PHYSICAL REVIEW 8 VOLUME 52, NUMBER 3 15 JULY 1995-I Electronic structure of the semimetals Bi and Sh Yi Liu and Roland E. Allen Department ofPhysics, Texas ACkM University, Coliege Station, Texas 77843-4242 (Received 22 December 1994; revised manuscript received 13 March 1995) We have developed a third-neighbor tight-binding model, with spin-orbit coupling included, to treat the electronic properties of Bi and Sb. This model successfully reproduces the features near the Fermi surface that will be most important in semimetal-semiconductor device structures, including (a) the small overlap of valence and conduction bands, (b) the electron and hole efFective masses, and (c) the shapes of the electron and hole Fermi surfaces. The present tight-binding model treats these semimetallic proper- ties quantitatively, and it should, therefore, be useful for calculations of the electronic properties of pro- posed semimetal-semiconductor systems, including superlattices and resonant-tunneling devices. I. INTRQDUCTIQN energy gaps in the vicinity of the Fermi energy for Bi and Sb make a multiband treatment necessary; (ii) the band Recently, several groups have reported the successful alignment of these semimetal-semiconductor superlattices fabrication of semimetal-semiconductor superlattices, in- is indirect in momentum space, so a theoretical treatment cluding PbTe-Bi, ' CdTe-Bi, and GaSb-Sb. Bulk Bi and must represent a mixture of bulk states from different Sb are group-V semimetals. They have a weak overlap symmetry points of the Brillouin zone '" (iii) the carrier between the valence and conduction bands, which leads effective masses (particularly along the [111] growth to a small number of free electrons and holes. They also direction) will play an important role in transport proper- exhibit small energy gaps in the vicinity of the Fermi en- ties (including resonant tunneling and the SMSC transi- ergy. Connected with these properties are high carrier tion), so a theoretical calculation must treat the effective mobilities, small effective masses, and a large characteris- masses correctly. tic length, which makes Sb and Bi ideal for studying and The envelope-function approximation uses the effective employing quantum confinement effects. The three above masses, band gaps, and momentum matrix elements as in- semimetal-semiconductor superlattices are all grown in puts. It can, therefore, satisfy requirement (iii), and is the [111] direction, along which the electron efFective useful for a superlattice state whose energy is close to a masses of Bi and Sb are relatively small and the quantum band edge of one of the bulk materials, when only one or confinement length (scaling as m' ') consequently large. two bulk states in each material are of dominant impor- Also, there is only a small lattice mismatch in each case tance. ' However, this method fails to satisfy require- between the (111) planes of semimetal and semiconduc- ment (i), because the boundary conditions are extremely tor. ' These three systems are, therefore, particularly complicated when many bands are involved. " It is also promising for quantum transport studies and quantum unsatisfactory with respect to requirement (ii), since it devices. cannot handle a superlattice state derived from two or The semimetals Bi and Sb have an indirect negative more bulk states with widely separated wave vectors. ' '" band gap, since the conduction-band minima (at the I. The empirical tight-binding (TB) method provides a points) lie (40 meV for Bi, 180 meV for Sb) lower than the better theoretical framework for this problem. It takes valence-band maxima (at the T point for Bi, and H points the effect of the full band structure into account, and the for Sb). With decreasing thickness of confinement in Bi boundary conditions for connecting wave functions and Sb, a semimetal-to-semiconductor (SMSC) transition across the interface are straightforward. Also, the quali- should occur when the energy shift becomes great enough tative behavior of the TB bands near and below the Fer- to raise the lowest electron subband to an energy higher mi level is usually in good agreement with experiment. than that of the uppermost hole subband. Hoffman Requirement (iii) above can be approximately satisfied by et al. reported a SMSC transition in CdTe-Bi at a criti- carefully adjusting the TB parameters to obtain the cal Bi-layer thickness of the order of 300 A. correct effective masses of the bulk materials. There is a These indirect narrow-gap semimetal-semiconductor tradeoff, of course: Some properties of the bands are superlattices and heterostructures have been suggested treated