Comprehensive study on band-gap variations in sp^3-bonded semiconductors: roles of electronic states floating in internal space
CComprehensive study on band-gap variations in sp -bonded semiconductors: roles ofelectronic states floating in internal space Yu-ichiro Matsushita and Atsushi Oshiyama Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan (Dated: January 3, 2017)We have performed electronic structure calculations to explore the band-gap dependence on poly-types for sp -bonded semiconducting materials, i.e., SiC, AlN, BN, GaN, Si, and diamond. In thiscomprehensive study, we have found that band-gap variation depending on polytypes is common in sp -bonded semiconductors; SiC, AlN, and BN exhibit smallest band gaps in 3 C structure, whereasdiamond does in 2 H structure. We have also clarified that the microscopic mechanism of the band-gap variations is attributed to peculiar electron states floating in internal channel space at theconduction-band minimum (CBM), and that internal channel length and the electro-static potentialin channel affect the energy level of CBM. INTRODUCTION
The development of the modern society has beenmostly attributed to semiconductor technology. In mostsemiconductors from elemental to compound, each atomforms sp bonds and take four-fold coordinated tetrahe-dral structure. It is well known that the crystal structureconsisting of sp bonds exhibit hundreds of polytypes [1].Their structural difference is in the stacking of the tetra-hedral units along the (cid:104) (cid:105) direction in cubic structure,and (cid:104) (cid:105) direction in hexagonal structure. Zincblendeand wurtzite structures are the most famous examplesof them. The Zincblende structure is represented bythe stacking sequence of ABC and the wurtzite by AB.Nomenclature adopted vastly is introduced here: eachpolytype is labeled by the periodicity of the stacking se-quence n and the symmetry (cubic or hexagonal) such as2 H (wurtzite), 3 C (zinblende), 4 H , 6 H , etc.These structural differences have been assumed to beminor in the electronic properties. It is because thereare no differences in local atomic structure up to the 2ndnearest neighbor. Valence bands consist of sp -bondingorbitals and conduction bands sp -antibonding. How-ever, it is reported that the stacking sequence affectselectronic properties considerably in silicon carbide (SiC)[2, 3]. SiC is indeed a manifestation of the polytypes:Dozens of polytypes of SiC are observed. Yet, surpris-ingly, the band gaps vary by 40 %, from 2.3 eV in 3Cto 3.3 eV in 2H despite that the structures are locallyidentical to each other in all the polytypes [3]. This phe-nomenon was difficult to be understood in conventionalchemical pictures.We have recently reported [4–6] the microscopic mech-anism of band-gap variations in SiC polytypes basedon the density functional theory (DFT) [7, 8]. It isfound that continuum states exist in conduction bandsof sp -bonded materials, and furthermore, such pecu-liar electron state appears at the conduction-band min-ima (CBM) in SiC polytypes. The wavefunction at theCBM is not distributed near atomic sites, but extends FIG. 1. (Color online) Residual norms of the wavefunctionsof the energy bands of 3 C -SiC (a) and 2 H -SiC (b). Theresidual norms are represented by the color and the size ofthe dots. The energy of the valence-band top is set to be0. The residual norm which is a measure of the floatingnature is calculated in the following procedure: From thepseudo-atomic orbitals { φ isolated i } of isolated silicon and car-bon atoms, we have composed orthonormal basis set { φ atom i } with the Gram-Schmidt orthonormalization. Then we havecalculated the squared residual norm, which is defines as (cid:12)(cid:12) | φ n k (cid:105) − (cid:80) i (cid:12)(cid:12) φ atom i (cid:11) (cid:10) φ atom i | φ n k (cid:11)(cid:12)(cid:12) for each band n at k point. (or f loats ) in interstitial channels without atomic orbitalcharacter. This fact is clearly shown in Fig. 1. The figureshows the calculated residual norms of the wavefunctionof 3 C -SiC after projecting it to the s - and p -atomic or-bitals. While the wavefunction in valence bands is welldescribed by s − and p − atomic orbitals, conduction elec-trons cannot. This f loating character at the CBM is animportant key to reveal the microscopic mechanism of theband-gap variations. Channel length changes dependingon polytypes: 3 C structure has infinite channel lengthalong the (cid:104) (cid:105) direction, while similar channel structureis seen also in 6 H structure with the length of about7 a / √ (cid:104) ¯2201 (cid:105) which is slanted relative to (cid:104) (cid:105) direction with a a lattice constant. F loating states atthe CBM extends in the internal space. Therefore, chan- a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n FIG. 2. (Color online) Calculated band gaps as a functionof channel length for 24 SiC polytypes with Heyd-Scuseria-Ernzerhof (HSE) functional [9–13]. The channel length isdefined as the number of bilayers along the longest interstitialchannels. The orange curve represents a fitting function of y = 2 .
