ELECTRONIC STRUCTURE of the SEMIMETALS Bi and Sb 1571

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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 properties 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 rhombohedral 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. V is the total crystal potential, and o. close, so second-neighbor interactions are expected to be represents the Pauli spin matrices. significant. Our experience in fitting the bands demon- We denote the s and p orbitals on the two atoms of the strates that they do indeed play an important role. primitive cell by ~sio), ~x. io ), ~yicr ), ~zio ). The site in- The peculiarity of the band structures of these semime- dex i is either 1 or 2, and the spin index o either 1 or $. tallic materials is the overlap between the highest valence In this basis, the matrix elements of Ho with the same band and the lowest conduction band, which creates spin can be easily calculated by use of the two-center ap- ' small and equal numbers of free electrons at point I. and proximation given by Slater and Koster. The spin-orbit free holes at point T (for Bi) or point H (for Sb). The component H„can be easily represented in the total an- ' question of producing this indirect negative band gap is a gular momentum basis: for example, delicate one for a tight-binding model. For sp -bonded materials with the diamond and zinc-blende structures, (12) Dow' Vogl, Hjalmarson, and added an excited s state to where k is the spin-orbit coupling parameter. The trans- the sp basis set for each atom and successfully repro- formation ' between these two bases allows the total duced the lowest conduction bands even in indirect-gap Hamiltonian to be expressed in either basis. In the Ap- semiconductors. This s* state the lower antibond- repels pendix, we give the expression for the Hamiltonian ma- ing p-like conduction band and presses the indirect trix elements in the ~sio ), ~xio ), ~yio ), ~zio ) basis. conduction-band minima down in energy. The difBculties Our TB model has 14 adjustable parameters: E„E, in extending a TB model to distant neighbors with many rz' I I II Il ~ ~ " ~ ~ ~ ~ ~ adjustable parameters were avoided introducing an s* ~sscr~ Vspo. ppo. Vppm~ sso. spo. ppo. ppvr~ sso. spa by V", V" . and E are the on-site orbital energies. state instead. However, we have found in our calcula- E, The unprimed, primed, and double-primed parameters tions that the interaction between the s* and states is p are, respectively, for first, second, and third neighbors. not adequately selective in the A7 structure. The sp s* The spin-orbit coupling parameter k is taken to be ap- model fails to give the fine details of the band structure proximately 0.6 eV for Sb and 1.5 eV for Bi.' ' near the Fermi surface in Bi and Sb. IV. DETERMINATIC)N GF THE TB PARAMETERS

