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Band gap tuning in HgTe through uniaxial strains
Solid State Communications 166 (2013) 1–5
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Band gap tuning in HgTe through uniaxial strains Huxian Zhao a,n, Xiaoshuang Chen a, Jianping Lu b, Haibo Shu a, Wei Lu a a b
National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Science, Shanghai 200083, People's Republic of China Department of Physics and Astronomy, Curriculum in Applied and Materials Sciences, University of North Carolina, Chapel Hill, North Carolina 27599, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 11 March 2013 Accepted 12 April 2013 by X.C. Shen Available online 23 April 2013
The impact of uniaxial strain along the [111] direction on the structural and electronic properties of bulk HgTe in the zinc blende is studied by DFT. Uniaxial strain can effectively manipulate the local lattice structure along the same direction. The transformation may be caused to form the graphite-like structure in large compression and the layered one under stretching. Meanwhile, the conductive band minimum (CBM) and valence band maximum (VBM) are gradually shifted to form the indirect band structures. Further, the band gap is opened in HgTe for the significant stretching, The uniaxial compression only changes the coordination of HgTe by maintaining the semi-metallic properties. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. HgTe B. DFT C. Uniaxial strain D. Electronic properties
1. Introduction It is well known that the transformations in structure of semiconductors can lead to the change of optoelectronic properties. Application of appropriate strain whether external stresses [1] (such as hydrostatic pressure) or internal strains [2,3] (through lattice or thermal mismatch strains in thin films), is able to alter the atomic structures, and modulate the material properties [4]. In fact, as the functional materials, the “strain engineering” is an important way to obtain the desired physical and electronic properties. In semiconductors, the electronic structures and band gap are highly relevant to the inter-atomic distances and the relative positions of atoms. The change of structural distribution makes it possible to control and tune the band gap (Eg) through the structural fine transformations. The concept has been employed to investigate some compounds of III–V and II–VI: for GaN, the relatively small compressive misfit cannot tune Eg effectively, but for small stretching strain, the Eg can be decreased significantly [4]. And for ZnX (X¼O, S, Se, Te), under uniaxial strain along the [0001] direction, the Eg decreases for both compressive and tensile in-plane strains [5]. Moreover, a phase transition from strainfree to graphite-like at large compressive strains has been predicted both in GaN and ZnX (X¼O, S, Se, Te). In IIB–VI zinc blende compounds, besides the wide band gap semiconductors such as zinc compounds, there are narrow gap semiconductors, such as CdTe and HgTe. Among these compounds, HgTe is of special electronic properties and technological interest [6,7]. HgTe and its alloys are employed to fabricate a wide array of electro-
n
Corresponding author. Tel.: +86 21 3505 2062. E-mail addresses: [email protected] (H. Zhao), [email protected] (X. Chen). 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.04.013
optic devices, high-performance detectors, and room-temperature radiation detectors [8]. For example, its alloys with CdTe, Hg1-xCdxTe (MCT), have a direct band gap from −0.26 eV to 1.61 eV by changing the component x at 77 K, covering the entire IR region. Accompanying with the small effective electron mass, it is widely used for infrared detecting field [9,10,11]. HgTe itself also has a number of other special properties related to the low-dimensional structures in the material system and its low- dimensional properties are much different from the bulk ones. Filling the single wall carbon nanotube, the onedimensional HgTe also can be opened with a band gap at least 1.2 eV [12]. Moreover, HgTe is a candidate for spintronics applications, and a quantum spin Hall Effect has been discovered in its quantum wells [13,14,15]. It is shown, both theoretically and experimentally, that HgTe behaves as an insulator in its interior or bulk, and a topological insulator, permitting the movement of charge on its surface with topologically protection by time reversal symmetry and particle number conservation. In order to discover more applications of HgTe, we have undertaken a study of bulk HgTe with the DFT under uniaxial strain to reveal the strain-induced structural and electronic modulations. 2. Methods and models 2.1. Methods Here, we study the electronic properties of HgTe with different structures, especially the band characteristic near the Fermi energy, based on the density functional theory (DFT). We use a plane-wave expansion of the eigenfunctions and non-normconserving pseudopotentials as implemented in the Vienna Ab initio Simulation Package (VASP) [16]. For exchange–correlation interaction, the generalized
