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Densities of electronic states for tight-binding two-dimensional square-lattice ternary alloys AxB1−xC
J. Phy.7 Chem. Solid.v Vol. 45. No. 7. pp. 731-732. F’rinted in the U.S.A.
W22-3697/M $3.00 + .OO Pergamon Pm8 Ltd.
1984
DENSITIES OF ELECTRONIC STATES FOR TIGHTBINDING TWO-DIMENSIONAL SQUARE-LATTICE TERNARY ALLOYS A,B,,C DAVID V. FROELICHand JOHN D. Dow Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, U.S.A. and
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. (Received 24 October
1983; accepted 1 December 1983)
Abstract-The
densities of electronic states for model two-dimensional A$& calculated for l(r-atom crystals, using the negative eigenvalue theorem. 1. INTRODUCDON With the increasing technological importance of semiconducting ternary alloys for light-emitters and highmobility, modulation-doped electronic devices, there is a clear need for improved theories of the electronic states of ternary alloys. Currently, the most widely used alloy theories are the coherent potential approximation, (CPA) [ 11,and variants of it [2], and the recursion method[3]. The CPA is known to produce poor spectra whenever clusters of two or more minority alloy constituents are important[2], and is best suited for treating the long-wavelength properties of alloys. The recursion method is well adapted to treating local properties of alloys, but must be applied to very large clusters before it correctly predicts some longwavelength properties. Thus several workers are attempting to develop a new theory of alloys that combines the major advantages of the CPA and the recursion method without having the deficiencies of either method. However, there are presently no fiducial standards against which one can measure such a new theory to determine if its predictions are good or not. The need for such standards for ternary alloys motivates the present work. Here we calculate the exact densities of states of 104-atom, square-lattice, nearest-neighbor tight-binding, diagonally disordered, two-dimensional, crystalline ternary alloys, A$-$. The
2. CALCULATION electronic density of states is D(E)=
Tr 6 (E-H) N
where 6 is the Dirac delta function and His the tightbinding Hamiltonian,
tight-binding alloys are
neighbors, in which case it takes on the constant value V. The on-site energy depends on the particular atomic species located at R, and is W, for R being a C-site (cation) and either W, or W, if R is an A or B (anion) site. The probability of W, being W, or W, for an anion site is x or l-x, respectively. If x = 0 or 1 the material is the crystal BC or AC and is periodic; therefore a band structure can be obtained:
Here we have on-site energies W, and W, and nearestneighbor interaction energy V. In these perfect-crystal limits we can calculate the density of states directly[rl]. For other values of x the A’s and B’s are randomly mixed, therefore no periodicity exists. In these cases we use the negative eigenvalue theorem[5], which gives an exact integrated density of states S(E) for a finite crystal, E
D(E’) dE’.
S(E) = s -m
The density of states D(E) can be obtained by numerical differentiation of S(E). The numerical uncertainty of this calculation is visible as “noise” resulting from this differentiation. We assumed the following values for our work: W, = 3, W, = 2, and V = 1, and W, = 0 or 1 for the asymmetric or symmetric case. For the symmetric case, wc- w*= W,- W,, therefore results were not computed for x > 0.5. 3. RFsuLls
H = 1 [R)%t(R[ II
+ c IR)P(R, R’)(R’I. &R
Our results are given in Fig. 1 for the asymmetric case. Several features of the spectra merit special mention. For x = 0, the density of states spectrum exhibits
Here IR) is an orbital localized at the site R of a square lattice and V is nonzero only if R and R’ are nearest731
132
D. V. FKOELICH and J. D. Dow
k
Fig. 1. The calculated densities of states D(E) vs energy E of the tight-binding two-dimensional alloy A$, _ ,C for the indicated values of x and for W, = 0, W, = 3, W, = 2, and v= 1.
