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Electronic structure and optical properties of Si, Ge and diamond in the lonsdaleite phase

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 045801 (http://iopscience.iop.org/0953-8984/26/4/045801) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 045801 (16pp)

doi:10.1088/0953-8984/26/4/045801

Electronic structure and optical properties of Si, Ge and diamond in the lonsdaleite phase Amrit De1,2 and Craig E Pryor1 1

Department of Physics and Astronomy and Optical Science and Technology Center, University of Iowa, Iowa City, IA 52242, USA 2 Department of Physics and Astronomy, University of California, Riverside, CA 92506, USA E-mail: [email protected] Received 22 October 2013 Published 8 January 2014 Abstract

Crystalline semiconductors may exist in different polytypic phases with significantly different electronic and optical properties. In this paper, we calculate the electronic structure and optical properties of diamond, Si and Ge in the lonsdaleite (hexagonal diamond) phase using a transferable model empirical pseudopotential method with spin–orbit interactions. We calculate their band structures and extract various relevant parameters. Differences between the cubic and hexagonal phases are highlighted by comparing their densities of states. While diamond and Si remain indirect gap semiconductors in the lonsdaleite phase, Ge transforms into a direct gap semiconductor with a much smaller bandgap. We also calculate complex dielectric functions for different optical polarizations and find strong optical anisotropy. We further provide expansion parameters for the dielectric functions in terms of Lorentz oscillators. Keywords: band structures, semiconductors, optical dielectric functions, empirical pseudopotentials (Some figures may appear in colour only in the online journal)

1. Introduction

crystal, known as lonsdaleite (LD). This form of diamond was not discovered until 1967 when it was found in a meteorite [2]. It was synthesized soon thereafter [3] and is now believed to be the hardest known substance [4]. The high pressure phases of Si and Ge have been investigated [1, 5] and it has been determined that both undergo a series of structural phase transitions from cubic, to β-Sn, to simple hexagonal, to an orthorhombic phase, to hexagonal close packed, to a face centered cubic phase [1, 5]. On the other hand, III–V semiconductors typically crystallize in β-Sn, nickeline (NiAs) or rocksalt structures when subjected to high temperature and pressure [1, 5]. More recently, the growth of WZ phase bulk GaAs was achieved under extreme conditions [6]. Extreme temperatures and pressures are not the only things that can be used to achieve WZ or LD growth. Laser ablation has been used to synthesize stable LD phase

It is well known that under extreme conditions a crystalline material may undergo a structural transition to a phase which remains effectively stable upon returning to standard thermodynamic conditions. The existence of such phases, polymorphism in compounds and allotropy in elemental crystals, results in materials with differing electrical and optical properties, such as diamond and graphite. Polytypism is a particular case of polymorphism in which the coordination number does not change. For example, III–V and group IV semiconductors can crystallize in their non-naturally occurring polytypic phase while maintaining their tetrahedral coordination [1]. For example, III–V semiconductors can crystallize in either the zincblende (ZB) phase or the hexagonal wurtzite (WZ) phase. Under extreme conditions cubic diamond (CD) transforms to a hexagonal wurtzite 0953-8984/14/045801+16$33.00

1

c 2014 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 26 (2014) 045801

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inclusion of SO interactions, the optical selection rules state that a direct 06 –03 transition is not allowed [52] but when the SO interactions are included, then a 09 –08 direct transition is allowed. Experimentally, very strong photoluminescence signals are seen from such pseudo-direct transitions in WZ-GaP [53] and WZ-GaAs [54]. In this paper, we present bulk electronic band structures and optical dielectric function calculations for diamond, Si and Ge in the LD phase using empirical pseudopotentials with the inclusion of SO interactions. These calculations are based on transferable continuous model pseudopotentials assuming an ideal LD structure. The spherically symmetric ionic model potentials are first obtained by fitting the calculated bulk energies of the cubic polytype to experimental results. The band structure of the LD polytype is then obtained by transferring the spherically symmetric model pseudopotentials to the LD crystal structure. Model potentials with a continuous smooth decay in momentum are best suited for transfer between polytypes. The transferability of the model potentials can be justified on the basis of the similarities of the crystal structures of the two polytypes. Like the cubic structure, the LD structure is built from tetrahedrons of the same atom but stacked differently. In both structures all of the nearest neighbors and nine out of the twelve second-nearest neighbors are at identical crystallographic locations [55] and all the second-nearest neighbors are equidistant. Hence the local electronic environments should be very similar in the two structures. This method has proven to be quite successful in obtaining the bulk band structures of various semiconductor polytypes in the past [48, 56–63]. We have used this method to predict the band structures of WZ phase III–V semiconductors [64] and our calculations are in excellent agreement with experiment for the cases for which the WZ bandgaps are known (namely GaAs, InP and InAs). Since then, a number of recent experiments have provided further conformations of the predicted bandgaps, their respective symmetries [65–69] and even for the effective mass of WZ-InAs [70]. In addition, our predictions for the direct gaps of WZ-GaP and WZ-AlAs (which are indirect in the ZB phase) and their symmetries are in excellent agreement with recent experiments [53, 71]. One practical difficulty of our method is that it requires continuous atomic form factors (for best transferability) and that the calculated transition energies must be accurately fitted to experiment to maximally constrain the form factors. This type of a multi-variable fit can be extremely challenging due to the presence of multiple local minima. We have used a modified simulated annealing method to adjust the pseudopotential form factors so that our calculated transition energies are in very good agreement with experiment at various high symmetry points. We calculate the dielectric functions for diamond, Si and Ge in the LD phase in the linear response regime within the electric dipole approximation for light polarized parallel and perpendicular to the c axis of the crystal. These calculations are in part motivated by recent experimental results showing that the photoluminescence (PL) intensity in nanowires is

Si [7] and III–V nanowires tend to crystallize in the WZ phase [8–10]. This has been attributed to various factors such as the small nanowire radii [11, 12], growth kinetics [13], interface energies [14] and electron accumulation at the catalyst’s interstitial site [15]. It is now believed that the tendency of nanowires to crystallize in the WZ/LD phase may hold for group IV semiconductors as well [16]. While typically for III–V semiconductors the WZ phase is more stable in the bulk limit, contrary to expectations, it has been experimentally found [17] that Si nanowires with a radius in excess of 10 nm tend to crystallize in the WZ/LD phase, while the cubic phase occurred about less than half of the time when the radius was below 10 nm. Theoretical investigations on the other hand indicate that the cubic phase is not stable for Si [18, 19] and Ge [19] nanowires below a certain critical radius and that the LD and polycrystalline phases are more stable for Si in the 1D limit [20]. However in [21] it has been argued that the occurrence of LD phases in nanowires is determined more by the growth kinematics and not by the energetic factorability of the crystallographic phase. Technologically, any new phase for Si and Ge should be of considerable interest as these are arguably the most important semiconductors. Semiconductor nanowires themselves have attracted much interest due to their potential applications such as in photovoltaic cells [22–24], nanomechanical resonator arrays [25], THz detectors [26, 27], single-photon detectors [28–30], field-effect transistors [31, 32], single-electron transistors and other devices [33–39]. More recently, strong signatures of Majorana bound states have been observed in semiconductor nanowires with large spin–orbit couplings, placed in close proximity to a superconductor [40–43]. In general, the quasi-one-dimensional nature of a nanowire allows materials with large lattice mismatches to be combined to form heterostructures [44–47] that are not possible in planar structures. The design and characterization of such devices requires an understanding of the electronic and optical properties of WZ/LD phase semiconductors. Even though bulk LD phase Si, Ge and diamond have been synthesized in the laboratory, their electronic structure still remains experimentally unverified. For any application pertaining to spintronics, it is important to include spin–orbit (SO) interactions in any electronic structure calculations. Though the band structures and dielectric functions for LD phase diamond, Si and Ge have been obtained before using empirical pseudopotentials [48], these calculations were all done without the inclusion of SO interactions and with discrete form factors with abrupt momentum cutoffs. The group IV LD phase band structures have also been calculated using density functional theory (DFT) in the local density approximation (LDA) [49–51]. However LDA-DFT has well known shortcomings in predicting energies of excited states. The inclusion of SO interaction is also important for calculating optical properties, especially when they involve pseudo-direct transitions. A cubic semiconductor that has an indirect L valley bandgap will have a direct zone center gap in the WZ/LD phase due to zone folding. Without the 2