For the purpose of studying electronic transport and SMSC transitions in semimetal-semiconductor superlattices and heterostructures, we concentrate on features in the energy bands near the Fermi surface instead of the global band structures. We first set out to reproduce (i) the overlap between the highest valence and lowest conduction bands, and (ii) the Fermi energy of the free carriers. These are the basic properties which determine the semimetallicity of Bi and Sb. In our calculations, however, we found that the overlap and Fermi energy are in- sensitive to the TB parameters, and can be obtained correctly from different sets of parameters. Another important consideration is (iii) the efFective masses of the free carriers, especially along the [111]direction. This is the growth direction for the semimetal-semiconductor superlattices in which we are interested. The effective mass is important for determining the SMSC transition and the transport properties, and must, therefore, be fitted care- FIG. 2. Brillouin zone for the rhombohedral structure. fully. Still another important factor is (iv) the shape of 52 ELECTRONIC STRUCTURE QF THE SEMIMETALS Bi AND Sb 1569 the Fermi surfaces for electrons and holes. The external at high-symmetry points of the BZ. momenta and external areas are very sensitive to the TB Within the TB model described previously, the explicit parameters. The density of free carriers (n or p with expressions for two of the bands along the I T(-[111]) n =p) is determined by the volume of the electron and direction are given by hole pockets. The final important feature is (v) the band — gaps near the Fermi level. In Bi, the I, point, where the E(145 T45 )=A+3( Vpq + Vpp„)+1/3A. electron pocket is located, has a direct of 13.6 energy gap +Qm +I +2ml cos(a, . (13) meV (Ref. 24) at low temperature. Nonparabolicity is ex- k), pected to be important in this case. We also were con- where k=(u, u, u) with 0 u 0.5, and cerned with the band structure farther away from the — Fermi level, but this is the least important aspect for our m =2 cos a V~ +(3 2cos a) V~~ (14) and was taken into account when purposes, only there 1=2cos a'V' +(3—2cos a')V' (15) was no risk of inordinately compromising the first five properties. For the I Point, k=(0, 0,0), Eq. (13) gives Several semiconductors with the diamond or zinc- E(l )=E +3(V + ", . blende structure contain Sb or elements near to Sb and Bi V, .)+1/3~+lm+Il (16) in the periodic table. These include a-Sn, InSb, and For the T Point, k=(0.5, 0.5, 0.5), Eq. (13) gives GaSb, for which TB parameters have been determined by ' several authors. ' To obtain the first and second- E(T45)=E~+3(Vp~ + V" )+1/3A+im —li . neighbor-interaction parameters for Sb and Bi, we as- sume that the parameters of e-Sn, InSb, and GaSb and The bandwidth associated with (16) and (17) those of Sb and Bi are connected by a d scaling rule. 6(T4~ —I 4~)=4cos a'V~~ +(6—4cos a')V' . (18) (The crystal symmetries are different, of course: a-Sn, — InSb, and GaSb are cubic, while Bi and Sb are rhom- Note that b.( T4~ I 4~ ) is only determined by the bohedral. ) The parameters obtained from this scaling are second-neighbor interaction parameters V' and V' then adjusted slightly to fit the theoretical curves to the In Bi, the hole pocket is at the T point with symmetry existing experimental data, with emphasis on properties T~~.'4'8 From Eq. (13), we can derive an explicit expres- (i) —(v) discussed above. The third-neighbor interactions sion for the effective mass of the hole at the maximum of are regarded as small perturbations, adjusted freely the band along the [111]direction: without imposing the d rule (see Fig. 3). For materials having the diamond or zinc-blende struc- (19) ture, with only nearest-neighbor interactions included, it is a simple matter to relate the TB parameters to the en- Equation (19) is quite useful when we fit the effective ergy bands at high-symmetry points in the Brillouin masses of the free carriers. ' ' zone. ' In this case, the TB parameters can be easily We find, to an excellent approximation, that V„and fit the our howev- adjusted to experimental data. In fits, V,', can be related to the energies of the two lowest-lying rhom- er, we have to deal with (a) the lower-symmetry valence bands at the I and T points. At I, the relations bohedral structure, (b) third-neighbor interactions, and are (c) the spin-orbit coupling. The Hamiltonian matrices of E(l )=E, " +A+B " Bi and Sb are consequently much more complicated even 6 +6V,, /[E, +6V,, E— —6V~~„+( A+C)], (20) where A =3(V„+V,', ), (21) (22) B =3(V,& cosy+ V,~ cosy'),

C=3(V cos y+ V~ sin y)

+3( V~ cos y'+ V„'~ sin y') . (23)

If ~A+C~, ~B~ ((~E, E~~ is well satisfied, w—e can obtain a simple expression for the energy difference of the I 6+ and I 6 states: — ', . (24) E(l;) E(r,+)=~6(V„.+V, .)~ FIG. 3. Positions of the first-, second-, and third-nearest neighbors projected onto the plane perpendicular to the trigonal At T, the relations are axis. The three first- and three second-nearest neighbors are in " E(T6 )=E, +6V,", A'+B' /[E, +6V,, the planes above and below the plane in which the central atom E— lies, while the six third-nearest neighbors are in the same plane —6V"„+(A '+ C') ], as the central atom. ( —central atom and third neighbors; 0 —nearest neighbors; 6—second neighbors. ) (25) 1570 YI LIU AND ROLAND E. ALLEN 52