2
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
gradient approximation (GGA) of Perdew, Becke, and Ernzerhof (PBE) [17] is used. Electron–ion interaction is described by using the projector augmented wave method (PAW) [18]. We use the standard PAW distributed with VASP package. The semi-core electrons of cation atoms (5d10 and 6s2 of Hg) are included explicitly in the calculations. The set of plane waves used is extended up to a kinetic energy cutoff of 500 eV for all compounds. The large cutoff energy is required in order to achieve highly converged results within the PAW and USPP scheme. A Γ-centered k-points mesh of 7n7n7 is used to sample Brillouin Zone (BZ) in structural calculation, and in the electronic property, the mesh is increased to 9n9n9. The high density grid of BZ integrations is in order to ensure highly converged results in compounds. For the calculation of total energy, we use the tetrahedron method with Blöchl corrections. The method also gives a good account for the electronic density of states (DOS). In the calculations of band structures, the Gaussian smearing method is used. We also use an accurate algorithm in the calculations for allowing us to obtain highly converged force. The spin–orbit coupling is directly related to atomic number in the system. And especially for a system containing an active d orbital, just as HgTe, the SO coupling has a very significant role on the electronic properties. Accordingly, the SO coupling is considered in the calculations of self-consistent charge density and other properties.
2.2. Models The strain-free phase of HgTe is in zinc blende structure. In order to simulate the situations in practice, we impose the uniaxial strain along the [111] direction. The strain axis is the most preferred nanowire growth direction and the system is subjected to the direction [19,20]. Therefore, in the calculation, the adjustment of primitive unit cell is required. We redefine the c direction of new lattice as [−1 −1 3] in terms of old one. So the used cell in our calculation contains three HgTe pairs. For the convenience of comparison, we define the c to be the 2/3 of the height of the new cell in Fig. 1. As the anticipation of the tendency to ‘graphite-like’ structure, the calculations in which the symmetry of the system is constrained are performed by using the unit cell, as shown in Fig. 1(b). As defined in Fig. 1, the length of the bond along the c-direction is bc, the other one is ba. The bond angle in the plane, parallel with c-axis, is γ, and α is out of the plane. In all the calculations, for lattice vectors, c is fixed at a specific value. Corresponding to each fixed c value, a is optimized. Each structure is fully relaxed to its equilibrium configuration by calculating the forces on atoms and the stress tensor. In the relaxed equilibrium configuration, the forces are less than 10−4 eV/Å. Highly
Fig. 1. (a) Strain-free Zinc Blende unit cell and (b) graphite-like semiconductor structure unit cell under large compressive uniaxial strains. The c-axis is along [111] direction in each case.
converged results on forces are required for the calculation of structures and electronic property.
3. Results 3.1. Structural transformations In the presence of strains induced structural transformations, a certain value of strain is specified corresponding to a new lattice parameter in the strain direction. The strain in the vertical direction, along which we impose the uniaxial strain, is given by ε ¼(c–c0)/c0. Therefore, the vertical lattice parameter c (such as ε) is taken as the parameter of calculations and varied in our simulations from 4.013 to 6.475 Å, corresponding to −30.92% oε o11.45%. For all equilibrium configurations, as shown in Fig. 2(a), the calculated variations of a [such as εa ¼ (a−a0)/a0] display approximately a linear response as a function of ε while vertical compression strain ε is within 15%, and it can be easily understood by the Hooke's law. When ε is greater the value, the expansion variation of a gradually slows down. When ε is beyond −30%, the variation of a comes to irregularity. The inset of Fig. 2(a) shows the change of bond lengths, ba and bc, such as εbond ¼(bx −bx0)/bx0. The uniaxial strains can only affect bc along the strain axis, and the influence of the strain on bc is not monotonic. When ε is around −15%, which is the limit of Hooke's law, the bc comes to minimum. The variation of ba appears to be saturated for most of the strain range considered. When the variation of a comes to irregularity beyond compression strains of about −30%, the variation of ba appears similar anomalous distribution.