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involving local field effects, our simple approach fixes 0 without shifting peak positions. This paper is organized as follows. Section 2 gives an outline of the empirical pseudopotential method, followed by a description of both the cubic and the hexagonal band structures, and a discussion of the respective symmetries of the two polytypes in section 3. Our results are in section 3.2 where we present the calculated band structures, densities of states (DOS), effective masses for zone center states and transition energies at various high symmetry points. The optical dielectric functions and reflectivity spectra are given in section 4. Finally, we summarize our results in section 5.

strongly polarization dependent [9, 65, 72–76], including for Si nanowires [77]. Nanowire polarization effects can arise from the dielectric mismatch at the wire surface [72] and electron–hole recombination selection rules [78–80], as well as from the underlying WZ-type crystal structure [81–86]. The dominant effect could be more easily identified if the optical dielectric functions for these semiconductors were easily available. However, experimentally measuring optical dielectric functions for LD phase semiconductors is difficult due to the extreme conditions required for growing bulk samples. Our dielectric function calculations are carried out within the one-particle picture. Much has been done in recent years to include the effects of two-particle contributions (such as electron–hole interactions) in the dielectric function [87–93]. In the case of first-principles calculations, this improves agreement with experiment for the low energy part of the optical spectrum [87]. However, the dominant contribution to the dielectric function comes from one-particle calculations, which themselves tend to be in good agreement with experiment [94–96]. Moreover, while two-particle corrections improve ab initio dielectric functions [63], empirical methods include some two-particle effects through their fit to the experimental data (which includes such effects). Typically, dielectric functions calculated using EPMs are in good agreement with experiment [63]. The required momentum matrix elements, required for our calculations, are obtained from the pseudopotential wavefunctions. In general, momentum matrix elements for pseudopotential calculations need to be corrected for the missing core states. Such corrections are typically done using nonlocal terms [97–100], which can also account for the exchange and correlations effects as well [101–103]. In our calculations, nonlocal effects are included in the form of spin–orbit interactions only. Another problem with ab initio dielectric function calculations is that 0 is often overestimated [104–106]. This is then improved upon by the inclusion of local field effects [88] such as electron–phonon interactions, which typically affect the low frequency part of the dielectric function (terahertz regime). However, this is not necessary in our case since local field effects shift the peak positions [89]. The EPM includes such effects through the fittings to experiments which necessarily include the effects. We instead adopt a simpler approach to take the missing core states and local field effects into account. We correct the static dielectric function, 0 , for the unknown polytype by making use of the known 0 of the cubic phases of diamond, Si and Ge. First, the optical dielectric functions for these cubic group IV semiconductors are calculated. The optical sum rules are then used to obtain a set of constants which normalize the calculated cubic 0 to their respectively known experimental values. Since the constituent element is the same for each polytype, corrections to account for the missing core states should be nearly the same and transferable between polytypes. These normalization constants are then used to correct 0 for the unknown polytypes (LD), which therefore also corrects the LD phase dielectric functions as well. Unlike calculations

2. Empirical model pseudopotentials

We use the empirical pseudopotential formalism of Cohen and Chelikowsky [63]. However, rather than using discrete form factors we use continuous model potentials, so they are transferable between polytypes. The pseudopotential Hamiltonian consists of the kinetic, local pseudopotential (Vpp ) and SO interaction (Vso ) terms, −h¯ 2 K2 + Vpp + Vso . (1) 2m In a periodic crystal, Vpp can be expanded in terms of plane waves as H=

Vpp (r) =

N 1 XX V FF (G)eiG·(r−τα,j ) N G,α j=1 α

(2)

where the Gs are reciprocal lattice vectors, α labels the atom type, VαFF (G) is the form factor of the αth type of atom, N is the number of atoms per unit cell of a given type, and τα,i is the position of atom number j of type α. For a compound with only one type of atom, the pseudopotential is simply hG0 |Vpp |Gi = V FF (G0 − G)S(G0 − G) where the structure factor is 1 X S(G) = exp(−iG · τj ). N j

(3)

(4)

V FF can be obtained in various ways [63, 107]. In the empirical pseudopotential approach the atomic form factors are adjusted such that the calculated energies at various high symmetry points fit experiment. In order for pseudopotentials to be transferable between polytypes (having different Gs) V FF (G) should be a continuous function of G. A wide variety of such model potentials have been used in the literature [59, 107–111], and they use potentials of the form h i−1 V FF (G) = (x1 G + x2 ) 1 + exp(x3 G 2 + x4 ) (5) where G = |G × a/2π |2 , and the xj are adjustable parameters used to fit each material’s cubic phase band structure to experiment. See appendix A for more details on the fitting parameters and on how the model pseudopotential compares 3

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˚ and c = 6.520 A, ˚ the ideal lattice constants are a = 3.993 A which are within 1% of the experimental values [117] of ˚ and c = 6.57 A. ˚ a = 3.96 A When viewed along the [111] direction, the inter-layer atomic bonds in LD lie in an eclipsed conformation, defining the axis of hexagonal symmetry, while in the CD structure the inter-layer atomic bonds are in a staggered conformation, making all four body diagonals of the cube equivalent. The nearest neighbors are the same in the two polytypes (due to their tetrahedral symmetry), and nine of the twelve next nearest neighbors are in identical positions in both crystals and the remaining three next nearest neighbors are equidistant. These structural similarities suggest that the local electronic environments will be very similar in the two crystals, and hence the atomic form factors should be nearly identical in the two polytypes. The LD crystal structure has a space group symmetry classification of D46h (P63 /mmc or space group #194). It has inversion symmetry in addition to all the symmetries of WZ. The irreducible representations of the space group of 0 are just the representations of the point group D6h (which has all the symmetries of C6v as well as inversion symmetry). While moving along the kz direction, at the 1 point, the symmetry is lowered to C6v . A, K and H all have point group symmetries of D3h [118]. D3h is isomorphic to C6v and is a symmorphic invariant subgroup of D6h . The L and M points have D2h symmetry. The point group operations must be followed by appropriate translations in order to obtain the irreducible representations of the wavefunctions at the high symmetry points. This is further discussed in the appendix. Lonsdaleite has lower symmetry than the cubic diamond structure, and the SO interaction leads to additional lifting of orbital degeneracies. In the absence of spin–orbit coupling, the hexagonal crystal field of LD splits the p-like 015 state of cubic structure into fourfold-degenerate 06 and doubly degenerate 01 . In terms of the p orbitals, these states are pz → 01 and px , py → 06 . With the inclusion of spin–orbit coupling, 06v splits into the 09v heavy hole and the 07v light hole. Therefore, all zone center states in LD belong to 07 , 08 , or 09 with either even or odd parity since LD has a center of inversion (see the character tables in appendix B). Also unlike for WZ, there are no spin splittings in LD due to its inversion symmetry.