0 and second-nearest-neighbor distance (in and TB pa- TABLE II. Nearest-neighbor distance d & d2 A) rameters for Bi, Sb, a-Sn, InSb, and GaSb. Those of a-Sn, InSb, and GaSb are obtained from Ref. 12. For the compounds, the upper value of E, or Ep corresponds to the anion, and the lower to the cation. The upper (lower) V,p corresponds to the s orbital on the anion {cation), and a p orbital on its nearest neighbor.

d2 E, Vspo Vppm Bi 3.062 3.512 —10.906 —0.486 —0.608 1.320 1.854 —0.600 Sb 2.902 3.343 —10.068 —0.926 —0.694 1.554 2.342 —0.582 a-Sn 2.81 —5.670 1.330 —1.417 1.953 2.373 —0.687 InSb 2.81 —8.016 0.674 —1.380 1.640 2.289 —0.619 —3.464 2.916 1.987 GaSb 2.64 —7.321 0.855 —1.539 2.148 2.459 —0.637 —3.899 2.915 2.021

I I II ", VII Vsscr Vspo Vppm V, Vsp cr Bi —0.384 0.433 1.396 —0.344 0.156 Sb —0.366 0.478 1.418 —0.393 0.352

~'+&'I && — where If I IB'I IE, E~l is well satisfied, it follows that A'= —3(V„—V,', ), (26) — — B'= 3( V, cosy Vz cosy'), (27) T6+ — = ', E( ) E( T6 ) i 6( V„—V, ) i . (29) C'= —3( V cos'y+ V»~in'y) cos2y'+ V'~ y') +3(V' sin . (28) For Bi and Sb, since the s levels lie about 9 eV lower than ' the p levels, the above two inequalities are usually true. Equations (24) and (29) are used to fit the two lowest valence bands, while Eqs. (13), (16), and (17) are used for a—Sn Bi(1) the higher-lying valence and conduction bands. Equation + InSb Bi(2) (19) gives the exact formula for the effective mass of the ~ GQSb Sb(1) hole at the maximum of the valence band along the [111] sb(2)- direction. V =2.335 /md The TB parameters, which give the best agreement with experimental data, are presented in Table II, along with the parameters for a-Sn, InSb, and GaSb. The values of V„and V,', of Bi and Sb obtained from these three cubic semiconductors by assuming d scaling fail to produce the experimental results for the two lowest- X V =1.836 /md lying valence bands. The values in Table II are calculat-

Ol ed from Eqs. (24) and (29), instead of the d scaling rule. E O However, when the values of and V' 0 0 V, , V, V', „ O. are plotted against the bond length in Fig. 4, d scaling is found to be approximately satisfied. We mention that

x V =-0.64fl /md TABLE III. Energy levels of Bi (in eV) at the symmetry points T, I, L, andX. T6 —12.2442 I 6+ —14.0000 L, —11.9545 X, —12.1649 T+ —11.1331 I —8.0870 L, —11.5966 X, —10.3162 + 0.08 0.10 0.1 2 0.14 0.16 0.18 T —1.1798 I —2.5364 L, —1.7896 X, —5.6525 d '(L') —1.1196 r, —1.1289 L, —1.6669 X, —4.1632 T 0.0111 I"+ —0 8238 L, —0.0403 X —3.1047 FIG. 4. Interatomic TB parameters V» (Vpp ), V» (Vpp ), T 0.3813 I 0.2430 —0.0267 2.3508 and Vp multiplied by the square of the bond length, vs the L, X, T6 0.9529 I 6 1.6893 L, 0.8024 X, 3.2956 bond length d. V,p for the compounds InSb and GaSb is ob- T4+5 1.4754 I 1.7878 9201 tained by averaging the two values in Table II. 4g L, 0. X, 4.4031 52 ELECTRONIC STRUCTURE OF THE SEMIMETALS Bi AND Sb 1571

