formula: DH = Hnew phase/n1 � Hdiamond silicon/n2, wheren1 and n2 are the number of atoms in the unit cell of eachsilicon phase. Diamond silicon is the most stable phase atambient pressure. As shown in Table 2, among the six newallotropes, the P21/m phase is the most stable with only 0.049 eVper atom higher in energy than diamond silicon at ambientpressure, while the least favourable phase is I4/mmm, which is0.366 eV per atom higher in energy than diamond silicon. Thesequence of the six predicted silicon phases with low to high

Fig. 1 Crystal structures of the new silicon allotropes in a unit cell.

Table 1 The structural parameters of the new silicon allotropes

Phases Wyckoff position

Im%3m Si1: 16f (�0.66000, �0.34000, �0.66000)C2/c Si1: 8f (�0.23602, �0.37204, 0.08295)

Si2: 4e(0.00000, �0.94753, 0.25000)I4/mcm Si1: 16l (0.35284, �0.85284, 0.18656)I4/mmm Si1: 32o (0.10776, �0.73776, �0.83427)P21/m Si1: 2e (0.93282, 0.75000, 0.03906)

Si2: 2e(0.50707, 0.75000, 0.76767)Si3: 2e(0.00226, 0.75000, 0.24905)Si4: 2e(0.57602, 0.75000, 0.96986)Si5: 2e (0.80568 0.75000 0.68542)Si6: 2e(0.67862, 0.75000, 0.30690)Si7: 2e(0.60625, 0.75000, 0.49738)

P4/mbm Si1: 8j (0.96655, 0.19657, 0.50000)Si2: 8k (0.39945, 0.10054, 0.21778)

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energy is P21/m, I4/mcm, C2/c, P4/mbm, Im%3m, and I4/mmm. Themass density of the new phases was also calculated and it wasfound that all the new phases have less mass density comparedto diamond silicon.

To confirm the stability of the novel phases, the phononspectra of all the new phases were calculated and the results areshown in Fig. 2. As seen in Fig. 2, these new structures arethermodynamically stable because no imaginary frequencies areobserved throughout the whole Brillouin zone.

The mechanical properties and stability of the six new siliconallotropes were also studied thoroughly. The elastic constantswere calculated via the strain–stress method and the results are

listed in Table 3. Among the six phases, three have tetragonalsymmetry (P4/mbm, I4/mcm, and I4/mmm), two are monoclinicstructures (P21/m and C2/c), and one cubic (Im%3m). For tetragonalstructures, the criteria for the mechanical stability are given bythe following four equations23

Cii 4 0, i = 1,3,4,6, (1)

(C11 � C12) 4 0, (2)

(C11 + C33 � 2C13) 4 0, (3)

[2(C11 + C12) + C33 + 4C13] 4 0. (4)

For monoclinic structures, the elastic constants need to satisfythe following generalized Born’s mechanical stability criteria:23

Cii 4 0, i = 1–6, (5)

[C11 + C22 + C33 + 2(C12 + C13 + C23)] 4 0, (6)

(C33C55 � C352) 4 0, (7)

(C44C66 � C462) 4 0, (8)

[C22(C33C55�C352) + 2C23C25C35�C23

2C55�C252C33] 4 0,

(9)

2 C15C25ðC33C12 � C13C23Þ þ C15C35ðC22C13 � C12C23Þ½

þC25C35ðC11C23 � C12C13Þ� � C152ðC22C33 � C23

2Þ�

þC252ðC11C33 � C13

2Þ þ C35ðC11C22 � C122Þ�þ C55g 4 0

(10)

g = C11C22C33 � C11C232 � C22C13

2 � C33C122 + 2C12C13C23.

(11)

And the stable cubic structure needs to follow the criteria forthe mechanical stability given by23

C11 4 0, C44 4 0, C11 4 |C12|, (C11 + 2C12) 4 0(12)

As shown in Table 3, our calculated elastic constants ofdiamond silicon are in great agreement with the literature. Andthe elastic constants under normal pressure of the six newphases all comply well with the Born’s mechanical stabilitycriteria mentioned above, indicating the mechanical stability ofthese new silicon phases.

The modulus of elasticity can be used as an indicator todemonstrate the ease of structural elastic deformation. It is areflection of the bonding strength between atoms, ions or mole-cules at the microscopic level. A larger modulus value meansgreater rigidity of the structure. The bulk modulus (B) andshear modulus (G) were calculated using the Voigt–Reuss–Hillapproximation.24 The Young’s modulus (E) and the Poisson’sratio (n) were obtained using the formula:

E = 9BG/(3B + G), (13)

v = (3B � 3G)/[2(3B + G)]. (14)

Table 2 Crystal lattice parameters, mass density and relative enthalpy todiamond silicon of the new structures

Phases a (Å) b (Å) c (Å) b (1)r(g m�3)

Relative enthalpy(eV per atom)

Im%3ma 7.494 1.773 0.322C2/ca 5.794 8.857 6.280 123.790 2.089 0.210I4/mcma 7.590 6.380 2.030 0.131I4/mmma 11.018 7.229 1.701 0.366P21/ma 6.631 3.851 11.784 100.680 2.208 0.049P4/mbma 8.421 5.454 1.929 0.223Diamondsilicona

5.469 2.329 0.000

Diamondsiliconb

5.431

a This work. b Ref. 16–18.