to discrete form factors. In general, model potentials that yield an accurate band structure of a known polytype should reliably predict the band structure for the unknown polytype if the electronic environments in the two crystals structures are similar. We take spin–orbit coupling into account by including the interaction [112, 113] hK0 , s0 |Vso |K, si = (K0 × K) · hs0 |σ|si X λl P0l (cos θK0 ·K )S(K0 − K). (6) × l

It is not necessary to expand equation (6) beyond l = 2 since group IV semiconductors do not have core shells filled beyond d orbitals. For Ge the terms up to l = 2 in equation (7) are included, while for Si and diamond we only go up to l = 1. Expanding equation (6) up to l = 2, the spin–orbit coupling term is ˆ 0 × K) ˆ · hs0 |σ|si hK0 , s0 |Vso |K, si = −i(K 0 ˆ 0 · K)S(G ˆ × [(λp + λd K − G)] (7)

and λl is a coefficient that can be written in terms of the core wavefunctions: λl = µl βnl (K0 )βnl (K) ∞ p βnl (K) = C il 4π(2l + 1)jnl (Kr)Rnl (r)r2 dr

(8)

Z

(9)

0

where σs are the Pauli matrices, K = G + k, θ is the angle between K and K0 , and the µl are adjustable parameters used to fit the spin–orbit splitting energies to experiment [113]. The overlap integral, βnl , is constructed from the radial part of the core wavefunction, Rnl , which is an approximate solution to the Hartree–Fock equations which we obtain from Herman–Skillman tables [114]. C is a normalization constant such that βnl (K)/K approaches unity in the limit as K goes to zero. Spin–orbit interactions are included for only the outermost p shells (n = 4 in Ge, n = 3 in Si, and n = 2 in diamond) and d shells (n = 3 in Ge). 3. Band structures and crystal symmetries 3.1. Lonsdaleite crystal structure and symmetries

The LD/WZ crystal structure is constructed from two interpenetrating hexagonal close-packed (HCP) lattices, just as the diamond/ZB structure is constructed from two interpenetrating FCC lattices. For ideal crystals, the lonsdaleite lattice constant √ is related to the diamond lattice constant as a = acubic / 2 and the lattice constant along the c √ axis (the [111] direction) is given by c = 1/ua. We assume an ideal LD crystal with u = 3/8 in this paper, giving a = ˚ and c = 4.119 A ˚ for diamond, both of which are 2.522 A ˚ in agreement with the experimental values of a = 2.52 A ˚ and c = 4.12 A [3]. In the case of LD phase Si, a wide range of measured lattice constants have been reported with ˚ 6.280 A) ˚ [7], (3.84 A, ˚ 6.180 A) ˚ [115], (a, c) = (3.84 A, ˚ ˚ (3.837 A, 6.316 A) [116]. Assuming an ideal LD crystal ˚ and c = 6.264 A ˚ for Si. For Ge structure, we use a = 3.836 A

3.2. Predicted lonsdaleite band structures

We use spherically symmetric local form factors fitted to the CD band structure, which are then transferred to the LD crystal. The potentials should be transferable since the local electronic environments of the two polytypes are very similar. The lonsdaleite primitive unit cell has four atoms. We choose the origin such that the atoms are located at v1 = 13 a1 + 32 a2 , v2 = 23 a1 + 13 a2 + 21 a3 , v3 = v1 + ua3 and v4 = v3 + √ ua3 , where a1 = (1, 0, 0)a, a2 = (−1, 3, 0)a/2 and a3 = (0, 0, c) are the primitive lattice vectors. Substituting these atomic positions into equation (4), we obtain the following 4

J. Phys.: Condens. Matter 26 (2014) 045801

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Figure 1. (a) Band structure for diamond in the lonsdaleite phase.

Figure 3. (a) Band structure for Ge in the lonsdaleite phase.

(b) Density of states.

(b) Density of states.

Due to the similarities of the two crystals, many of the high symmetry points in the Brillouin zones of cubic and LD have a one-to-one correspondence with each other (just as in the case of ZB and WZ). This one-to-one correspondence is particularly useful in interpreting their respective band structures. The volume of LD’s first Brillouin zone is about half of that of its cubic counterpart. Therefore, if one were to take an intersection of the two Brillouin zones such that each of their 0 points coincide, then the L point in the cubic structure also coincides with the 0 point in LD [50]. Thus, in the free electron model, the zone center LD/WZ energies can be directly predicted from the cubic 0 and L point energies. In single-group notation (in the absence of SO interactions), the cubic 01 , L1 and L3 states correspond to 01 , 03 and 05 respectively in LD. However, the presence of the crystal potential will perturb the exact one-to-one correspondence of the high symmetry point energies. Because of this zone folding of the L valley, indirect gap cubic materials with L valley conduction band minima could be expected to have a direct gap in the LD phase (unless the energy of that state was significantly shifted by the crystal potential). See figure B.1 in appendix B for more details on how the zone center states are related. There also are similar correspondences between the high symmetry directions of the two crystals. The 3(0 → L) line in the cubic structure corresponds to the 1(0 → A) line in the LD [55] structure. Note that there are eight equivalent L directions in the cubic structure. Only the ones that are along the c axis map onto the 0hex point. The other six along with Xcub map onto a point on the Uhex line, two thirds away from the Mhex point. We label this point as M0hex . The 1cub line maps onto a line joining M0hex and 0hex . The 1cub line is especially important as the bandgaps for diamond and Si both lie along this line close to Xcub . We list the energies at various high symmetry points along with the corresponding irreducible representations of these states in tables 1–3. The irreducible representations of the zone center states were determined by transforming

Figure 2. (a) Band structure for Si in the lonsdaleite phase.

(b) Density of states.

structure factor: S=

    iGy a iGz uc 1 Gz uc exp − √ − cos 2 2 2 3    iGx a iGy a iGz c × 1 + exp − + √ − 2 2 2 3

(10)

where the Gj (j = x, y, z) are the components of the reciprocal lattice vector G. The calculated band structures and the corresponding densities of states (DOS) for diamond, Si and Ge in LD phase are shown in figures 1–3. The electronic band structures are calculated in the irreducible wedge of the Brillouin zone. The LD band structure is more complicated than its CD counterpart due to its lower crystal symmetry. For a given energy range, there are roughly twice as many bands for the LD phase. The points A and H are special points where the energy levels stick together because the structure factor is zero there. 5

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Table 1. Band energies (w.r.t. the top of the valence band) at various high symmetry points and the respective irreducible representations (IR) for the lonsdaleite phase of diamond. The 0 point effective masses are also listed for directions parallel and perpendicular to the c axis. Note that this is an indirect gap semiconductor with the conduction band minima in the K valley.