21 + 21 . / ( x + 1 . . Specific values of each plot areshown in Appendix. nel length and channel shapes are decisive in the positionsof the CBM in energy space. In fact, we have found thatthe energy level of the floating state strongly dependson channel length via quantum confinement effect (seeFig. 2) [5].We have already clarified the microscopic mechanismof band-gap variations in SiC cases as shown in Fig. 1and Fig. 2. From the similarity of the crystal structures,however, it is expected that the similar phenomena canbe seen also in other sp -bonded semiconductors. In thisstudy, we have investigated the possibility of band-gapvariations in wide range of sp -bonded semiconductors,the effects of each atomic character on the electronicproperties, and the differences between elemental andcompounds semiconductors for Si, diamond, AlN, BN,and GaN. CALCULATION CONDITIONS
Total-energy band-structure calculations were per-formed based on the DFT [7, 8] in this study using theplane-wave-basis-set ab initio program package, TAPP[14–16]. Our calculations have been performed in thegeneralized gradient approximations (GGA) [17, 18]. Nu-clei and core electrons are simulated by either norm-conserving [19] pseudo-potentials in the TAPP code. Wegenerate norm-conserving pseudo-potential to simulatenuclei and core electrons, following a recipe by Troul-lier and Matins [19]. The core radius r c is an essentialparameter to determine transferability of the generatedpseudo-potential. We have examined r c dependence ofthe calculated structural properties of benchmark mate-rials and adopted the pseudo-potentials generated with the following core radii in this paper: 0.85 ˚A for Si 3 s ,and 1.16 ˚A for Si 3 p , 1.06 ˚A for Ga 4 s and 4 p , and 1.48˚A for Ga 4 d , 0.64 ˚A for N 2 s and 2 p , 0.85 ˚A for C 2 s and2 p , 1.06 ˚A for Al 3 s , 3 p , and 3 d , 0.847 ˚A for B 2 s and 2 p . RESULTS AND DISCUSSION
We first present our calculated band gaps for struc-turally optimized polytypes in the next subsection. Inthe following subsections, we describe the floating statesin each polytype and the roles of floating states on bandgap variations.