TABLE IV. Comparison of characteristic energy levels (in eV) for Bi with other calculations and with experimental data. Present Golin XU Experiment' Band calculation (Ref. 18) (Ref. 15) Value Ref. r,+(1) —14.00 —10.50 —13.32 —14.0 29 r,-(1) —8.09 —6.55 —7.53 —8.1 29 T6+ (1)-T6 (1) 1.11 1.78 2.39 1.18+0.08 29 L, (1)-L,(1) 0.36 0.58 0.74 X,(2) —5.65 —3.70 —5.2 29 r, (2)-r,+,(1) 1.07 0.83 1.61 0.65-0.71 30,31 I (2)-I +(3) 1.37 0.82 1.97 0.72-0.81 30,31 T6+ (3)-T45(1) 0.370 0.505 0.756 0.18-0.41 32-36 (3)-T45(1) 0.94 0.87 0.99 0.80-0.88 31 T6+ (2)-T4-, (1) —1.13 —1.60 —1.23 L, (3)-L,(3) —0.014 —0.015 —0.874 —0.011-—0.015 24, 33,37-39 L,(2)-L,(3) —1.64 —1.82 —2.24 —1.92- —2. 10 30,31 L.'(4)-L.'(3) 0.84 1.13 1.73 1.05-1.15 30,31 L,(4)-L,(3) 0.83 1.12 0.86 T ( 1)-L,(3) 0.038 0.039 0.040 0.036-0.039 32,33,39,40 'See the references for experimental conditions.

the difference in E, and E for Sb atoms in bulk Sb, Table III. The ordering of energy levels is the same as GaSb, and InSb can be interpreted as arising from the that of Rose and Schuchardt, but different from that of ' different local environments in these three materials. Golin, with the I 6+(3) and I 45(1) interchanged. Table IV presents a comparison with other theoretical results V. BAND STRUCTURE OF Bi and with experimental data. For the two lowest-lying valence bands, our results are The calculated band structure of Bi is shown in Fig. 5, in better agreement with experimental data than those with the zero of taken to be Fermi level. The en- energy from the pseudopotential calculation of Golin. ' (Xu ergies at the points and are in symmetry T, I, I, X given et al. ' fitted their model to the calculation of Gonze, Michenaud, and Vigneron' instead of the experimental data. ) This demonstrates that a TB model can provide a good representation of the lower-lying valence bands. It can be seen that the Fermi level intersects the valence band at the T point, and the conduction band at L. The overlap between them, E[T~~(1)—L, (3)], is 38 rneV. There are many band energies near the Fermi level. The smallest band gap, at the I. point, is only about 14 meV. This is E[L,(3)—L,(3)]. The band gap at T is about 370 meV [E[T6+(3)—T~~(1)]J. We can repro-

LA (3 lZ WX C) Electron— LLJ C)

0 C O L oN CO

C) C)

I 0.05 Bisectrix T W L AU I L U T

FICx. 5. Band structure of Bi along various symmetry lines. FICs. 6. Fermi surface of electrons in the trigonal-bisectrix The notation used here is the same as that of Golin (Ref. 18). plane for Bi. The unit of reciprocal length is g = 1.386 A YI LIU AND ROLAND E. ALLEN 52

TABLE V. Theoretical and experimental parameters for Fermi surface and effective masses' of Bi. The notation for direction is that used by Edel'man (Ref. 41). Rose and Present Schuchardt Experiment" work (Ref. 28) Values Ref. Overlap Eo (meV) 37.8 24.0 36.0-38.5 32,33,39,40 Fermi energy Ef (meV) 26.7 16.0 25.0-29.7 32,33,39,40,42

Electrons Tilt angle 8 6.0' 8.2 6'23'+1'

P, (10 ' gcm/s) 8.615 6.60 7.88+0.2 % P2 0.589 0.599 0.559+0.2 /o 41 P3 0.714 0.675 0.740+0.2 %

I) (m, ) 0.198 0.235 0.261 m2 0.001 47 0 00194 0.001 13 40 Pl 3 0.002 15 O.OO246' 0.002 59

Holes Tilt angle 0 00 oo 00 41

P, (10 ' gcm/s) 1.543 1.56 1.47 P2 1.534 1.56 1.47 41 P3 4.547 5.89 4.88

m& (m, ) 0.0675 0 097 0.064 m2 0.0675 0.096 0.064 33,40,41,43 m3 0.612 137 0.690-0.702