IR

E (eV)

07+ 08− 08− 08+ 09+ 07+ 07− 09− 07− 09− 07− 09− 09+ 08+ 07+

−26.4861 −20.6056 −16.081 −5.2053 −5.1929 −1.6965 −0.003 0 5.7365 5.741 8.8245 8.8246 9.0085 9.0095 9.3188

mk

m⊥

IR

E (eV)

IR

E (eV)

IR

E (eV)

IR

E (eV)

IR

E (eV)

1.1376 0.1915 0.1482 1.1656 1.1697 0.1810 0.3414 0.3418 0.7860 0.7877 0.2504 0.2504 0.9974 0.9979 0.2782

1.1947 1.1724 1.8260 0.3104 0.3425 0.6821 0.2437 0.3226 0.3785 0.3588 0.9981 2.6106 0.5130 0.2464 0.5695

A7 A8 A8 A7 A8 A9 A9 A7 A7 A9 A8 A9 A7 A8 A7

−24.6126 −24.5697 −8.9149 −8.8447 −3.3676 −3.3601 −3.1847 −3.1772 7.2555 7.2574 7.4665 7.4685 12.2676 12.5631 15.361

L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5

−19.7797 −19.73 −15.4592 −15.313 −6.9353 −6.6979 −6.1246 −5.8708 5.5528 5.6029 9.6269 9.8392 13.1219 13.3848 24.929

M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5

−21.5188 −18.0857 −14.9875 −12.2387 −11.2565 −8.6464 −7.101 −4.0212 5.2688 6.5429 9.0309 10.1738 18.184 18.2476 20.3734

H7 H8 H8 H8 H9 H7 H8 H9 H9 H7 H9 H7 H9 H8 H9

−18.5436 −18.3647 −15.7359 −15.3608 −10.5802 −10.3175 −5.303 −5.03 7.5599 7.6419 10.4545 10.5574 17.4962 17.5338 19.3015

K7 K8 K8 K8 K9 K8 K9 K7 K9 K7 K9 K7 K9 K8 K9

−19.6131 −19.0582 −12.7624 −12.6617 −12.0976 −9.0102 −8.8838 −8.1377 4.7676 13.1856 13.4011 14.0873 17.1210 17.5896 18.8105

Table 2. Band energies (w.r.t. the top of the valence band) at various high symmetry points and the respective irreducible representations (IR) for the lonsdaleite phase of Si. The 0 point effective masses are also listed for directions parallel and perpendicular to the c axis. Note that this is an indirect gap semiconductor with the conduction band minima in the M valley.

IR

E (eV)

07+ 08− 08− 08+ 09+ 07+ 07− 09− 08− 09− 09− 07+ 07+ 09+ 08+

−12.8713 −10.4924 −7.5715 −1.7207 −1.6793 −0.3496 −0.0279 0 1.4814 2.6962 2.7151 3.4206 4.2135 5.1671 5.1779

mk

m⊥

IR

E (eV)

IR

E (eV)

IR

E (eV)

IR

E (eV)

IR

E (eV)

1.1365 0.2697 0.184 1.3479 1.3598 0.1028 0.5481 0.5637 1.0483 0.6496 0.6548 0.1527 4.0888 0.3732 0.3735

1.1522 1.1129 2.4867 0.1843 0.1916 0.7687 0.201 0.2128 0.1224 1.096 0.9096 1.0882 0.1375 0.9400 0.6606

A7 A8 A8 A7 A7 A9 A8 A9 A8 A7 A8 A9 A7 A9 A8

−12.0498 −12.0433 −4.406 −4.3857 −1.0552 −1.0322 −1.0192 −0.9962 2.3662 2.3878 3.7479 3.7614 3.7624 3.7759 7.7044

L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5

−9.9375 −9.9237 −7.6756 −7.6573 −2.5751 −2.5516 −2.5085 −2.4906 1.2958 1.3036 2.4036 2.4125 6.695 6.705 9.1484

M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5

−10.6599 −8.992 −8.1319 −6.321 −4.7847 −3.2709 −3.01 −1.4971 0.7957 2.9848 3.4078 3.8669 5.8173 7.2281 7.6228

H7 H8 H8 H7 H9 H7 H8 H9 H9 H8 H9 H8 H9 H8 H7

−9.3824 −9.3576 −7.8786 −7.8319 −4.6241 −4.5975 −2.0413 −2.0303 2.3229 2.3319 4.1826 4.1891 6.8291 6.8401 7.3607

K7 K8 K8 K7 K9 K8 K9 K7 K8 K9 K9 K9 K7 K7 K8

−9.6803 −9.6430 −7.0424 −7.0031 −5.3748 −3.6509 −3.6304 −3.3739 1.2191 5.7005 6.4501 6.4750 6.8974 6.9190 6.9399

the pseudo-wavefunctions under the symmetry operations of the respective crystallographic point group. The zone center (0 point) effective masses for diamond, Si and Ge for k parallel and perpendicular to the c axis are also shown in these tables. From the band structure calculations, it is seen that in the LD phase, diamond and Si have indirect bandgaps. Diamond has a bandgap of about 4.767 eV with band minima occurring in the K valley. This is in agreement with earlier results obtained by Salehpour and Satpathy [49] where the band minima were also shown to occur at K. Their estimated bandgap from LDA calculations (with corrections) was about 4.5 eV. The LD bandgap for diamond is also significantly smaller than its cubic phase bandgap of 5.4 eV.

In the case of the LD phase Si, the calculated bandgap is 0.796 eV and the band minima are in the M valley. The LD phase gap for Si is a lot smaller than its cubic bandgap of about 1.1 eV. In our calculations, Ge is the only direct gap group IV semiconductor in the LD phase. It has a 08 conduction band minimum. The 08 symmetry is due to the fact that cubic phase Ge has a conduction band minimum in the L valley, which folds over to 08 in the hexagonal phase. Si, like Ge, has a 08 band minimum at the zone center, whereas diamond has a 07 band minimum. With the inclusion of SO interactions for LD, the top three valence states are typically (in order of decreasing energy) 09 , 07 , 07 (normal ordering) [78, 119] or 07 , 09 , 07 (anomalous ordering) which results from a negative spin–orbit energy. In 6

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Table 3. Band energies (w.r.t. the top of the valence band) at various high symmetry points and the respective irreducible representations (IR) for the lonsdaleite phase of Ge. The 0 point effective masses are also listed for directions parallel and perpendicular to the c axis. Note that this is a direct gap semiconductor.