Optimized structures and their band gaps
First, we have theoretically determined lattice con-stants in the hexagonal plane and along the stacking di-rection, a and c in the GGA. The obtained a and the ratio c/na of each polytype are listed in Table I, where n repre-sents the periodicity of stacking bilayers. The differencesof c/na among the polytypes are found to be extremelysmall. This fact means that the distortion along the c-axis is quite small. Our calculated lattice constants agreewith available experimental data with an error of at most2 %. Table I also shows the calculated total energy dif-ferences (∆ E ) among the geometry-optimized polytypes.The table includes some polytypes not observed yet, .e.g,6 H -AlN. Yet, it is likely that these polytypes are synthe-sizable since the total energy difference is small, being inthe range of 50 meV or less per molecular unit.The most energetically favorable polytype in SiC isthe 6 H followed by the 3 C with the energy increase of1.2 meV per SiC molecular unit. It is said that 4 H isalso one of the most energetically favorable polytypes[29]. Yet, the 6 H structure is an often observed polytypein experiments, and our calculations show quite smalldifference in total energy than that of 4 H polytype by0.1 meV. Therefore, we discuss the 6 H polytype in thisstudy. The least energetically favorable polytype is 2 H whose total energy is higher than 6 H by 7.1 meV perSiC. We have found that, compared with other materi-als, SiC exhibits smaller energy difference among poly-types. This is derived from the balance of ionicity andcovalency. The materials with dominantly ionicity preferhexagonal structure. SiC is a exquisite material possess-ing a delicate balance of ionicity and covalency to exhibithundreds of polytypes [30]. As for the other materials,most stable structure of each material is 2 H -AlN, 2 H -BN, 2 H -GaN, 3 C -Si, and 3 C -C, respectively. The moststable structures in other materials are commonly ob-served in experiments.Next we have calculated electronic band structure foreach material. The calculated results are shown in Fig. 3.Remark that we have adopted a unit cell of the 6 H struc- TABLE I. Calculated hexagonal lattice constant a and the ratio c/na for different polytypes labeled as either nH or nC ( n = 3) of the various sp -bonded semiconductors. Calculated total energies per formula unit are also shown. The values arerelative to the energy of the corresponding the most stable structure. Materials a [˚A] c/na ∆E [meV]this work Expt. this work Expt.2 H -SiC 3.085 3.076 (Ref. [20]) 0.8217 0.8205 (Ref. [20]) 7.13 C -SiC 3.091 3.083 (Ref. [21]) 0.8165 0.8165 (Ref. [21]) 1.26 H -SiC 3.091 3.081 (Ref. [22]) 0.8180 0.8179 (Ref. [22]) 02 H -AlN 3.117 3.110 (Ref. [23]) 0.8103 0.8005 (Ref. [23]) 03 C -AlN 3.112 3.090 (Ref. [24]) 0.8165 0.8165 (Ref. [24]) 41.96 H -AlN 3.112 − − H -BN 2.556 2.553 (Ref. [25]) 0.8252 0.8265 (Ref. [25]) 35.53 C -BN 2.561 2.557 (Ref. [25]) 0.8165 0.8165 (Ref. [25]) 06 H -BN 2.556 2.500 (Ref. [26]) 0.8203 0.8293 (Ref. [26]) 9.62 H -GaN 3.255 3.189 (Ref. [27]) 0.8156 0.8130 (Ref. [27]) 03 C -GaN 3.255 3.175 (Ref. [28]) 0.8165 0.8165 (Ref. [28]) 15.36 H -GaN 3.255 − − H -Si 3.853 − − C -Si 3.863 3.863 (Ref. [28]) 0.8165 0.8165 (Ref. [28]) 06 H -Si 3.858 − − H -C 2.503 − − C -C 2.514 2.519 (Ref. [28]) 0.8165 0.8165 (Ref. [28]) 06 H -C 2.508 − − ture even for 2 H and 3 C structures to facilitate the com-parison among the polytypes. From the figures, the va-lence bands of the three polytypes resemble each other ineach material. The tiny differences are attributed to thedifference of the symmetries by which degenerate statesin the high-symmetry structure split. The valence-bandtop is located at Γ point in all the polytypes in all thematerials. In contrast, the conduction bands are quali-tatively different among polytypes in spite of their struc-tural similarity in the local atomic arrangement. In theSiC