Fig. 2. The calculated densities of states D(E) vs energy E of the tight-binding two-dimensional alloy A,B,_,C for the indicated values of x and for W, = 1, W, = 3, W, = 2. and v= 1.
singularities
Similar results hold for the applicable x values in the symmetric case, Fig. 2, except no resonantantiresonant behavior is exhibited for x E 0.5. We propose, that to be judged successful, approximate theories of ternary alloys be capable of reproducing the spectra of Figs. 1 and 2 with adequate precision.
associated
with
van
Hove’s
critical
At E = -3.12 V and E = 5.12V there are step-function M, and M, singularities; and there are logarithmic M, singularities at the band gap (extending from E = 0 to E = 2 V). As x increases, the singularities evolve until at x = 1 the M, and Mz singularities are at E = - 1.53V and E = 6.53 V, whereas the gap extends from E = 2 V to E = 3 V. With increasing x (i.e. x z 0.1) an impurity (A) band appears in the gap of BC (near E = 1.2 V), with sidebands related to pairs and clusters of A atoms. This band grows with increasing x until it fills the band gap (x E 0.5). The band bottom moves up in energy with increasing X, as does the top of the band. The sharp peak at E = 0 for x = 0 broadens into a resonance with increasing x and exhibits some antiresonance behavior (near E = 0 for x = 0.34.5) and some structure attributable to clusters of B atoms near E = 2.1 V for x = 0.9. The band bottom edge peak of the upper band at E = 2 V for x = 0 persists for all values of x, becoming the M, singularity at the top of the lower band for x = 1. As x approaches unity a new (AC) bandgap is formed and van Hove singularities emerge. “Bound” B impurity bands at E < - 2 V and E = 2 V are visible as the gap forms from an antiresonance (x E 0.9). points[6].
Acknowledgemenfs-This work was supported by the Division of Materials Science, Department of Energy Contract DE-AC02-76ERal198. One of us (D.V.F.) thanks the Office of Naval Research for a fellowship. REFERENCES 1. Butler W. H. and Nickel B. G., Phys. Reo. Left. 30. 373 (1973) and Refs. [9-271 of Ref. [2]. 2. Myles C. W. and Dow J. D., Phys. Rev. Lett. 42, 254 (1979). 3. Haydock R., Solid Sf. Phys. (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 35, pp. 215-294. Academic Press, New York (1980): Kelly M. J., Solid-St Phys., pp. 295-383. 4. Lehman G. and Taut M.. Phvs. Status Solidi (h) 54. 469 ( 1972). 5. Dean P., Rev. Mod. Phys. 44, 129, 167 (1972). See also Payton D. N. and Visscher W. M., Phys. Rev. 154, 802 (1967) and 156, 1032 (1967) and 175, 1201 (1968) and O’Hara M. J., Myles C. W., Dow J. D. and Painter R. D.. J. Phys. Chem. Soli& 42, IO43 (198 1). 6. L. van Hove, Phys. Ret?. 89, 1189 (1953).
W22-3697/M $3.00 + .OO Pergamon Pm8 Ltd.
1984
DENSITIES OF ELECTRONIC STATES FOR TIGHTBINDING TWO-DIMENSIONAL SQUARE-LATTICE TERNARY ALLOYS A,B,,C DAVID V. FROELICHand JOHN D. Dow Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, U.S.A. and
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. (Received 24 October
1983; accepted 1 December 1983)
Abstract-The
densities of electronic states for model two-dimensional A$& calculated for l(r-atom crystals, using the negative eigenvalue theorem. 1. INTRODUCDON With the increasing technological importance of semiconducting ternary alloys for light-emitters and highmobility, modulation-doped electronic devices, there is a clear need for improved theories of the electronic states of ternary alloys. Currently, the most widely used alloy theories are the coherent potential approximation, (CPA) [ 11,and variants of it [2], and the recursion method[3]. The CPA is known to produce poor spectra whenever clusters of two or more minority alloy constituents are important[2], and is best suited for treating the long-wavelength properties of alloys. The recursion method is well adapted to treating local properties of alloys, but must be applied to very large clusters before it correctly predicts some longwavelength properties. Thus several workers are attempting to develop a new theory of alloys that combines the major advantages of the CPA and the recursion method without having the deficiencies of either method. However, there are presently no fiducial standards against which one can measure such a new theory to determine if its predictions are good or not. The need for such standards for ternary alloys motivates the present work. Here we calculate the exact densities of states of 104-atom, square-lattice, nearest-neighbor tight-binding, diagonally disordered, two-dimensional, crystalline ternary alloys, A$-$. The
2. CALCULATION electronic density of states is D(E)=
Tr 6 (E-H) N
where 6 is the Dirac delta function and His the tightbinding Hamiltonian,
tight-binding alloys are
neighbors, in which case it takes on the constant value V. The on-site energy depends on the particular atomic species located at R, and is W, for R being a C-site (cation) and either W, or W, if R is an A or B (anion) site. The probability of W, being W, or W, for an anion site is x or l-x, respectively. If x = 0 or 1 the material is the crystal BC or AC and is periodic; therefore a band structure can be obtained:
Here we have on-site energies W, and W, and nearestneighbor interaction energy V. In these perfect-crystal limits we can calculate the density of states directly[rl]. For other values of x the A’s and B’s are randomly mixed, therefore no periodicity exists. In these cases we use the negative eigenvalue theorem[5], which gives an exact integrated density of states S(E) for a finite crystal, E
D(E’) dE’.