Fig. 2. (a) Variation of lattice parameter a [εa ¼ (a–a0)/a0] as a function of ε. Inset shows the variation of the change in the bond lengths [ba and bc, defined in Fig. 1, εbond ¼(bx−bx0)/bx0] with respect to ε. (b) bond angles (γ and α, defined in Fig. 1) with different ε. The open circle and triangle represent the strain-free values.
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
The more dramatic feature of structural changes along with the uniaxial strains is the transformation of the systems from the typical regular tetrahedron configuration with sp3 hybridization to a graphitelike one. From Fig. 2(b), the transformation is indicated by seeing the variation of the two bond angles (α and γ, defined in Fig. 1). Close to the strain-free state, α and γ are both 109.51, which is one of the characteristics of sp3 hybridization in zinc blende phrase. Under the uniaxial compression, they both gradually approach to the bond angles in graphite, 901 and 1201, respectively. Fig. 2(b) shows that for uniaxial compressive strains greater than 30%, the transformation is almost complete. However, the Te and Hg atoms are not coplanar as exactly as the C atoms within the grapheme, and the distance between Te layer and Hg layer is less than 0.1 Å. Under large uniaxial stretching strain, α and γ are limited to 1061 and 1121, respectively. Thus at large c, HgTe layers are formed, but the Hg and Te atoms are not in the same plane. The Te layer is about 1.124 Å higher than Hg layer. However, when ε is beyond 9.32%, the variations of both angles are changed. Up to now, we can conclude that the uniaxial strain can effectively tune the local structural parameters along the same direction. The other parameters, not in the direction, are little affected by the uniaxial strain. Thus, the symmetry group of system reduces from Td to R3m (No. 160). In addition, for the case of large ε, the irregular variation of lattice parameter a under compressive strains and the changed variations of bond angles under stretching strains both imply that phase transition may happen to the system. 3.2. Electronic structure distributions At ambient pressure, due to the symmetry of the crystal filed of HgTe, the valence band and conduction bands are degenerated at Γ
3
point [21]. Therefore, the symmetry-lowering perturbations such as uniaxial pressure can split the degenerate states, and electronic properties of HgTe are modified by the uniaxial strain. Four band structures under different uniaxial strains are showed in Fig. 3, and the bottom panels (a2), (b2), (c2) and (d2), show the corresponding detailed structures around Fermi level (−2 eV oEnergy o2 eV). The band structures of (b), (d) correspond to the systems with relatively small deformation. No matter compression or stretching, the valence band minimum (VBM) and conduction band maximum (CBM) are both around the Γ point. The band dispersion becomes anisotropic. The valence band and conduction bands acquire extrema at k≠0, which is consistent with the earlier studies [22] and suggest that the isoenergetic surfaces near the extremum of the band are toroidal. However, the big deformation system shows an obviously semi-metallic band structure with “indirect” zero band gap (Fig. 3(a)). The VBM is still around Γ point, but the CBM now have shifted to H point. The similar situation has also been found in the GaN. However, in GaN, the VBM shifts to H point and the CBM is always at Γ point. It can be found in the detailed band structures (Fig. 3(a2), (b2), (c2) and (d2)) that influence of strain on the electronic states at Γ point is greater than that around it. So in the case of compression, the band structures form two saucer-shaped structures below and above the Fermi energy. Moreover, the greater the compression is, the deeper the drop valley in saucer-shaped structures is. However, the influence on the states around Γ point is different for the stretching and compression. Therefore, in the case of stretching, only one saddled-shaped structure is formed in valence bands and the conduction bands maintain the parabolic-like shape. Also under the stretching, the CBM is always at Γ point, and is not moved.
Fig. 3. (Color online) Band structures of HgTe for (a) ε ¼−28.79%; (b) ε ¼−8.56%; (c) ε ¼0%; (d) ε ¼ +9.32%. The bottom ones (a2), (b2), (c2) and (d2), are the band structures around the Fermi level (−2 eV o Energy o 2 eV). The horizontal lines in magenta at zero indicate Fermi level.
