Abstract

The theoretical refractive index of the direct-gap II–VI compounds CdSe, CdS, and CdSexS1–x with the wurtzite lattice structure is calculated for frequencies below the fundamental absorption edge. Use is made of the band structure of the Kane theory, modified to include the dependence of the refractive index on the effective hole masses of the three valence bands, the lattice constants a(Å) and C(Å) of the hexagonal structure, and the direction of polarization of the incident wave. An extension of the previous result for the refractive index of a cubic zinc blende crystal, in which birefringence is absent, to the hexagonal wurtzite structure, where birefringence occurs, is given. Results are obtained for the refractive indices of the ordinary and extraordinary rays in terms of experimentally available quantities, which include the direct-band-gap energy G, the effective electron mass mn, the three effective hole masses mp1,mp2, and mp3 of the three valence bands at k = 0, and lattice constants a(Å) and C(Å). Numerical values of the refractive index for values of x in the range 0 ≤ x ≤ 1 are given and compared with experimental results.

© 1986 Optical Society of America

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References

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  1. B. Jensen, “Quantum theory of the complex dielectric constant of free carriers in polar semiconductors,” IEEE J. Quantum Electron. QE-18, 1361–1370 (1982).
    [CrossRef]
  2. E. O. Kane, “Band structure of InSb,” J. Phys. Chem. Solids 1, 249–261 (1957).
    [CrossRef]
  3. B. Jensen, “Quantum theory of free carrier absorption in polar semiconductors,” Ann. Phys. (NY) 80, 284–360 (1973).
    [CrossRef]
  4. B. Jensen and A. Torabi, “Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors,” IEEE J. Quantum Electron. QE-19, 448–457 (1983); “Dispersion of the refractive index of GaAs and Alx Ga1–x As,” IEEE J. Quantum Electron. QE-19, 877–882 (1983).
    [CrossRef]
  5. B. Jensen and A. Torabi, “Linear and nonlinear intensity dependent refractive index of Hg1–x Cdx Te,” J. Appl. Phys. 54, 5945–5949 (1983).
    [CrossRef]
  6. I. P. Kaminow, An Introduction to Electro-Optic Devices (Academic, New York, 1974).
  7. A. Yariv, Introduction To Optical Electronics (Holt, Rinehart and Winston, New York, 1976).
  8. M. Neuberger, “II–VI ternary compound data tables” (Hughes Aircraft Company, Culver City, Calif., 1972).
    [CrossRef]
  9. B. Jensen and A. Torabi, “Temperature and intensity dependence of the refractive index of a compound semiconductor,” J. Opt. Soc. Am. B 2, 1395–1401 (1985).
    [CrossRef]
  10. E. D. Palik and A. Addamiano, in Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press Handbook Series (Academic, New York, 1985).
  11. Y. S. Park and J. R. Schneider, “Index of refraction of ZnO,” J. Appl. Phys. 39, 3049–3052 (1968).
    [CrossRef]
  12. M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
    [CrossRef]

1985 (1)

1983 (2)

B. Jensen and A. Torabi, “Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors,” IEEE J. Quantum Electron. QE-19, 448–457 (1983); “Dispersion of the refractive index of GaAs and Alx Ga1–x As,” IEEE J. Quantum Electron. QE-19, 877–882 (1983).
[CrossRef]

B. Jensen and A. Torabi, “Linear and nonlinear intensity dependent refractive index of Hg1–x Cdx Te,” J. Appl. Phys. 54, 5945–5949 (1983).
[CrossRef]

1982 (1)

B. Jensen, “Quantum theory of the complex dielectric constant of free carriers in polar semiconductors,” IEEE J. Quantum Electron. QE-18, 1361–1370 (1982).
[CrossRef]

1973 (1)

B. Jensen, “Quantum theory of free carrier absorption in polar semiconductors,” Ann. Phys. (NY) 80, 284–360 (1973).
[CrossRef]

1969 (1)

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

1968 (1)

Y. S. Park and J. R. Schneider, “Index of refraction of ZnO,” J. Appl. Phys. 39, 3049–3052 (1968).
[CrossRef]

1957 (1)

E. O. Kane, “Band structure of InSb,” J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Addamiano, A.

E. D. Palik and A. Addamiano, in Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press Handbook Series (Academic, New York, 1985).

Gudymenko, L. F.

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

Jensen, B.