Carrier density (cm )

n' 3.09 X 10' 3. 12 X 10' (2.75-3.O2) X 1O" 33,40,41,43

n/p 0.99 'Effective masses for the principal axes. See the references for experimental conditions. 'Effective masses at the bottom of the band. Obtained from Lax model. See Ref. 28. 'Obtained by assuming that the electron and hole pockets are ellipsoids.

duce these three characteristic energy values in good Hole agreement with both the experimental data and the pseudopotential calculations, ' and as shown in Table IV. The small direct band gap at L, which is only about O C: I I I I o I TABLE VI. Energy levels of Sb (in eV) at the symmetry points T, I, L, and X. C) — — T6 12.022 I 6 13.446 L, —11.659 X, —11.607 CO T+ — — — — C) 10 378 I 7.150 L, 11.156 X, 9.916 I I T6 —1.252 I 6+ —2.684 L, —2.421 X, —6.571 —0.05 T4s —0.814 I —1.759 L, —2.145 X, —4.730 Bisectrix T6+ —O.332 r4+, —1.454 L, —0.786 X, —4.217 T6 0732 I 6 1.076 L, —0.0899 X, 2.579 1.125 1.836 166 FIG. 7. Fermi surface of holes in the trigonal-bisectrix plane T6 r6 L, 0. X, 3.229 T4+, 1.474 I 2.114 0.991 4.135 for Bi. The unit of reciprocal length is g = l. 386 A 45 L, X, ELECTRONIC STRUCTURE OF THE SEMIMETALS Bi AND Sb 1573

Electron C3 C) U 0C

C) D

) —0.2 0.2 Bisectrix

FIG. 9. Fermi surface of electrons in the trigonal-bisectrix plane for Sb. The unit of reciprocal length is =1.461 A LA g C3

LLJ W

suits with other theoretical and experimental values for the Fermi surface and carrier effective masses is summa- rized in Table V. Figures 6 and 7 show the calculated electron and hole pockets. It is clear that there is excellent agreement. The previous TB calculation of Xu et al. ' gave a large value for the band gap at I. (874 meV). The related electron effective masses are expected to be far from the T W L L U TH away experimental data. This casts doubt on the reliability of their model for treating transport and other properties in FIG. 8. Band structure of Sb along various symmetry lines. semimetal-semiconductor quantum structures. The notation used here is the same as that of Falicov and Lin

(Ref. 17)~ VI. BAND STRUCTURE OF Sb

The calculated band structure of Sb is shown in Fig. 8. one-third of the overlap energy, has an important The zero of energy is taken to be the Fermi level. The en- inhuence on the transport properties of Bi. The strong ergies at the symmetry points T, I, I., and X are listed in coupling between the conduction and valence bands, and Table VI. The ordering of energy levels is essentially the the resulting nonparabolic band dispersion, should be same as that of Rose and Schuchardt; only T6 (3) and taken into account when calculating superlattice states. T~~(1) are interchanged. A comparison with other Also note that the small effective masses are usually asso- theoretical results and with experimental data is present- ciated with small energy gaps. For bismuth, the electron ed in Table VII. effective mass along the trigonal axis at the I. point is As in the case of Bi, our results for the two lowest- smaller than 3X10 m„which means that large quan- lying valence bands are in better agreement with experi- tum confinement is expected for Bi (Sec. I). In fact, our mental data than those of a pseud opotential calculated effective masses are in very good agreement calculation —for Sb, that of Falicov and Lin. ' with the experimental data. The comparison of our re- In our calculation, the maximum of the fifth band