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H. Zhao et al. / Solid State Communications 166 (2013) 1–5
Fig. 4. (Color online) Density of states (DOS) of HgTe under uniaxial strain (ε) of (a) −8.56%, (b) 0%, and (c) +9.32%. The top panel of each figure displays the DOS arising from the Te s, px, py and pz orbital, and the bottom tow panels display the DOS due to the different orbital of Hg. For clarity, the five components of d orbital are profiled in individual panel. The horizontal dash lines in olive at zero indicate Fermi level.
Fig. 4 shows the DOS of HgTe under uniaxial strain ε of (a) −8.56%, (b) 0%, and (c) +9.32%. From previous publications, it is noted that, although the DFT calculation underestimates the absolute value of energy band gap, the trends in some physical variables can well be predicted in agreement with others, and the resulting pressure coefficients of band gap are also in agreement with experimental ones. At strain-free, as shown in the DOS of Fig. 4(b), the Hg d and Te p orbital construct the main part of the VBM states, and the Hg s orbital contribute the CBM states. The perfect overlap of the px, py, and pz orbital, as well as s orbital near the VBM are evidences of the sp3 hybridization. Also, the overlap between different d orbitals and p orbitals suggests the stronge p–d hybridization in HgTe. As the same as the situation in other sp3 hybridization compounds [4,5], it could be seen in Fig. 4(a) and (c) that the uniaxial strain reduces the overlap of the DOSs: for Te, the pz orbital DOS splits away from the curves of px and py orbital; for Hg, besides the pz splitting, the dz2 orbital curve also moves away from the overlaps. For the orbital in the plane perpendicular to the strain direction, such as px and py, the effect of the uniaxial strain on the orbital is identical. So they maintain the degeneration as the same as that in strain-free. The same situation also happens between the dxy and dx2−y2, and dyz and dzx. Therefore, we regard that the coordination variation is consistent with the changes in the form of hybridization, from sp3 to dsp3. The dsp3 hybridization also could be seen as the sp2 hybridization in the xy plane, combined with the hybridization of pz and dz2 along the z direction. Thus, it can also be caused that there is the weakness of p–d hybridization, and the splitting of dz2. At strain-free, it is well known that the band structures of HgTe are inverted, which means the Γ8 state (CBM) is above the Γ6 state (VBM). To better understand the factors underlying the above
mentioned electronic property variations, we briefly consider details of the state energy. Due to the z component splitting, the Γ8 state (CBM), which is four-fold degenerated, splits into two double-degenerated states. The energy of state, which is mainly contributed by pz orbital of Te, rapidly increases with uniaxial stress increasing. The Γ6 state (CBM) energy, which is mainly occupied by the s orbital with spherical symmetry, only has a little change during our calculation. Thus, the relative energy of these two states has no inversion.
4. Summary We investigate the structural and electronic properties of bulk HgTe system over a wide range of uniaxial strains along the zinc blende [111] axes. Structurally, the uniaxial strain can effectively tune the local structural parameters along the strain direction. Under uniaxial compression, the Td structure with sp3 hybridization transformed to ones which display graphite-like features. The transformation is evident from the bond angles approaching to the values of 1201 and 901, respectively. In the uniaxial stretching, the transformation can be caused to form individual HgTe layer. For the electronic properties, under uniaxial compression, the VBM and the CBM both gradually move away from Γ point. In the big deformation, such as graphite-like structure, the CBM shifts to H point forming the semimetallic band structure with typical “indirect” zero band gap. Unlike the band gap vanishing under the uniaxial pressure in GaN and ZnX (X¼ O, S, Se, Te), here the stretching can open a positive band gap in HgTe. Although our research is purely bulk calculation results and the range of strain considered here is extremely large, this investigation has identified the opportunities for tuning the band
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
gap of HgTe through strain. Furthermore, it is indicated that the way of strain can provide the lattice-distortion in core/shell nanoparticles and epitaxial thin films. [23,24,25]
[9] [10] [11] [12]
Acknowledgments
[13] [14]
This work was supported in part by the State Key Program for Basic Research of China (2011CB922004), the National Natural Science Foundation of China (61006090, 10990104, and 60976092), and the Fund of Shanghai Science and Technology Foundation (10JC1416100).