IR

E (eV)

07+ 08− 08− 09+ 08+ 07+ 07+ 09− 08− 07+ 07+ 09− 07+ 08− 09+

−12.4692 −10.4783 −7.3386 −1.7267 −1.4617 −0.4896 −0.1293 0 0.3103 0.7659 2.5368 2.6714 3.3102 4.7797 4.8536

mk

m⊥

IR

E (eV)

IR

E(eV)

IR

E (eV)

IR

E (eV)

IR

E (eV)

1.1887 0.3375 0.2153 1.347 1.4218 0.0587 0.1484 0.6035 1.0563 0.0516 0.6213 0.6604 4.5651 0.4158 0.4177

1.1988 1.1500 11.7707 0.1550 0.1404 0.3416 0.0871 0.0672 0.0852 0.0410 1.0951 1.5539 0.4412 0.7988 0.8216

A7 A8 A8 A7 A9 A8 A7 A9 A8 A7 A7 A9 A7 A9 A7

−11.7515 −11.751 −4.6049 −4.5941 −1.1146 −1.1069 −0.8801 −0.8723 1.0951 1.1023 3.538 3.5433 3.6279 3.6332 6.0862

L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5 L5

−9.9534 −9.9492 −7.6576 −7.6503 −2.4754 −2.4673 −2.3285 −2.3199 0.7849 0.7877 2.4102 2.4151 6.3148 6.3199 8.3515

M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5 M5

−10.5753 −8.9518 −8.431 −6.548 −4.3358 −3.023 −2.6832 −1.4302 0.6828 2.0997 3.3933 3.7561 5.0137 6.6232 6.947

H7 H8 H8 H7 H7 H9 H9 H8 H7 H9 H8 H7 H7 H7 H9

−9.5329 −9.4188 −8.0632 −7.8404 −4.1686 −4.0578 −1.9988 −1.9369 1.7489 1.8076 4.1879 4.2033 6.5626 6.6076 6.6965

K7 K8 K8 K7 K8 K9 K7 K9 K8 K7 K9 K7 K7 K7 K8

−9.7671 −9.5896 −7.4710 −7.3368 −4.7870 −3.3445 −3.3327 −2.9510 1.5267 5.5903 5.6603 5.6979 5.7016 5.7153 6.4529

our calculations, the top three valence band states of all three semiconductors have normal ordering in the LD phase. The parities for these states are different for Ge, as compared to Si and diamond, because of the presence of substantially larger spin–orbit interactions. We can verify this by switching off the spin–orbit interactions for Ge, in which case we obtain 09− , 07− , 07+ for the valence band maxima—which is the same as for Si and diamond. The spin–orbit splitting energy, 1so , and the crystal field splitting, 1cr , can be extracted using the quasi-cubic approximation which assumes the WZ/LD structure to be equivalent to a [111]-strained zincblende structure [120]. 1so 1 light hole and 0 2 light hole and 1cr are related to the 07v 7v energies by 1,2 E(07v ) − E(09v ) = −

1so + 1cr 2

 1 1 1so 1cr 2 (11) (1so + 1cr )2 − 2 u √ where u = a/c is 3/8 for an ideal WZ/LD structure assumed here. The band ordering and irreducible representations of the zone center states need to be identified before using this equation. We have compiled a shorter table (table 4) for the LD phase semiconductors, that lists the bandgaps, 1so , 1cr , and the offset between the valence band edges for each of the polytypes. The effective masses for the conduction band minima of the indirect gap semiconductors are also listed in table 4. A comparison of the DOS between the cubic and hexagonal phases of these semiconductors reveals some interesting similarities and differences as shown in figure 4. For this comparison, the DOS was normalized so that the total integrated energy below 50 eV was the same for both polytypes. The DOS for diamond is more qualitatively similar for its polytypic phases than those for Ge or Si. While ±

Figure 4. A comparison between the densities of states of the cubic

and lonsdaleite phases for (a) diamond, (b) Si and (c) Ge.

LD phase diamond has an overall lower DOS around the bandgap, the DOS for LD-Si and LD-Ge are shifted lower in energy. This is because for LD-diamond the top of the valence band shifts upwards while the conduction band minima shift downwards, whereas for Si and Ge both the conduction band minima and valence band maxima shift to lower energies. 7

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

Table 4. Key parameters for the lonsdaleite phase for diamond, Si and Ge. Shown are the bandgaps (with the symmetry of the conduction band minima indicated in parentheses), the spin–orbit splitting energies (1so ), the crystal field splitting energies (1cr ), and the energy cubic hex difference between the tops of the valence bands of the hexagonal and cubic polytypes, 1EVB = EVB − EVB . Also listed are the effective masses (about the conduction band minima), and the static dielectric functions for directions parallel and perpendicular to the c axis. The specific directions about which the conduction band effective masses were extracted are indicated in parentheses.

Material

Eg (eV)

1so (eV)

1cr (eV)

Diamond Si Ge

4.7672 (K9 ) 0.7957 (M5 ) 0.310 (08 )

0.0045 0.044 0.404

1.6950 0.3336 0.1730

1EVB (eV) 0.0286 −0.1484 −0.1454

m⊥

0

0⊥

0.9333 (K → 0) 1.0846 (M → 0) 1.0563 (0 → K)

0.2740 (K → H) 0.7892 (M → L) 0.0852 (0 → A)

6.3192 11.8518 14.2896

5.7988 12.1808 16.0066

The momentum matrix elements for band-to-band transitions between the initial and final states are given by

Also LD phase Ge has the sharpest DOS at the gap as it is the only direct gap semiconductor out of the six. Qualitatively, the valence band DOS of Si and Ge are very similar for both phases.

Mcv (k) = hφv,k |ˆp|φc,k i

(15)

where pˆ is the momentum operator. We calculate Mcv using the pseudopotential wavefunctions, which can be written as X φv,k (r) = Cv (k, G) exp[i(k + G) · r] (16)

4. Optical properties 4.1. Calculations

G

where Cv (k, G) are the eigenvector coefficients, for the vth state, obtained by diagonalizing the pseudopotential Hamiltonian at a given wavevector k. Using equation (16), Mcv can therefore be rewritten in terms of the expansion coefficients as follows: X Mcv (k) = i Cv∗ (k, G)Cc (k, G)[(k + G) · eˆ ] (17)

At normal incidence, for any given polarization, the reflectivity acquires the simple form R = |(1 − ni )/(1 + ni )|2 , where ni (i = x, y or z depending on the surface normal) is the complex √ index of refraction. In the linear response regime, n(ω) = (ω), and the complex dielectric function can be separated into real and imaginary parts: (ω) =  0 (ω) + i 00 (ω), which are related to each other by the Kramers–Kronig relations. All dielectric function calculations in this paper are carried out in the long wavelength limit (i.e. assuming only direct band-to-band transitions (the same k). We obtain  00 (ω) using our empirical pseudopotential wavefunctions. In the electric dipole approximation, the optical response function for direct transitions between the initial valence band state, v, and the final conduction band state, c, is given by  2 2XZ h¯ π e 00  (ω) = |Mcv |2 δ(Ec (k) m2 ω2 c,v BZ

G

where eˆ is the optical polarization vector. The frequency dependent imaginary part of the dielectric function,  00 (ω), is then calculated using equation (17), for light polarized parallel and perpendicular to the c axis. In order to evaluate the Brillouin zone integral in equation (12), we have used a set of 4.5 × 104 special k points. These special k points are generated using the scheme of Monkhorst and Pack [122]. The momentum matrix elements calculated using the pseudopotential wavefunctions need to be corrected for the missing core states [107]. One way to do this is to include the commutator of the nonlocal pseudopotential and the position operator [123, 124], while other proposed methods involve the inclusion of a core repair term [97]. However it should be noted that both techniques cause small changes to the dielectric function (typically less than 5%). Some first-principles calculations have even shown that there is almost no difference between dielectric functions calculated using ab initio pseudopotential wavefunctions and those calculated using true electron wavefunctions [100]. Our efforts to calculate the dielectric functions of the LD phase for group IV semiconductors were however greatly aided by the fact that the cubic phase dielectric functions of diamond, Si and Ge are already well known. We take advantage of the fact that the pseudopotentials are being transferred between polytypes. Due to the similarity of the electronic environments around an atom in the two polytypes, the corrections to account for the missing core states should be nearly the same and transferable between polytypes. This method also accounts for local field effects and static screening effects.