polytypes, the CBM is located at K point in the 2 H -structure, whereas it is at M point in the 3 C -, and 6 H -structure. The X point in the cubic Brillouin zone (BZ)is folded on the M point in the hexagonal BZ. Further-more, the lowest conduction band in the 3 C structure isisolated and shifts downwards substantially, making theband gap narrower by 0.7 - 0.9 eV than those in the 6 H and 2 H polytypes. The calculated energy bands for othercompounds clearly show the same feature as in SiC, i.e.,the CBM in the 3 C -AlN, 3 C -BN is located at M point,whereas that in the 2 H -BN, and 2 H -diamond is locatedat the K point.The calculated and experimental band gaps for thepolytypes are given in Table II. Overall features of thecalculated band-gap variation are in accord with the ex-perimental values. It is clearly seen that the GGA un-derestimates energy gaps by about 50% because of theshortcoming inherent in the GGA. If necessary, the quan-titative description of the energy gaps is possible us-ing more sophisticated schemes of the GW [31–33] for quasiparticle-self energy or HSE functional [9–13] for theexchange-correlation energy. Yet, the relative differencein the energy gap calculated by the GGA among the poly-types is well reproduced, i.e., calculated results show theband gap of the 3 C -SiC is smaller than that of the 2 H -SiC by 0.936 eV, which corresponds to the experimentalone, 0.93 eV.From the Table II, it has been found that the largeband-gap variation is not limited to the SiC polytypes.For AlN and BN, the energy gap decreases substantiallyin the 3 C structures by 0.9 eV and 0.8 eV, respectively.In the case of AlN, the CBM at M point shifts down-wards substantially, so that the transition between thedirect gap in the most stable 2 H -structure and the indi-rect gap in the metastable 3 C -structure takes place. Thisresult gives good agreement with the observed experi-mental facts. In contrast, for the diamond polytypes, theband-gap decrease can be seen not at the 3 C -structure,but at the 2 H -structure: the energy gap varies from 4.521eV in the 6 H -structure to 3.406 eV in the 2 H -structure. Floating states in C structure We discuss the microscopic mechanism of the band-gap variation in this subsection. As we have clarified inthe previous papers [4–6], continuum-state like characterat CBM in SiC polytypes plays important roles in theband-gap variation. As shown in Fig. 4 (a), the CBMof 3 C -SiC extends (or f loats ) in internal channel cavity FIG. 3. (Color online). Band structures calculated by theGGA. The energy of the valence-band top is set to be 0. Inthese calculations, we adopted supercell calculations, so thatthe number of electrons is equal to each other for easy com-parison and they have the same Brillouin zone. Note that 3 C structures are also calculated in the hexagonal supercell, thusthe X point in cubic cell being folded to M point. without atomic orbital character, i.e., (cid:104) (cid:105) channels. Siatoms are positively charged because of the differencesin electronegativity between Si and C atom in SiC crys-tals (see Fig. 5(a)). Thus, this charge transfer causes theelectro-static potential at the tetrahedral ( T d ) interstitialsites surrounded by 4 Si atoms lower. These T d intersti-tial sites construct the (cid:104) (cid:105) channels where the floating TABLE II. Calculated, (cid:15) gap , and experimental, (cid:15) expt . , energygaps of the 2 H , 3 C and 6 H structures for various sp -bondedsemiconductors. Experimental data are taken for SiC fromRef. [3], for 2 H -AlN from Ref. [34], for 3 C -AlN from Ref. [35],for 3 C -BN from Ref. [36], for GaN, Si, and diamond fromRef. [28]. Materials (cid:15) gap (cid:15) expt . (eV) (eV)2 H -SiC 2.355 (indirect) 3.33 (indirect)3 C -SiC 1.419 (indirect) 2.40 (indirect)6 H -SiC 2.077 (indirect) 3.10 (indirect)2 H -AlN 4.233 (direct) 6.23 (direct)3 C -AlN 3.328 (indirect) 5.34 (indirect)6 H -AlN 3.817 (indirect) − H -BN 5.251 (indirect) − C -BN 4.487 (indirect) 6.4 (indirect)6 H -BN 5.190 (indirect) − H -GaN 1.622 (direct) 3.28 (direct)3 C -GaN 1.489 (direct) 3.47 (direct)6 H -GaN 1.533 (indirect) − H -Si 0.477 (indirect) − C -Si 0.660 (indirect) 1.17 (indirect)6 H -Si 0.639 (indirect) − H -diamond 3.406 (indirect) − C -diamond 4.246 (indirect) 5.48 (indirect)6 H -diamond 4.521 (indirect) − state extends having the maximum amplitude at T d sites.This lowering of electro-static potential at T d interstitialsites shifts the energy level of the floating state down-wards.First we discuss the character of the CBM in other sp -bonded semiconductors in 3 C structure in Fig. 4. Asclearly seen in Fig. 4, the Kohn-Sham (KS) orbitals atthe CBM at M point in 3 C structure on (0¯11) planeobtained in the GGA calculations are similar to that ofSiC, indicating that the CBMs at M point in other sp -bonded semiconductors are also floating states extendingin (cid:104) (cid:105) channels. These structures have similar channelfeatures as SiC: T d interstitial sites surrounded by cationsform (cid:104) (cid:105) channels, rendering the energy level of floatingstates lower.In contrast, elemental semiconductors, such as Si anddiamond, exhibit no such band-gap variation in 3 C struc-ture. It is because there is no charge transfer unlike thecompound ones. This fact makes no potential loweringat T d interstitial sites, causing no band-gap narrowing inelemental semiconductors. FIG. 4. (Color online) Contour plots of the calculated Kohn-Sham (KS) orbitals of the conduction-band minimum at M point for 3 C -SiC (a), 3 C -AlN (b), 3 C -BN (c), and 3 C -GaN(d) on the (0¯11) (left panel labeled (i)) and the (110) (rightpanel labeled (ii)) plane. The M point which we discuss corre-sponds to X = (0 , , π/a ) in cubic BZ. The mark ’X’ depictsthe tetrahedral ( T d ) interstitial sites surrounded by cations.In Fig. (a), brown(large) and white(small) balls depict sili-con and carbon atoms, respectively. In Fig. (b), brown(large)and white(small) balls depict aluminum and nitrogen atoms,respectively. In fig. (c), green and white balls are boron andnitrogen atoms, respectively, in Fig. (d), green and white isgallium and nitrogen atoms, respectively.FIG. 5. (Color online) Sketches of two tetrahedral ( T d ) in-terstitial sites in the 3 C -polytypes (a), and the 2 H -polytypes(b): One is surrounded by 4 cations and the other is by 4anions. The blue balls represent cations, and the red onesanions. In the 3 C -polytype (a), the cation-surrounded inter-stitial site is spatially separated from the anion one. On theother hand, they overlap each other in the 2 H -polytype. Next we discuss why band-gap variations are not seenin 2 H structure. As mentioned above, charge transferplays important roles in the substantial band-gap de-crease in the 3 C structures. On the other hand, in the2 H -structure such a cation-surrounded channel is absent.The internal space surrounded by cations overlaps con-siderably with that by anions in the 2 H structure [SeeFig. 5]. The cation-surrounded interstitial site is veryclose to the anion-surrounded one with the separation FIG. 6. (Color online) Contour plots of the calculated Kohn-Sham (KS) orbitals of the conduction-band minimum at M point for 6 H -SiC on (11¯20) plane in (a), and (0001) in plane(b). The brown and white balls depict Si atoms and C atoms,respectively. The mark ’X’ represents the tetrahedral ( T d )interstitial sites surrounded by Si atoms. of d/