Fig. 2 Phonon spectra of the six new silicon phases Im %3m (a), C2/c (b),I4/mcm (c), I4/mmm (d), P21/m (e), and P4/mbm (f).

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The computed moduli B, G, E, Poisson’s ratio n, and B/G ratio ofthe new silicon phases are also shown in Table 3. From Table 3,we can see that the calculated moduli of the new phases areslightly smaller than that of diamond silicon. Among these newstructures, the elastic modulus E of the P21/m phase is veryclose to that of diamond silicon. It was found that all the sixnew phases have a larger Poisson’s ratio n than diamondsilicon. The ratio of the bulk to shear modulus (B/G) iscommonly used25 to measure a material’s ductility and brittle-ness. That is to say, if B/G o 1.75, the material is brittle;otherwise, the material exhibits good ductility.26 From Table 3,one can see that diamond silicon is the most brittle with thesmallest B/G ratio. The two phases I4/mcm and P21/m can bedescribed as brittle as well. The other four phases C2/c, Im%3m,I4/mmm, and P4/mbm demonstrate good ductility with a B/Gratio larger than 1.75.

Crystal anisotropy means that atoms arranged in differentcrystalline directions with different periodicities and densitieslead to distinct orientation-dependent physical and chemicalproperties. It is well known that elastic anisotropy plays anindispensable role in engineering science and crystal physics.In this paper, we determine the directional dependence27 of theanisotropy. The three-dimensional (3D) figures of the Young’smodulus for these new phases are shown in Fig. 3. All the newallotropes show some degree of anisotropy but to a differentextent. For example, the three tetragonal structures of theI4/mcm (Fig. 3(c)), I4/mmm (Fig. 3(d)), and P4/mbm phases(Fig. 3(f)) demonstrate high anisotropy as seen from their 3Dmodulus surface largely deviated from a sphere.28 Similarly, thetwo monoclinic structures of the C2/c (Fig. 3(b)) and P21/m(Fig. 3(e)) phases also show significant anisotropy upon obser-ving their contour lines of Young’s modulus in the xy, yz, and zx

Table 3 The calculated elastic constants, various moduli (GPa), Poisson’s ratio, and band gap (eV) of the new phases and diamond silicon

Phases C11 C12 C13 C22 C23 C33 C44 C55 C66 B G E n B/G Eg (HSE06) Eg (PBE)

Im%3ma 84 51 51 84 51 84 32 32 32 61 24 64 0.327 2.559 0.658 0.100C2/ca 143 52 43 92 32 123 26 55 30 66 34 87 0.279 1.928 1.470 0.898I4/mcma 120 76 27 120 27 191 41 41 64 76 45 112 0.255 1.702 1.240 0.620I4/mmma 109 52 35 109 35 43 26 26 11 48 19 52 0.320 2.463 0.671 0.178P21/ma 158 42 34 190 31 196 43 45 63 84 57 140 0.221 1.458 1.207 0.618P4/mbma 114 46 33 114 33 155 15 15 40 67 27 73 0.319 2.418 1.395 0.814Diamond silicona 151 60 77 89 62 151 0.219 1.443 1.310 0.611Diamond siliconb 165 64 87 88 64 155 1.380 1.280 0.720

a This work. b Ref. 3.

Fig. 3 The directional dependence of the Young’s modulus for the six new silicon phases. Im %3m (a), C2/c (b), I4/mcm (c), I4/mmm (d), P21/m (e) andP4/mbm (f). The lines in the xy, yz, zx plane are the contours of Young’s modulus.

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planes deviated from a circle. Lastly, the cubic Im%3m (Fig. 3(a))phase shows a moderate orientation dependence on its Young’smodulus.

The calculated maximum and minimum of Young’s mod-ulus (Emax and Emin) and the ratio of Emax/Emin for the six newphases and diamond silicon are listed in Table 4. Diamondsilicon has a Emax/Emin ratio of 1.71. Among the six newlydiscovered allotropes, only the P21/m phase has a smallerEmax/Emin ratio, and all the other five phases have a greaterEmax/Emin ratio than diamond silicon.