[15] [16]
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Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Band gap tuning in HgTe through uniaxial strains Huxian Zhao a,n, Xiaoshuang Chen a, Jianping Lu b, Haibo Shu a, Wei Lu a a b
National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Science, Shanghai 200083, People's Republic of China Department of Physics and Astronomy, Curriculum in Applied and Materials Sciences, University of North Carolina, Chapel Hill, North Carolina 27599, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 11 March 2013 Accepted 12 April 2013 by X.C. Shen Available online 23 April 2013
The impact of uniaxial strain along the [111] direction on the structural and electronic properties of bulk HgTe in the zinc blende is studied by DFT. Uniaxial strain can effectively manipulate the local lattice structure along the same direction. The transformation may be caused to form the graphite-like structure in large compression and the layered one under stretching. Meanwhile, the conductive band minimum (CBM) and valence band maximum (VBM) are gradually shifted to form the indirect band structures. Further, the band gap is opened in HgTe for the significant stretching, The uniaxial compression only changes the coordination of HgTe by maintaining the semi-metallic properties. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. HgTe B. DFT C. Uniaxial strain D. Electronic properties
1. Introduction It is well known that the transformations in structure of semiconductors can lead to the change of optoelectronic properties. Application of appropriate strain whether external stresses [1] (such as hydrostatic pressure) or internal strains [2,3] (through lattice or thermal mismatch strains in thin films), is able to alter the atomic structures, and modulate the material properties [4]. In fact, as the functional materials, the “strain engineering” is an important way to obtain the desired physical and electronic properties. In semiconductors, the electronic structures and band gap are highly relevant to the inter-atomic distances and the relative positions of atoms. The change of structural distribution makes it possible to control and tune the band gap (Eg) through the structural fine transformations. The concept has been employed to investigate some compounds of III–V and II–VI: for GaN, the relatively small compressive misfit cannot tune Eg effectively, but for small stretching strain, the Eg can be decreased significantly [4]. And for ZnX (X¼O, S, Se, Te), under uniaxial strain along the [0001] direction, the Eg decreases for both compressive and tensile in-plane strains [5]. Moreover, a phase transition from strainfree to graphite-like at large compressive strains has been predicted both in GaN and ZnX (X¼O, S, Se, Te). In IIB–VI zinc blende compounds, besides the wide band gap semiconductors such as zinc compounds, there are narrow gap semiconductors, such as CdTe and HgTe. Among these compounds, HgTe is of special electronic properties and technological interest [6,7]. HgTe and its alloys are employed to fabricate a wide array of electro-
n
Corresponding author. Tel.: +86 21 3505 2062. E-mail addresses: [email protected] (H. Zhao), [email protected] (X. Chen). 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.04.013
optic devices, high-performance detectors, and room-temperature radiation detectors [8]. For example, its alloys with CdTe, Hg1-xCdxTe (MCT), have a direct band gap from −0.26 eV to 1.61 eV by changing the component x at 77 K, covering the entire IR region. Accompanying with the small effective electron mass, it is widely used for infrared detecting field [9,10,11]. HgTe itself also has a number of other special properties related to the low-dimensional structures in the material system and its low- dimensional properties are much different from the bulk ones. Filling the single wall carbon nanotube, the onedimensional HgTe also can be opened with a band gap at least 1.2 eV [12]. Moreover, HgTe is a candidate for spintronics applications, and a quantum spin Hall Effect has been discovered in its quantum wells [13,14,15]. It is shown, both theoretically and experimentally, that HgTe behaves as an insulator in its interior or bulk, and a topological insulator, permitting the movement of charge on its surface with topologically protection by time reversal symmetry and particle number conservation. In order to discover more applications of HgTe, we have undertaken a study of bulk HgTe with the DFT under uniaxial strain to reveal the strain-induced structural and electronic modulations. 2. Methods and models 2.1. Methods Here, we study the electronic properties of HgTe with different structures, especially the band characteristic near the Fermi energy, based on the density functional theory (DFT). We use a plane-wave expansion of the eigenfunctions and non-normconserving pseudopotentials as implemented in the Vienna Ab initio Simulation Package (VASP) [16]. For exchange–correlation interaction, the generalized