3, where d is the bond length between silicon andcarbon atoms. In fact, the electro-static potential at thecation-surrounded interstitial sites is almost the same asthat at anion-surrounded ones within 0.1 eV in the caseof SiC. Therefore, in the 2 H structures, charge transferdoesn’t cause the static potential lowering, leading to noband-gap variation. Floating states in H structure In this subsection we discuss the electronic structure atCBM in 6 H structure. As mentioned above, in the 3 C structures, the (cid:104) (cid:105) channels with infinite length playimportant roles in the variations in energy gaps. Similarchannel structure is seen also in 6 H structure. There ischannels with the length of about 7 a / √ (cid:104) ¯2201 (cid:105) which is slanted relative to (cid:104) (cid:105) direction with a alattice constant. The calculated KS orbital at the CBMof SiC is shown in Fig. 6, where the wavefunction hasthe maximum amplitude at the tetrahedral T d intersti-tial sites, and f loats in the finite-length channels. Dueto quantum confinement of the wavefunction, however,the kinetic energy at the CBM is greater than that in3 C structure and the band gap of 6 H structure becomeswider [See Fig. 3 and Table. II] by 0.66 eV. The relationsbetween the channel length and band gap is clearly shownin Fig. 1. Similar tendency is observed also in other sp compound semiconductors. AlN, and BN in 6 H struc-ture exhibit 0.49 eV, and 0.7 eV wider band gap thanthat in 3 C , respectively. Floating states in H structure In this subsection we discuss the electronic structureat CBM in 2 H structure, and give an explanation whydiamond exhibits smallest band gap at 2 H structure. FIG. 7. (Color online) Contour plots of the calculated Kohn-Sham(KS) orbitals of the conduction-band minimum at K point for 2 H -SiC (a), 2 H -AlN (b), 2 H -BN (c), 2 H -GaN (d),and 2 H -Si on (1¯100) (left panel) and (0001) (right panel)planes. In Fig. (a), the brown and white balls depict siliconand carbon atoms, in Fig. (b), the pink and sky blue balls arealuminum and nitrogen atoms, in Fig. (c), green and whiteballs are boron and nitrogen atoms, in Fig. (d), green andwhite are gallium and nitrogen atoms, respectively, and inFig. (e), the blue balls are silicon atoms. The floating state at M point in the 3 C -polytypes isdistributed along the (cid:104) (cid:105) channel which is slanted rel-ative to (cid:104) (cid:105) direction. In the 2 H structure, such aslanted channel is absent. Instead, there are channelsalong (cid:104) (cid:105) and (cid:104) (cid:105) directions. We have found thatthe CBM at K point in the 2 H structure floats in the (cid:104) (cid:105) channel in SiC case (Fig. 7(a)). This floatingstate is distributed solely in the (cid:104) (cid:105) channel with itsphase changing consecutively by exp( i π/ K point.We then expect that the existence of the floating statesis common to 2 H structure in most sp -bonded semi-conductors. We have therefore examined 2 H -AlN, 2 H -BN, 2 H -GaN, 2 H -Si, and 2 H -diamond. Fig. 7 shows theCBM at K point of them. We clearly see the floatingstates in all materials. Most remarkable case is diamond.Diamond polytypes show substantial band-gap decreasein the 2 H structure. Fig. 7(e) shows the KS orbital atthe CBM of the 2 H -diamond. The KS orbital is alsodistributed not near atomic sites, but f loats in internalspace. Fig. 8(a) shows the residual norm of wavefunc-tion after the projection to the s -, and p -atomic orbitals. FIG. 8. (Color online) (a) Residual norms of the wave-functions of the energy bands of 2 H -diamond. The resid-ual norms are represented by the color and the size of thedots. The energy of the valence-band top is set to be0. The residual norm which is a measure of the float-ing nature is calculated in the following procedure: Fromthe pseudo-atomic orbitals { φ isolated i } of isolated carbonatoms, we have composed orthonormal basis set { φ atom i } with the Gram-Schmidt orthonormalization. Then we havecalculated the squared residual norm, which is defines as (cid:12)(cid:12) | φ n k (cid:105) − (cid:80) i (cid:12)(cid:12) φ atom i (cid:11) (cid:10) φ atom i | φ n k (cid:11)(cid:12)(cid:12) for each band n at k point.