The conversion efficiency of photovoltaic materials is closelyrelated to their band gap.29,30 Therefore, we conducted electronicband structure calculations for the new silicon phases. The bandstructures were first calculated by the standard DFT using the PBEfunctional. Since the conventional DFT calculations usually under-estimate the band gaps of semiconductors,6 an advanced andmore computationally demanding hybrid functional HSE06 wasimplemented to better predict the gap. The band structures of thesix new silicon allotropes are presented in Fig. 4. The calculatedband gaps are also listed in Table 3. As shown in Table 4, ourcalculated gap for diamond silicon at the HSE06 level is in goodagreement with a previous study. It is also noted that despite theunderestimation of the band gap at the PBE level both the PBEand HSE06 functionals predict the same ranking order in theband gap values for the six new phases and diamond silicon,indicating that the ranking order of the total seven silicon allo-tropes in the band gap from the smallest to the largest is Im%3m,I4/mmm, diamond Si, P21/m, I4/mcm, P4/mbm, and the C2/c phase.

The band structures in Fig. 4 clearly demonstrate that all thenew predicted silicon phases in this work have direct bandgaps. The HSE06 predicted band gaps are in the range of0.658–1.470 eV, suggesting that all the new allotropes have greatpotential as photovoltaic materials compared to diamondsilicon. Among them, the four phases C2/c, I4/mcm, P21/m, andP4/mbm have the band gaps within the range of 1.0–1.5 eV,which are suitable for thin-film solar cell applications. Moreover,two phases Im%3m and I4/mmm have a band gap close to 0.60 eV,indicating promising photovoltaic materials in tandem solar cellapplications which require a 0.50–0.60 eV band gap31 absorber tocapture the low-energy photons.

To further explore the light absorption performance of thesenew structures, we calculated the imaginary part of their dielec-tric functions at the HSE06 level. For comparison, diamondsilicon was also calculated. The results are shown in Fig. 5.

The solar spectrum can be divided into three regions, infrared,visible, and ultraviolet, according to their photon energies. Sincesolar radiation is concentrated in the visible region (about1.6–3.2 eV), favourable photovoltaic materials need to have goodabsorption of visible light as well as infrared. As mentionedpreviously, the direct band gap of 3.4 eV for diamond silicon istoo large and absorbs mostly ultraviolet photons which representonly a small portion of the solar spectrum. However, the six newsilicon allotropes predicted in this work have much strongerabsorption in the visible region due to their direct band gaps inthe range of 0.658–1.470 eV. From Fig. 5, one can see that amongthe six new phases and diamond Si, the C2/c phase with a directband gap of 1.470 eV shows the best light absorption performancein the visible photon energy range of 1.6–1.9 eV and 2.7–3.2 eV,

Table 4 The calculated maximum and minimum values of Young’smodulus (Emax, Emin), and the Emax/Emin ratio of the new structures anddiamond silicon

Phases Emax (GPa) Emin (GPa) Emax/Emin

Im%3m 80.8 44.9 1.80C2/c 146.7 55.4 2.65I4/mcm 183.6 71.2 2.58I4/mmm 75.4 27.5 2.74P21/m 185.5 114.6 1.62P4/mbm 141.6 47.9 2.96Diamond silicon 182.6 106.8 1.71

Fig. 4 Electronic band structures of the new silicon phases using func-tional HSE06.

Fig. 5 Imaginary part of the dielectric function for the six new siliconallotropes and diamond Si calculated at the HSE06 level.

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while the I4/mmm phase with a direct gap of 0.671 eV exhibitsthe highest optical absorption in the range of 1.9 B2.7 eV. Itwas demonstrated that a maximum photoelectric conversionefficiency of 33.7% can be achieved when the band gap is1.34 eV for a single-gap photovoltaic device.32 A tandem solar cellsystem was proposed by stacking several homojunction cellstogether to form arrays, and it was found that a three- or four-stacked cell can boost the efficiency up to more than 60%, whichrequires a 0.5–0.6 eV band gap absorber to capture the low-energyphotons.31 The six new Si phases have a direct bandgap in therange of 0.658 to 1.470 eV and are promising materials inphotovoltaic applications.

Conclusions

Six new silicon allotropes with direct band gaps were predictedby performing Si substitution of carbon in the known carboncrystal structures through high-throughput calculations. Thesesix structures are in the Im%3m, C2/c, I4/mcm, I4/mmm, P21/m, andP4/mbm space groups, respectively. All of the new phases wereproved to be thermodynamically stable via phonon spectrumcalculations and mechanically stable by evaluating their elasticconstants at ambient pressure. More importantly, the new phasesall have direct band gaps in the range of 0.658–1.470 eV which issuperior for photovoltaic applications. Moreover, the new siliconallotropes demonstrate significantly improved photon absorptioncompared to diamond silicon upon investigating the imaginaryparts of their dielectric functions, which suggests that the newphases are promising photovoltaic materials for thin-film andtandem solar cell applications.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was financially supported by the National NaturalScience Foundation of China (Grant No.: 11204007), the NaturalScience New Star of Science and Technologies Research Planin Shaanxi Province of China (Grant No. 2017KJXX-53), andthe Natural Science Basic Research plan in Shaanxi Provinceof China (Grant No. 2019JM-353). We also acknowledge thecomputing facilities at the High Performance Computing Centerof Xidian University.

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