− Ev (k) − h¯ ω) d3 k (12) R where PBZ is an integration over the entire Brillouin zone (BZ), c,v is a sum over all initial valence band and final conduction band states, and Ev (k) and Ec (k) are the valence and conduction band energies at their respective ks. We use the following delta function where the tail is exponentially suppressed: δ(1E − h¯ ω) ≈ 2 (1 + cosh[γ (1E − h¯ ω)])−1

k

mk

(13)

where γ is an adjustable damping parameter that can be used to phenomenologically incorporate lifetime broadening effects. We used γ = 100 eV−1 which gives a transition linewidth of about 35 meV [121]. The real part of the dielectric function,  0 (ω), is then obtained using the Kramers–Kronig relation Z ∞ 0 00 0 ω  (ω ) 2 0  (ω) = 1 + P dω0 (14) π ω02 − ω2 0 where P is the Cauchy principle value. 8

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

Figure 5. Comparison between calculated and experimental reflectivity spectra for the cubic phase for Si and Ge at normal incidence.

Figure 6. Real and imaginary parts of the complex dielectric function as a function of the optical frequency are shown here for polarized

light: (a) E⊥ in lonsdaleite phase diamond, (b) Ek in lonsdaleite phase diamond, (c) E⊥ in lonsdaleite phase Si, (d) Ek in lonsdaleite phase Si, (e) E⊥ in lonsdaleite phase Ge and (f) Ek in lonsdaleite phase Ge.

and k00 are obtained by appropriately scaling their respective

First the dielectric functions for cubic diamond, Si and Ge are evaluated. We then normalize the calculated (ω = 0) to the experimentally known static dielectric constant by making use of the optical sum rule Z 2 ∞  00 (ω) dω (18) o = 1 + κ π 0 ω

k

static dielectric functions o⊥ and o . These scaled o⊥ and k o are obtained using the f sum rule (equation (18)) and the constant C (which is already determined from the cubic phase calculations). The static dielectric function needs to be corrected for the missing core states and other effects such as the influence of local fields and static screening. The local field effects that often arise from the inhomogeneity of the crystal are known to alter the static dielectric function [126, 127]. Typically they

where κ is a scaling constant which is adjusted such that the calculated 0 matches experimental results from [125]). Next, the LD phase dielectric functions are calculated on the basis 00 of their respective band structures. The final LD phase ⊥ 9

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

are difficult to account for in dielectric function calculations. Our scaling method could be viewed as an empirical way to include the local field effects as well. The use of this same scaling factor in the LD structure can be further justified by the similarity of its local electronic environment to that of its cubic phase. To give an example, we compare the experimental reflectivity spectra at normal incidence with our calculations for cubic Si and Ge in figure 5. They are seen to be in good agreement with experiment. The measured refractive indices for Si and Ge were obtained from [128]. 4.2. Predicted dielectric functions

As mentioned earlier, all zone center states belong to 07 , 08 or 09 representations. The allowed interband transitions for LD zone center states are as follows [78, 79]. For light polarized parallel to the c axis (Ek ), the optical selection rules only allow transitions between same symmetry states, i.e. 07 ↔ 07 , 08 ↔ 08 and 09 ↔ 09 . For light polarized perpendicular to the c axis (E⊥ ), the allowed transitions are 07 ↔ 07 , 08 ↔ 08 , 09 ↔ 07 and 09 ↔ 08 . Note that the 07 ↔ 08 transition is forbidden for all polarizations. It should be mentioned that without the inclusion of SO interactions, the optical selection rules state that a direct 06 –03 transition is not allowed [52] (see figure B.1). Only when SO interactions are included is a 09 –08 direct transition allowed for E⊥ . Though the 09 –08 transition is often refereed to as a pseudo-direct transition, experimentally very strong photoluminescence signals are seen from such transitions in WZ-GaP [53] and WZ-GaAs [54]. The other allowed high symmetry point transitions for Ek are A7,8 ↔ A7,8 , A9 ↔ A9 , K4,5 ↔ K4,5 , K6 ↔ K6 , H4,5 ↔ H4,5 and H6 ↔ H6 , whereas for E⊥ the allowed transitions are A7,8 ↔ A7,8 , A9 ↔ A7,8 , K4,5 ↔ K6 and H4,5 ↔ H6 . All M and L valley transitions are allowed for all polarizations (i.e. M5 ↔ M5 and L5 ↔ L5 ) [79, 129]. In our calculations, Ge is the only group IV semiconductor that has a direct bandgap in the LD phase. Its conduction band minimum has 08 symmetry while the top of the valence band has 09 symmetry and hence LD phase Ge will only be optically active for E⊥ . For Ek only transitions from deep within the valence band will be allowed. The LD phase diamond and Si on the other hand are indirect gap semiconductors and will not be optically active. We calculate the frequency dependent  0 (ω) and  00 (ω) for parallel and perpendicularly polarized light for diamond, Si and Ge, as shown in figure 6. As seen in figure 7, the corresponding reflectivity spectra for both polarizations show several peaks which originate from interband transitions along various high symmetry points. Each of these crystals has distinct spectral features, depending on the details of their electronic structure. For Ge and Si the most prominent features are typically seen up to about 4 eV, whereas for diamond the most prominent peak is seen at about 12 eV. All the group IV LD phase semiconductors are optically anisotropic as expected. However, diamond exhibits greater optical anisotropy than Si or Ge as seen in the reflectivity spectra (figure 7). Notice that around 12 eV, there is

Figure 7. Calculated reflectivity spectra at normal incidence for the

lonsdaleite phase for (a) diamond, (b) Si and (c) Ge.

significantly more absorption for E⊥ due to the much smaller ⊥ structures at the same frequency. We also fit the numerically calculated ⊥ and k , using a series of Lorentz oscillators, whose real and imaginary parts are  0 (ω) = 1 −  00 (ω) =

X

fj (ω2 − 2j )

j

(ω2 − 2j )2 + (γj ω)2 fj γj ω

X j

(ω2

− 2j )2 + (γj ω)2

.

(19) (20)

Here, fj , j and γj were used as fitting parameters and these are listed in table 5, where f represents the oscillator strength, γj represents the relaxation rate and j is a resonance frequency term. Our fits are in excellent agreement with the calculated dielectric functions shown in figure 6. The fits are fully self-consistent—they satisfy the KK relations. A total of eight oscillators were used to obtain the fits. In general we found that the number of oscillators required for obtaining a good fit increases with the atomic number, Z, as the dielectric functions tend to have more degrees of freedom. We found the calculated dielectric function of LD phase Ge to be very well 10

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

Table 5. Lorentz oscillator model, equations (19) and (20); fitting coefficients for the dielectric functions of the lonsdaleite phase for diamond, Si and Ge. The fits self-consistently satisfy the Kramers–Kronig relations.