2
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
gradient approximation (GGA) of Perdew, Becke, and Ernzerhof (PBE) [17] is used. Electron–ion interaction is described by using the projector augmented wave method (PAW) [18]. We use the standard PAW distributed with VASP package. The semi-core electrons of cation atoms (5d10 and 6s2 of Hg) are included explicitly in the calculations. The set of plane waves used is extended up to a kinetic energy cutoff of 500 eV for all compounds. The large cutoff energy is required in order to achieve highly converged results within the PAW and USPP scheme. A Γ-centered k-points mesh of 7n7n7 is used to sample Brillouin Zone (BZ) in structural calculation, and in the electronic property, the mesh is increased to 9n9n9. The high density grid of BZ integrations is in order to ensure highly converged results in compounds. For the calculation of total energy, we use the tetrahedron method with Blöchl corrections. The method also gives a good account for the electronic density of states (DOS). In the calculations of band structures, the Gaussian smearing method is used. We also use an accurate algorithm in the calculations for allowing us to obtain highly converged force. The spin–orbit coupling is directly related to atomic number in the system. And especially for a system containing an active d orbital, just as HgTe, the SO coupling has a very significant role on the electronic properties. Accordingly, the SO coupling is considered in the calculations of self-consistent charge density and other properties.
2.2. Models The strain-free phase of HgTe is in zinc blende structure. In order to simulate the situations in practice, we impose the uniaxial strain along the [111] direction. The strain axis is the most preferred nanowire growth direction and the system is subjected to the direction [19,20]. Therefore, in the calculation, the adjustment of primitive unit cell is required. We redefine the c direction of new lattice as [−1 −1 3] in terms of old one. So the used cell in our calculation contains three HgTe pairs. For the convenience of comparison, we define the c to be the 2/3 of the height of the new cell in Fig. 1. As the anticipation of the tendency to ‘graphite-like’ structure, the calculations in which the symmetry of the system is constrained are performed by using the unit cell, as shown in Fig. 1(b). As defined in Fig. 1, the length of the bond along the c-direction is bc, the other one is ba. The bond angle in the plane, parallel with c-axis, is γ, and α is out of the plane. In all the calculations, for lattice vectors, c is fixed at a specific value. Corresponding to each fixed c value, a is optimized. Each structure is fully relaxed to its equilibrium configuration by calculating the forces on atoms and the stress tensor. In the relaxed equilibrium configuration, the forces are less than 10−4 eV/Å. Highly
Fig. 1. (a) Strain-free Zinc Blende unit cell and (b) graphite-like semiconductor structure unit cell under large compressive uniaxial strains. The c-axis is along [111] direction in each case.
converged results on forces are required for the calculation of structures and electronic property.
3. Results 3.1. Structural transformations In the presence of strains induced structural transformations, a certain value of strain is specified corresponding to a new lattice parameter in the strain direction. The strain in the vertical direction, along which we impose the uniaxial strain, is given by ε ¼(c–c0)/c0. Therefore, the vertical lattice parameter c (such as ε) is taken as the parameter of calculations and varied in our simulations from 4.013 to 6.475 Å, corresponding to −30.92% oε o11.45%. For all equilibrium configurations, as shown in Fig. 2(a), the calculated variations of a [such as εa ¼ (a−a0)/a0] display approximately a linear response as a function of ε while vertical compression strain ε is within 15%, and it can be easily understood by the Hooke's law. When ε is greater the value, the expansion variation of a gradually slows down. When ε is beyond −30%, the variation of a comes to irregularity. The inset of Fig. 2(a) shows the change of bond lengths, ba and bc, such as εbond ¼(bx −bx0)/bx0. The uniaxial strains can only affect bc along the strain axis, and the influence of the strain on bc is not monotonic. When ε is around −15%, which is the limit of Hooke's law, the bc comes to minimum. The variation of ba appears to be saturated for most of the strain range considered. When the variation of a comes to irregularity beyond compression strains of about −30%, the variation of ba appears similar anomalous distribution.