(b) Energy analyses of KS orbitals in 2 H -diamond. Thekinetic-energy contribution (cid:15) kin and the Hartree-energy con-tribution (cid:15) H to the orbital energy of each KS state for K point. The abscissa represents the i th KS state from thevalence-band bottom and the 25th state is the conduction-band minimum. As much as 0.47 are floating character at the CBM inthe 2 H -diamond, while it is only 0.04 at the valence-band top. This fact shows that this KS orbital at theCBM does not consist of atomic orbitals neither. Themaximum amplitude is on the axis of the (cid:104) (cid:105) chan-nels, and the floating state is distributed in the horizontalchannels. We have found that the floating state inducesband-gap variations also in this system.In order to clarify the reason for the energy gain ofthe floating state, we show the energy analyses of KSorbitals at K point in Fig. 8(b). According to the fig-ure, floating state reveals the kinetic energy gain by ex-tending in the channels broadly. Another energy gainis Coulomb energy gain, because the floating state isdistributed far from atomic nuclei, which core electronsare distributed near around. In addition, we have foundthat the electro-static potential from the ions energy gainplays important roles in the decrease of the energy gapin the 2 H -diamond. In fact, the local atomic structurearound the interstitial sites is different between the 3 C and 2 H structure. The interstitial site in 2 H structure issurrounded by six nearest neighbor atoms, and six nextnearest neighbor atoms. In contrast, the interstitial sitein the 3 C polytype is surrounded by four nearest neigh-bor, and three next neighbor atoms. That is, in the 2 H polytypes, the number of neighbor atoms around the in-terstitial sites is larger compared with other polytypes.This structural difference makes the electro-static poten-tial at the interstitial sites in the 2 H -diamond lower thanthat in the 3 C -diamond by 0.589 eV from the DFT calcu-lation. That value corresponds to the band-gap variationin the 2 H structure, 0.8 eV smaller than 3 C . CONCLUSIONS
We have performed electronic structure calculations toexplore the band gap dependence on polytypes for sp -bonded semiconducting materials, i.e., SiC, AlN, BN,GaN, Si, and diamond. We have found that band-gapvariation is common in sp -bonded semiconductors; SiC,AlN, and BN exhibit smallest band gaps in 3 C structure,whereas diamond does in 2 H structure. We have alsoclarified that the microscopic mechanism of the band-gap variations is attributed to peculiar electron states f loating in internal channel space at the conduction-band minimum (CBM), and that channel length andelectro-static potential in channel space affect the energylevel of the floating states; In compound semiconductors,charge transfer causes the elecro-static potential in chan-nel lower in 3 C structure, while elemental semiconductorsshow lower electro-static potential in 2 H rather than in3 C . ACKNOWLEDGEMENTS
This work was supported by the Grant-in-Aid forYoung Scientists (B) conducted by MEXT, Japan, un-der Contract No. 93002181. This research (in part) usedcomputational resources of COMA provided by Interdis-ciplinary Computational Science Program in Center forComputational Sciences, University of Tsukuba, and theSupercomputer Center at the Institute for Solid StatePhysics, The University of Tokyo.
APPENDIX
Here we show the specific values of band gaps of SiCpolytypes in Table. III. [1] A. R. Verma and Krishna, Polymorphism and Polytyp-ism in Crystals (Wiley, New York, 1966)[2] T. Kimoto, J.A. Cooper,
Fundamentals of Silicon Car-bide Technology: Growth, Characterization, Devices, andApplications (John Wiley & Sons Singapore Pte. Ltd,2014).[3] For a review, G. L. Harris (ed.) ,
P roperties of SiliconCarbide , (INSPEC, London, 1995).[4] Y.-i. Matsushita, S. Furuya, and A. Oshiyama, Phys.Rev. Lett. , 246404 (2012).[5] Y.-i. Matsushita, and A. Oshiyama, Phys. Rev. Lett. , 136403 (2014). TABLE III. Calculated energy gaps, (cid:15) gap , of various SiC poly-types.
Polytypes (cid:15) gap
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