Diamond

f1 1 γ1 f2 2 γ2 f3 3 γ3 f4 4 γ4 f5 5 γ5 f6 6 γ6 f7 7 γ7 f8 8 γ8

Si

Ge

0 ⊥

k0

0 ⊥

k0

0 ⊥

k0

985.3673 32.8328 14.5859 853.9515 41.4835 0.0034 236.9051 21.9587 7.6482 233.2564 17.3520 4.2684 116.1930 12.3676 1.3144 106.8025 13.5408 1.8422 79.7343 14.9675 2.1050 74.1751 11.3129 1.3779

678.5831 40.6987 0.0013 578.0487 33.8467 10.6590 472.8091 24.3140 11.7097 184.8990 14.5275 2.7171 177.3834 17.1822 4.4543 117.7706 11.6698 1.1840 107.7722 11.0678 1.4464 94.0708 12.6235 1.8784

680.8746 18.3159 11.9052 72.5879 9.1403 4.0961 48.3910 4.0414 0.4695 46.3267 4.5743 0.7588 42.2792 5.8564 1.3023 38.8594 6.8823 1.6976 31.4714 5.0620 0.9031 3.6917 3.3768 0.4415

267.6021 16.5706 3.2055 147.4411 13.0696 6.0867 119.2238 7.5069 3.3057 46.0472 5.7108 1.5300 42.2002 4.1491 0.8118 35.6588 4.8775 0.7870 21.0394 3.4097 0.7051 10.7713 4.6176 0.2178

302.5362 20.4947 0.0001 274.2928 15.5813 9.0120 77.8386 3.5460 0.9091 67.3935 6.5916 2.7558 61.2418 9.2438 5.4133 46.0533 4.6447 0.9617 36.1245 5.4127 1.3409 10.1995 2.4327 0.6403

408.5224 16.3944 5.6344 66.7507 11.3004 4.8116 65.4694 3.5794 1.0019 50.1012 8.0090 2.5887 40.7816 5.4303 1.2101 31.8034 6.5723 1.6575 29.5812 4.6618 0.5825 20.1230 2.8069 0.5918

phase, with the inclusion of spin–orbit interactions, using transferable model potentials. The potentials should be accurate since the local electronic environments for the cubic and lonsdaleite polytypes are very similar. A number of parameters are tabulated, such as high symmetry point energies, and their irreducible representations and effective masses, which could be useful for constructing effective low energy models such as k · p theory. It is seen that while diamond and Si remain indirect in the lonsdaleite phase, Ge is transformed into a narrow direct gap semiconductor due to zone folding effects and is expected to be optically active. Even though the optical transitions in LD-Ge will be of the pseudo-direct type, the double-group selection rules and recent observations of strong photoluminescence from similar pseudo-direct WZ semiconductors [53, 54] suggest that LD-Ge could be extremely useful for optoelectronics. We have also calculated the frequency dependent complex dielectric functions up to 20 eV for light polarized parallel and perpendicular to the c axis in the dipole approximation. Expansion parameters are provided for the dielectric functions in terms of Lorentz oscillators so that they can be easily reproduced. Strong optical anisotropy is seen, which makes LD phase materials potentially useful as nonlinear crystals since their optical birefringence enables them to satisfy phase matching conditions.

reproduced with eight oscillators. These analytic dispersion relations could be useful for modeling optical devices and multi-layer thin-film structures. The static dielectric constants were calculated from the imaginary R parts of the dielectric functions ∞ (o = 1 + 2 0  00 (ω) dω/π ω), for light polarized parallel and perpendicular to the c axis, and are listed in table 4. In general, the semiconductor with the higher atomic number (Z) has the larger static dielectric constant. We also see that k 0⊥ < 0 in the case of diamond (which has a 07 direct gap), k whereas 0⊥ > 0 for Si and Ge (which have 08 direct gaps). This can be explained on the basis of the optical selection rules. For a given material, o depends on the number of allowed transitions and the oscillator strength of each transition. The closer the bands are, the stronger the dipole transitions will be. Consider a small region of ω around the direct bandgap where the transitions are the strongest for zone center states. For E⊥ a transition between the 09 heavy hole (HH) and the 07 (or 08 ) conduction band is allowed—all three materials, Si, Ge and diamond, will have allowed (1,2) transitions—whereas for Ek the 07 light hole/split-off hole to 07 conduction band is only allowed for diamond. Si and Ge are completely optically dark in this case. Hence for Ek , diamond has more allowed transitions than for E⊥ (assuming k similar oscillator strengths) and therefore its 0⊥ < 0 , whereas the opposite is true for Si and Ge.

Appendix A. Pseudopotential parameters 5. Summary

In this appendix we provide the details for our band structure calculations for the cubic phase for diamond, Si and Ge. The form factors of the cubic polytypes are required for

We have calculated the electronic band structures and dielectric functions for diamond, Si and Ge in the lonsdaleite 11

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

Table A.1. A comparison between targeted experimental transition energies taken from [125] and the calculated values for cubic diamond,

Si and Ge. For each material, the first column shows the targeted transitions, the second column shows the targeted energies and the third 0 0 column shows the converged results from fitting the pseudopotentials. The spin–orbit splitting energies are 1so = E8v − E7v and for Ge, 0 0 10so = E8c − E6c . Diamond Transition

Expt. (eV)

Si Calc. (eV)

− + 07c –08v − + 06c –08v + + 08v –06v − L+ 6c –L6v

7.3

7.3

12.9

12.854

26

26.779

10.5

10.508

X5c –X5v 1so 10so

12.9 0.006 —

12.89 0.006 —

Transition − + 07c –08v − + 06c –08v + + 08v –06v + L+ 6c –08 X5c –08+ 1so 10so

Ge

Expt. (eV) 3.35

Calc. (eV) 3.35

4.15

4.148

12.5

12.707

2.05

2.072

1.13 0.441 —

1.13 0.441 —

Transition − + 07c –08v − + 06c –08v + + 08v –06v + L+ 6c –08 X5c –08+ 1so 10so

Expt. (eV) 0.898 3.22 12.6

Calc. (eV) 0.898 3.223 12.286

0.76

0.76

1.16 0.297 0.2

1.081 0.297 0.2

Table A.2. Fitting parameters for the pseudopotential form factors (see equation (5)) and fitting parameters for spin–orbit splitting (see

equation (8)). Note that the form factors are in units of Ryd. Material

x1

Diamond Si Ge

444.305 8.2808 0.0791

x2 −1716.53 −54.1842 −0.5737

x3 0.0263 0.0116 0.0247

x4 6.2294 4.7922 −1.2269

µ1

µ2

12.515 0.0536 0.2413

— — 0

fitted to the discrete form factors of [113] and for diamond, initial fits to the discrete form factors of [130, 131] were tried. The adjustable parameters xi , and µl in equations (5) and (8) were then adjusted to fit the calculated band structure to the experimental energies of the band extrema of the cubic materials. A modified simulated annealing method was used for the fitting procedure (see [64] for more details). The five adjustable parameters for diamond and Si, and a total of six adjustable parameters for Ge were used to fit the calculated band structure to seven different experimental transition energies (which were obtained from [125]). Additional constraints were imposed to ensure the correct band ordering of valence states and conduction states. As shown in table A.1, our fits are in excellent agreement with experiment. The fitting parameters for the form factors, used for obtaining the results shown in table A.1, are given in table A.2, and the atomic form factors themselves are shown in figure A.1. The calculated ZB band structures of these group IV semiconductors are shown in figure A.2. Note that for cubic diamond and Si, the p-like conduction band