TABLE VII. Comparison of characteristic energy levels (in eV) for Sb with other calculations and with experimental data. Present Falicov and Lin Xu et al. Experiment' Band calculation (Ref. 17) (Ref. 15) Values Ref. +(1) —13.45 —12.10 —12.87 —13.3 29 r, — — — — r;(1) 7.15 6.15 5 ~ 17 7.0 29 T6+(1)-T6 (1) 1.64 1.71 2.9 1.67+0.06 29 L, (1)-L,(1) 0.50 0.54 0.94 X,(1)-X,(1) 1.69 1.90 'X,(2) —6.57 —4.79 —5.5 T6 (2)-T4s(1) 0.48 0.63 0.64 0.143 T (3)-T+ (2) 1.06 0.98 1.32 L,(3)-L.'(3) —0.70 —0.24 —0.34 —0. 101+0.003 44-46 L'. (2)-L', (3) —1.36 —1.11 —1.31 'See the references for experimental conditions. YI LIU AND ROLAND E. ALLEN 52

TABLE VIII. Theoretical and experimental parameters of Fermi surface and effective masses of Sb. The notation for directions is that used by Issi (Ref. 4). Rose and Present Schuchardt Experiment' work (Ref. 28) Values Ref. Overlap Eo (meV) 17.40 234 177.5 Fermi energy Ef (meV) 89.9 115 93.1

Electrons Tilt angle L9,„ —8.5' 60 40 70 47-49 ' P& ( 10 g cm/s) 4.30 5.2 3.95 P2 30.82 32.3 31.30 P3 4.62 3.6 4.22

m[]]]] (m, ) 0.073 0 093 4,47

Holes Tilt angle 0;„ 24' 56' 53 48,49

' P& (10 gcm/s) 5.11 4.22 P2 19.41 20.2 18.45 P3 3.00 44 4.43

m[»1] (me ) 0.12 0 091" 4,47

Carrier density (cm ') n' 5.32 X 10' p 5.17X 10' (3.74-5.50) X 10' 48,50,51

n/p 1.03 'See the references for experimental conditions. Calculated from the formula 1/m =(1,, /m&)+(A, 2/m2) +(A,3/m3). Here m&, m2, and m3 are the

efFective masses along the principal axes, and A, &, A,&, and A, 3 are the direction cosines of the trigonal axis in the principal-axis coordinates. 'Calculated by assuming the electron and hole pockets are ellipsoids. occurs at the point H, which is near the point T. (H is on respectively. The Fermi surface results are given in Table the mirror plane, with trigonal coordinates [0.4543, VIII, and are compared with other theoretical and exper- 0.3722, 0.3722].) The minimum of the sixth band occurs imental data. at points near or at I.. The overlap of these bands is 174 For the hole pocket, our calculated tilt angle is 24, meV. which is the only feature significantly different from ex- The cross sections of the electron pockets and hole periment data. The shapes of the hole pocket and elec- pockets on the mirror plane are shown in Figs. 9 and 10, tron pocket are generally in agreement with those given

TABLE IX. The matrix elements of H». slT sip ylT zlT xi) ylg z1$ " II tl s1T E +g2e V,, 0 g27 Vspcr g28 Vspo 0 0 0 ", tt II sly E, +g2e V, 0 0 g27 Vspo. gz8 Vspcr 0 ——'A, xlT Ep+g29 Vpp i 3 0 0 +g3o Vpp. +g31( Vpp Vpp ——'A, ylT Ep+g 3o Vppcr i 3 +g29 Vpp zlT —'X Ep+g2e Vpp 3 i 3 x1$ p g29 pp 3 g3& pp pp +g3o Vpp y1$ Ep+g3o Vpp y1$ g29 ppm z1$ Ep+g2e Vpp~ 52 ELECTRONIC STRUCTURE OF THE SEMIMETALS Bi AND Sb 1575

b Hole O bO bO + O + O b + b

O bO bO bO 0C + + + I—

hp I b bo+ I I O + O O O b W+ b b b b

CV 0.2 bO bO + bO +bO bO bO Bisectrix

b FIG. 10. Fermi surface of holes in the trigonal-bisectrix 00 = bO + plane for Sb. The unit of reciprocal length is 1.461 A I I g bO bO + b I O + O + b b b~ b b