Fig. 2. (a) Variation of lattice parameter a [εa ¼ (a–a0)/a0] as a function of ε. Inset shows the variation of the change in the bond lengths [ba and bc, defined in Fig. 1, εbond ¼(bx−bx0)/bx0] with respect to ε. (b) bond angles (γ and α, defined in Fig. 1) with different ε. The open circle and triangle represent the strain-free values.
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
The more dramatic feature of structural changes along with the uniaxial strains is the transformation of the systems from the typical regular tetrahedron configuration with sp3 hybridization to a graphitelike one. From Fig. 2(b), the transformation is indicated by seeing the variation of the two bond angles (α and γ, defined in Fig. 1). Close to the strain-free state, α and γ are both 109.51, which is one of the characteristics of sp3 hybridization in zinc blende phrase. Under the uniaxial compression, they both gradually approach to the bond angles in graphite, 901 and 1201, respectively. Fig. 2(b) shows that for uniaxial compressive strains greater than 30%, the transformation is almost complete. However, the Te and Hg atoms are not coplanar as exactly as the C atoms within the grapheme, and the distance between Te layer and Hg layer is less than 0.1 Å. Under large uniaxial stretching strain, α and γ are limited to 1061 and 1121, respectively. Thus at large c, HgTe layers are formed, but the Hg and Te atoms are not in the same plane. The Te layer is about 1.124 Å higher than Hg layer. However, when ε is beyond 9.32%, the variations of both angles are changed. Up to now, we can conclude that the uniaxial strain can effectively tune the local structural parameters along the same direction. The other parameters, not in the direction, are little affected by the uniaxial strain. Thus, the symmetry group of system reduces from Td to R3m (No. 160). In addition, for the case of large ε, the irregular variation of lattice parameter a under compressive strains and the changed variations of bond angles under stretching strains both imply that phase transition may happen to the system. 3.2. Electronic structure distributions At ambient pressure, due to the symmetry of the crystal filed of HgTe, the valence band and conduction bands are degenerated at Γ
3
point [21]. Therefore, the symmetry-lowering perturbations such as uniaxial pressure can split the degenerate states, and electronic properties of HgTe are modified by the uniaxial strain. Four band structures under different uniaxial strains are showed in Fig. 3, and the bottom panels (a2), (b2), (c2) and (d2), show the corresponding detailed structures around Fermi level (−2 eV oEnergy o2 eV). The band structures of (b), (d) correspond to the systems with relatively small deformation. No matter compression or stretching, the valence band minimum (VBM) and conduction band maximum (CBM) are both around the Γ point. The band dispersion becomes anisotropic. The valence band and conduction bands acquire extrema at k≠0, which is consistent with the earlier studies [22] and suggest that the isoenergetic surfaces near the extremum of the band are toroidal. However, the big deformation system shows an obviously semi-metallic band structure with “indirect” zero band gap (Fig. 3(a)). The VBM is still around Γ point, but the CBM now have shifted to H point. The similar situation has also been found in the GaN. However, in GaN, the VBM shifts to H point and the CBM is always at Γ point. It can be found in the detailed band structures (Fig. 3(a2), (b2), (c2) and (d2)) that influence of strain on the electronic states at Γ point is greater than that around it. So in the case of compression, the band structures form two saucer-shaped structures below and above the Fermi energy. Moreover, the greater the compression is, the deeper the drop valley in saucer-shaped structures is. However, the influence on the states around Γ point is different for the stretching and compression. Therefore, in the case of stretching, only one saddled-shaped structure is formed in valence bands and the conduction bands maintain the parabolic-like shape. Also under the stretching, the CBM is always at Γ point, and is not moved.
Fig. 3. (Color online) Band structures of HgTe for (a) ε ¼−28.79%; (b) ε ¼−8.56%; (c) ε ¼0%; (d) ε ¼ +9.32%. The bottom ones (a2), (b2), (c2) and (d2), are the band structures around the Fermi level (−2 eV o Energy o 2 eV). The horizontal lines in magenta at zero indicate Fermi level.