S(E) = s -m
The density of states D(E) can be obtained by numerical differentiation of S(E). The numerical uncertainty of this calculation is visible as “noise” resulting from this differentiation. We assumed the following values for our work: W, = 3, W, = 2, and V = 1, and W, = 0 or 1 for the asymmetric or symmetric case. For the symmetric case, wc- w*= W,- W,, therefore results were not computed for x > 0.5. 3. RFsuLls
H = 1 [R)%t(R[ II
+ c IR)P(R, R’)(R’I. &R
Our results are given in Fig. 1 for the asymmetric case. Several features of the spectra merit special mention. For x = 0, the density of states spectrum exhibits
Here IR) is an orbital localized at the site R of a square lattice and V is nonzero only if R and R’ are nearest731
132
D. V. FKOELICH and J. D. Dow
k
Fig. 1. The calculated densities of states D(E) vs energy E of the tight-binding two-dimensional alloy A$, _ ,C for the indicated values of x and for W, = 0, W, = 3, W, = 2, and v= 1.
Fig. 2. The calculated densities of states D(E) vs energy E of the tight-binding two-dimensional alloy A,B,_,C for the indicated values of x and for W, = 1, W, = 3, W, = 2. and v= 1.
singularities
Similar results hold for the applicable x values in the symmetric case, Fig. 2, except no resonantantiresonant behavior is exhibited for x E 0.5. We propose, that to be judged successful, approximate theories of ternary alloys be capable of reproducing the spectra of Figs. 1 and 2 with adequate precision.
associated
with
van
Hove’s
critical
At E = -3.12 V and E = 5.12V there are step-function M, and M, singularities; and there are logarithmic M, singularities at the band gap (extending from E = 0 to E = 2 V). As x increases, the singularities evolve until at x = 1 the M, and Mz singularities are at E = - 1.53V and E = 6.53 V, whereas the gap extends from E = 2 V to E = 3 V. With increasing x (i.e. x z 0.1) an impurity (A) band appears in the gap of BC (near E = 1.2 V), with sidebands related to pairs and clusters of A atoms. This band grows with increasing x until it fills the band gap (x E 0.5). The band bottom moves up in energy with increasing X, as does the top of the band. The sharp peak at E = 0 for x = 0 broadens into a resonance with increasing x and exhibits some antiresonance behavior (near E = 0 for x = 0.34.5) and some structure attributable to clusters of B atoms near E = 2.1 V for x = 0.9. The band bottom edge peak of the upper band at E = 2 V for x = 0 persists for all values of x, becoming the M, singularity at the top of the lower band for x = 1. As x approaches unity a new (AC) bandgap is formed and van Hove singularities emerge. “Bound” B impurity bands at E < - 2 V and E = 2 V are visible as the gap forms from an antiresonance (x E 0.9). points[6].
Acknowledgemenfs-This work was supported by the Division of Materials Science, Department of Energy Contract DE-AC02-76ERal198. One of us (D.V.F.) thanks the Office of Naval Research for a fellowship. REFERENCES 1. Butler W. H. and Nickel B. G., Phys. Reo. Left. 30. 373 (1973) and Refs. [9-271 of Ref. [2]. 2. Myles C. W. and Dow J. D., Phys. Rev. Lett. 42, 254 (1979). 3. Haydock R., Solid Sf. Phys. (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 35, pp. 215-294. Academic Press, New York (1980): Kelly M. J., Solid-St Phys., pp. 295-383. 4. Lehman G. and Taut M.. Phvs. Status Solidi (h) 54. 469 ( 1972). 5. Dean P., Rev. Mod. Phys. 44, 129, 167 (1972). See also Payton D. N. and Visscher W. M., Phys. Rev. 154, 802 (1967) and 156, 1032 (1967) and 175, 1201 (1968) and O’Hara M. J., Myles C. W., Dow J. D. and Painter R. D.. J. Phys. Chem. Soli& 42, IO43 (198 1). 6. L. van Hove, Phys. Ret?. 89, 1189 (1953).