less accurately when one focuses on the effective for many potential applications, because of their masses, but these are expected to be less important in the unique transport and optical properties. Resonant tun- superlattice calculations. neling and negative differential resistance have been ex- Tight-binding models have been widely used for semi- perimentally observed. conductors. ' ' There has been much less work on the There have been relatively few theoretical attempts to semimetals Bi and Sb, except for an early effort by Mase' ' study the electronic and optical properties of these het- and more recent work by Xu et al. Mase' correctly erostructures. Any model of the band structure faces determined the locations and symmetries of the free car- stringent requirements: (i) The weak overlap and small riers of Bi. However, his treatment was not sufhcient to 0163-1829/95/52(3)/1566(12)/$06. 00 1995 The American Physical Society ELECTRONIC STRUCTURE OF THE SEMIMETALS O'i ANDAN Sb producerodu a reasonablea e band struc ok ey ban an correct- 1 ""du"d 'h e overla P and s ructures. ention was paiai ior o' the ~ "'"'""'"" e ermi ener . T conse- ' e, tl do noot yieldld the ccorrect carrier e , w ic areim portant for furth er ' calculaations of superir atticett states. we ' p p ' resen mo el for Bi and Sb. Instead ofjust considering tQe overall1 band s b d h dt'ld C e ective 1 masses of elect at the electron and holeo e pockets, and sm The re tmod 1(d fi db t p 1''ng devices, and semicoonductor heterostructures FIGIG. 1. Crystal structus ructure of Sb and Bii, showing first d secondn neighborsn {lab 1 d a atom II. BUI.K CRYSTAL STRUCTURE re res tdb pen circle. Th 'mi ' axes, and primitivemiiveive translation vectcors a&1~ a Bulk Bi and Sb have thee r o bo edra 2~ et a3) are also shown g h h 11 d' o d f ce . e take and trigonal axes to be the x ns, respec- b=a/c . 1. Th he hree primitive translation vectec ors off the ' attice are The relaelative position ofo th e two basis's atoms is given b a)= —1 2a, —&3/6a, 1/3c), d=(0, 0, 2p)c . ' *' a2=(1/2a, —&3/6a, 1/3c (2) The values of a, c, p, and g are listedlis in Table I, along1 ' a3=(0,&3/3a, 1/3c) . (3) Bnllouin zone for Bi an in g. 1. ' e regarded as a distorted The tthree correspondinon ing reciprocal-latticere ice vectors, ree de6ned by d eig'hb ors. The vectorsors from a central b;.a =2m.6" o t e nearest neieig hb ors are a- i j ij — a d tho t th e secondd nei g 1 a2 are given by 3 ~ b, =( —1, &3/3, b g, — III. TIGHT-BINDING MODELM bz=(1, —&3/3, b )g, (6) The tight-bindin modelm we use is ofo the Slater-Kost er 13=(0,2&3/3, b )g, (7) ing p . For the group-Vp- e1ements , the external electr 3 where configuration is $2p, p us a completee d ssh ell.. Only thee s TABLE I.. Crystal structureure parametersaram of Bi and Sb (RefRefs. 17 and 18) at 4.2 K. Bi Sb Lattice constants s in heexagonal system a (A) 4.5332 4.3007 e(A) 11.7967 11.2221 Rh ombohedral anng 1e 57'19' 57'l4' ' Internal dis placacement parameter p 0.2341 0.2336 eciprocal-lattice coconstant g(A ) 1.3861 1.4610 earest-neighb or d'istance d, (A) 3.0624 2.9024 ext-nearest-neighbor d'isiance d2 (A) 3.5120 3.3427 YI LIU AND ROLAND E. ALLEN and p states strongly mix to produce the valence and con- Instead of adding an s* state, we developed an sp duction bands of the solid. The basis functions are taken model, which includes third-neighbor interactions, but to be orthogonal s and p Lowdin orbitals. The crystal which treats details near the Fermi surface more careful- structure of Bi and Sb is rhombohedral, with three ly than the model of Ref. 15. The semimetallicity of Bi nearest, three second-nearest, and six third-nearest neigh- and Sb makes it reasonable that the interaction between bors. The relative positions and distances of these neigh- Lowdin orbitals (or Wannier functions) should extend boring are shown in Fig. 2 and Table I. The first and further than in more covalent systems (with band gaps), second neighbors are, respectively, above and below the like group-IV and III-V semiconductors. plane of the central atom. The total Hamiltonian of Bi or Sb is' A reasonable TB model must include second-neighbor H interactions for two reasons: (i) Only the combination of =Ho+H nearest- and second-nearest neighbors is suficient to H„=(A/4 mc )[VVXP] o.. satisfy all the symmetry requirements of the rhom- bohedral structure [with the bonding between (111) bi- Ho is the Hamiltonian without spin-orbit coupling, and layers eliminated if there are no second-neighbor interac- H„ the spin-orbit component, which couples p orbitals tions]; (ii) the values of the distances d, and d2 are very on the same atom.