CV bO + + by experiment. Since we cannot quantitatively reproduce + the carrier effective masses along all three principal axes, we focus on those along the trigonal axis —i.e., the growth [111]direction for CdTe-Bi, PbTe-Bi, and GaSb- O I I bO+ Sb superlattices. Our calculated effective masses along O I b O O O Qh + b the [lllj direction, listed in Table VIII, are in good Ct)

agreement with the experimental measurements. bO + + + VII. CONCLUSIQNS ~ W c5 b ~a The electronic band structures of bismuth and an- E oo hp timony have been successfully represented with a tight- I I bO I I binding model. The orbital interaction parameters were + O b a„ ~O O O determined by fitting the experimental data, which include the following: (1) the overlap between the highest bO bO bp bo valence band and the lowest conduction band; (2) the Fer- + + + mi energy; (3) the effective masses of the carriers at the minima of the conduction band and at the maxima of the b valence band; (4) the shapes of the Fermi surfaces for bO + + I I small bO bO both electrons and holes; and (5) the band gaps. + b I b O b Particular attention was given to the behavior of the b b bands in the vicinity of the Fermi surface, since it is this bO behavior that will dominantly inhuence the electronic bo + + + properties important for device applications. b b b b

APPENDIX bO I I I O+ O O O b b b b for the This Appendix gives the explicit expressions C& bO bO bO bO Hamiltonian matrix elements in the representation ~si o ), I I I a.nd The si.te index i 2 distin- ~xio), ~yio ), ~zio). =1, b b guishes which represents the two atoms in the primitive cell. bO bO bO O I I O O O H)( H)2 + b H= H2( H~2 hp bO bO

I The diagonal matrix H&& =H22 and the off-diagonal matrix H, 2=H2& are presented in Tables IX and X, respectively. 1576 YI LIU AND ROLAND E. ALLEN 52

In these tables, gp, g, , . . . , g3, are functions of the ing a„a2, a3, cosa, cosP, and cosy in the expressions for reciprocal-lattice vector k =k1b1+k2b2+ k 3b3, with b1, gp g)2 by a2+a3, a)+ a3, a2+a), cosa', cosP', and cosy', 12, and b3 the primitive reciprocal-lattice vectors defined respectively, where cosa', cosp', and cosy' are the direc- in Sec. II. Among them, gp-g12 involve the first-neighbor tion cosines for the vector a1+a3 —d, from the central interactions g13-g25 the second neighbors and g26 g31 atom to one of the second-neighbor atoms. Finally, the third neighbors. The expressions for gp-g12 are g26 g 3 1 are given by ik ik 'k. ik (a) —d) ik.(a2 —d) ik-{s3—d) (a1 a2 (a2 ~1 82 3

— ik — ik k ik (a2 d) ik (a& d) (a3 a2 (al 3 3 1 g, =(e ' —e ' )cosa, +e '+e +e — — — ik (a2 . a&—) ik. (a& —a2) ik (a& d) ik (a2 d) ik (a3 d) 27= e —e

' ik a2 a.3) ik.(a3 a2 g3 =gpcosp + —, e e ik. — — — — (a' d) ik (a2' d) ' ik (a3 a&) —ik. (a& a3) g4=(e +e )cos a, + —, e e gs=gp ik.(a3 —a& ) ik.(a& —a3) g2() ——&3 e e i g6 =g, cosy, ik —d) ik —.d) ik —.d) g7=(e (a&' +e (a2' +4e (a3' cos ik (a& —a3) ik.(a3 —a&) gs =go g29=4 e +e =gpcoS 2 — ik a2 a3 ik.(a3 a2 g9 f + 4'e +e 2 — — g10 gOS1n p ik (a2 a&) ik.(a& a2) — ik. — ik. — g11 g2COSQ ' (a) a3) (a3 a& ) gso= —, e +e g)2 =g1cosp, ik.(a2 —a3) ik.(a3 —a2) 4 ) with and d defined in Sec. Here a„a2, a3, II. cosa, cosp, ik (a3 —a&) ik.(a& —a3) and cosy are the direction cosines for the vector a2 —d, from the central atom to one of the nearest-neighbor — — ik(a 3 a2 ) ik(a 2 a3 ) atoms. The functions g13 —g25 can be obtained by replac- e )

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