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Fig. 4. (Color online) Density of states (DOS) of HgTe under uniaxial strain (ε) of (a) −8.56%, (b) 0%, and (c) +9.32%. The top panel of each figure displays the DOS arising from the Te s, px, py and pz orbital, and the bottom tow panels display the DOS due to the different orbital of Hg. For clarity, the five components of d orbital are profiled in individual panel. The horizontal dash lines in olive at zero indicate Fermi level.
Fig. 4 shows the DOS of HgTe under uniaxial strain ε of (a) −8.56%, (b) 0%, and (c) +9.32%. From previous publications, it is noted that, although the DFT calculation underestimates the absolute value of energy band gap, the trends in some physical variables can well be predicted in agreement with others, and the resulting pressure coefficients of band gap are also in agreement with experimental ones. At strain-free, as shown in the DOS of Fig. 4(b), the Hg d and Te p orbital construct the main part of the VBM states, and the Hg s orbital contribute the CBM states. The perfect overlap of the px, py, and pz orbital, as well as s orbital near the VBM are evidences of the sp3 hybridization. Also, the overlap between different d orbitals and p orbitals suggests the stronge p–d hybridization in HgTe. As the same as the situation in other sp3 hybridization compounds [4,5], it could be seen in Fig. 4(a) and (c) that the uniaxial strain reduces the overlap of the DOSs: for Te, the pz orbital DOS splits away from the curves of px and py orbital; for Hg, besides the pz splitting, the dz2 orbital curve also moves away from the overlaps. For the orbital in the plane perpendicular to the strain direction, such as px and py, the effect of the uniaxial strain on the orbital is identical. So they maintain the degeneration as the same as that in strain-free. The same situation also happens between the dxy and dx2−y2, and dyz and dzx. Therefore, we regard that the coordination variation is consistent with the changes in the form of hybridization, from sp3 to dsp3. The dsp3 hybridization also could be seen as the sp2 hybridization in the xy plane, combined with the hybridization of pz and dz2 along the z direction. Thus, it can also be caused that there is the weakness of p–d hybridization, and the splitting of dz2. At strain-free, it is well known that the band structures of HgTe are inverted, which means the Γ8 state (CBM) is above the Γ6 state (VBM). To better understand the factors underlying the above
mentioned electronic property variations, we briefly consider details of the state energy. Due to the z component splitting, the Γ8 state (CBM), which is four-fold degenerated, splits into two double-degenerated states. The energy of state, which is mainly contributed by pz orbital of Te, rapidly increases with uniaxial stress increasing. The Γ6 state (CBM) energy, which is mainly occupied by the s orbital with spherical symmetry, only has a little change during our calculation. Thus, the relative energy of these two states has no inversion.
4. Summary We investigate the structural and electronic properties of bulk HgTe system over a wide range of uniaxial strains along the zinc blende [111] axes. Structurally, the uniaxial strain can effectively tune the local structural parameters along the strain direction. Under uniaxial compression, the Td structure with sp3 hybridization transformed to ones which display graphite-like features. The transformation is evident from the bond angles approaching to the values of 1201 and 901, respectively. In the uniaxial stretching, the transformation can be caused to form individual HgTe layer. For the electronic properties, under uniaxial compression, the VBM and the CBM both gradually move away from Γ point. In the big deformation, such as graphite-like structure, the CBM shifts to H point forming the semimetallic band structure with typical “indirect” zero band gap. Unlike the band gap vanishing under the uniaxial pressure in GaN and ZnX (X¼ O, S, Se, Te), here the stretching can open a positive band gap in HgTe. Although our research is purely bulk calculation results and the range of strain considered here is extremely large, this investigation has identified the opportunities for tuning the band
H. Zhao et al. / Solid State Communications 166 (2013) 1–5
gap of HgTe through strain. Furthermore, it is indicated that the way of strain can provide the lattice-distortion in core/shell nanoparticles and epitaxial thin films. [23,24,25]
[9] [10] [11] [12]
Acknowledgments
[13] [14]
This work was supported in part by the State Key Program for Basic Research of China (2011CB922004), the National Natural Science Foundation of China (61006090, 10990104, and 60976092), and the Fund of Shanghai Science and Technology Foundation (10JC1416100).
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