Figure A.1. Comparison of continuous atomic form factors, Vp , with the discrete form factors of [131] (for diamond) and [113] (for Si and Ge). Note that the y axis for diamond is on the right.

obtaining the band structures of the hexagonal polytypes. The Hamiltonian, given in equation (1), is diagonalized with a plane wave basis cutoff of G = |Ga/2π |2 ≤ 32. As a starting point, our continuous form factors for Si and Ge were first

Figure A.2. Calculated band structures, with the inclusion of spin–orbit interactions, for the cubic phase for (a) diamond, (b) Si and (c) Ge.

12

13

1

1

2

2

−2

−2

1

1

2

2

2

2

2

07±

08± 09±

−2

1

1

02± 03± 04± 05± 06±

1

1

01±

E

E¯

0

0

0

2

−2

−1

−1

1

1

C2 , C¯2 τ

−2

1

1

−1

−1

1

1

1

1

2C3

2

−1

−1

−1

−1

1

1

1

1

2C¯3

3

0

− 3

√

−1 √

1

−1

−1

1

1

2C6 τ

0

−1 √ − 3 √ 3

1

−1

−1

1

1

2C¯6 τ

0

0

0

0

0

−1

1

−1

1

3σv , 3σ¯ v

0

0

0

0

0

1

−1

−1

1

0 3C20 , 3C¯2 τ

±2

±2

±2

±2

±2

±1

±1

−1

1

3C200 , 3C¯2

00

group operation. The characters in the table will then be accompanied by an additional phase factor of exp(iτ · k).

∓2

∓2

∓2

±2

±2

±1

±1

±1

±1

I

0

0

0

±2

∓2

∓1

∓1

±1

±1

I¯ τ

∓2

±1

±1

∓1

∓1

±1

±1

±1

±1

σh , σ¯ h

±2

∓1

∓1

∓1

∓1

±1

±1

±1

±1

2S6

0

∓1 √ ± 3 √ ∓ 3

±1

∓1

∓1

±1

±1

2S¯ 6 τ

0

∓1 √ ∓ 3 √ ± 3

±1

∓1

∓1

±1

±1

2S3 τ

0

0

0

0

0

∓1

±1

∓1

±1

2S¯ 3

0

0

0

0

0

±1

∓1

∓1

±1

3σd , 3σ¯ d τ

Table B.1. Character table for the single and double groups of D6h . The symbol τ below the class operation here indicates that a translation by τ = [0, 0, c/2] is required in addition to the point

J. Phys.: Condens. Matter 26 (2014) 045801 A De and C E Pryor

J. Phys.: Condens. Matter 26 (2014) 045801

A De and C E Pryor

states are below the s-like state. In addition, these are also indirect gap semiconductors with the conduction band minima lying in very close proximity to the X valley along the 1 direction. In figure A.1 the final continuous form factors are compared with the popular discrete form factors of [113, 131]. The form factor for diamond is very close to the discrete ones for diamond [131]. However our form factors are much closer to the targeted experimental energies. For example the targeted zone center gap was 7.3 eV—which we exactly match, whereas the nonlocal form factors of [131] yield 8.22 eV. The discrete form factors are different for Si and Ge: they are similar at G = 3, 11 but notably differ at G = 4, which we attribute to the sharp cutoff for these discrete form factors. Overall our calculations reproduce the reflectivity spectra of these two semiconductors exceptionally well (see figure 5). Appendix B. Irreducible representations for lonsdaleite

For zone center states, sometimes 07 and 08 (or 05 and 06 in single-group notation) are used interchangeably [120, 132]. In this paper we have followed Koster’s convention [132] and the character tables for the zone center states are reproduced here for convenience. We also briefly mention how the irreducible representations are determined from the pseudopotential wavefunctions. For LD, the irreducible representations at the 0 point are those of D6h , which has all the symmetries of C6v , as well as inversion symmetry. The D6h character table for the single and double groups is as shown in table B.1 [132]. Due to the similarities in the structures, the high symmetry L and 0 points in cubic diamond have a one-to-one correspondence with those of the hexagonal structure. In single-group notation (following [132]) the cubic 01 , L1 and L3 states correspond to the hexagonal 01 , 03 and 05 points, as shown in figure B.1. The irreducible representations are determined as follows. For the pseudopotential wavefunctions, a twodimensional spinor representation can be used for the Bloch functions with spin. The spinor in the new set of coordinates, r0 , is related to the spinor in the old set of coordinates, r, by " # " # 0 ψk,↑ ψk,↑ (r) (r0 ) = D1/2 (θ ) · (21) 0 ψk,↓ (r0 ) ψk,↓ (r)

Figure B.1. Correspondence between the energy levels at the L and

0 points in fcc cubic crystals (zincblende and cubic diamond) to the 0 point in close-packed hexagonal structures (wurtzite and lonsdaleite) with and without spin–orbit interactions. The dashed lines indicate the correspondence between the states, and additional degenerate energy levels are shown in gray.

functions with spin, R∞ 0 Oi,j = R ∞ 0

0∗ (r0 ) · ψ (r) d3 r ψk,i k,j ∗ (r) · ψ (r) d3 r ψk,i k,j

(22)

where the spin indices {i, j} ∈ {↑, ↓}. The overlap integrals are evaluated over a unit cell. Now Tr[O] corresponds to the characters of the double-group character table for a given symmetry operation—which can be used to assign IRs. In case of WZ/LD, with the inclusion of spin the zone center states are 07 , 08 or 09 . These states are only distinguishable under a C6 operation which is a screw axis operation that involves a π/3 rotation about the c axis followed by [0, 0, c/2] translation. Note that C6 is not a symmetry operation if the origin is chosen to be at one of the atoms. The origin should be at the center of the triangle formed by nearest atoms of the same type. References

E ) and LE = where r0 = R(θ ) × r + τ, R(θ ) = exp(−iˆn · Lθ [Lx , Ly , Lz ] is a vector of SO(3) generators. Here D1/2 is the representation through which a spin-1/2 particle transforms: D1/2 (θ ) = exp(−iˆn · σE θ/2), where θ is the rotation angle about the axis nˆ . The single group, 0i , and double group, 0j , representations are related by 0i × D1/2 = αij 0j , where the αij are constants. The degenerate orbital states, as shown in figure B.1, split into as many states as there are irreducible representations in this direct product of 0i and D1/2 . To assign IRs for high symmetry points a 2 × 2 overlap matrix, O, is constructed whose matrix elements are the overlap integrals of the untransformed and transformed Bloch

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