B. Jensen and A. Torabi, “Temperature and intensity dependence of the refractive index of a compound semiconductor,” J. Opt. Soc. Am. B 2, 1395–1401 (1985).
[CrossRef]

B. Jensen and A. Torabi, “Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors,” IEEE J. Quantum Electron. QE-19, 448–457 (1983); “Dispersion of the refractive index of GaAs and Alx Ga1–x As,” IEEE J. Quantum Electron. QE-19, 877–882 (1983).
[CrossRef]

B. Jensen and A. Torabi, “Linear and nonlinear intensity dependent refractive index of Hg1–x Cdx Te,” J. Appl. Phys. 54, 5945–5949 (1983).
[CrossRef]

B. Jensen, “Quantum theory of the complex dielectric constant of free carriers in polar semiconductors,” IEEE J. Quantum Electron. QE-18, 1361–1370 (1982).
[CrossRef]

B. Jensen, “Quantum theory of free carrier absorption in polar semiconductors,” Ann. Phys. (NY) 80, 284–360 (1973).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, An Introduction to Electro-Optic Devices (Academic, New York, 1974).

Kane, E. O.

E. O. Kane, “Band structure of InSb,” J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Lisitsa, M. P.

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

Malinko, V. N.

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

Neuberger, M.

M. Neuberger, “II–VI ternary compound data tables” (Hughes Aircraft Company, Culver City, Calif., 1972).
[CrossRef]

Palik, E. D.

E. D. Palik and A. Addamiano, in Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press Handbook Series (Academic, New York, 1985).

Park, Y. S.

Y. S. Park and J. R. Schneider, “Index of refraction of ZnO,” J. Appl. Phys. 39, 3049–3052 (1968).
[CrossRef]

Schneider, J. R.

Y. S. Park and J. R. Schneider, “Index of refraction of ZnO,” J. Appl. Phys. 39, 3049–3052 (1968).
[CrossRef]

Terekhova, S. F.

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

Torabi, A.

B. Jensen and A. Torabi, “Temperature and intensity dependence of the refractive index of a compound semiconductor,” J. Opt. Soc. Am. B 2, 1395–1401 (1985).
[CrossRef]

B. Jensen and A. Torabi, “Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors,” IEEE J. Quantum Electron. QE-19, 448–457 (1983); “Dispersion of the refractive index of GaAs and Alx Ga1–x As,” IEEE J. Quantum Electron. QE-19, 877–882 (1983).
[CrossRef]

B. Jensen and A. Torabi, “Linear and nonlinear intensity dependent refractive index of Hg1–x Cdx Te,” J. Appl. Phys. 54, 5945–5949 (1983).
[CrossRef]

Yariv, A.

A. Yariv, Introduction To Optical Electronics (Holt, Rinehart and Winston, New York, 1976).

Ann. Phys. (NY) (1)

B. Jensen, “Quantum theory of free carrier absorption in polar semiconductors,” Ann. Phys. (NY) 80, 284–360 (1973).
[CrossRef]

IEEE J. Quantum Electron. (2)

B. Jensen and A. Torabi, “Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors,” IEEE J. Quantum Electron. QE-19, 448–457 (1983); “Dispersion of the refractive index of GaAs and Alx Ga1–x As,” IEEE J. Quantum Electron. QE-19, 877–882 (1983).
[CrossRef]

B. Jensen, “Quantum theory of the complex dielectric constant of free carriers in polar semiconductors,” IEEE J. Quantum Electron. QE-18, 1361–1370 (1982).
[CrossRef]

J. Appl. Phys. (2)

B. Jensen and A. Torabi, “Linear and nonlinear intensity dependent refractive index of Hg1–x Cdx Te,” J. Appl. Phys. 54, 5945–5949 (1983).
[CrossRef]

Y. S. Park and J. R. Schneider, “Index of refraction of ZnO,” J. Appl. Phys. 39, 3049–3052 (1968).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. Solids (1)

E. O. Kane, “Band structure of InSb,” J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Phys. Status Solidi (1)

M. P. Lisitsa, L. F. Gudymenko, V. N. Malinko, and S. F. Terekhova, “Dispersion of the refractive indices and birefringence of CdSx Se1–x, single crystals,” Phys. Status Solidi 31, 389–399 (1969).
[CrossRef]

Other (4)

E. D. Palik and A. Addamiano, in Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press Handbook Series (Academic, New York, 1985).

I. P. Kaminow, An Introduction to Electro-Optic Devices (Academic, New York, 1974).

A. Yariv, Introduction To Optical Electronics (Holt, Rinehart and Winston, New York, 1976).

M. Neuberger, “II–VI ternary compound data tables” (Hughes Aircraft Company, Culver City, Calif., 1972).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Theoretical values of the refractive index no (solid lines) along with the experimental values (crosses) as a function of photon energy w (eV) for the compounds CdS, CdSe, and CdSexS1–x.

Tables (3)

Tables Icon

Table 1 Values of Quantities YB, mn, m p 1 , m p 2 , m p 3, G(eV), and do for CdSeXS1–X

Tables Icon

Table 2 Theoretical Refractive Indek no and Experimental Refractive Index no* for the Compounds CdS, CdSe, and CdSeXS1–X as a Function of Photon Energy ω (eV) at 300 Ka

Tables Icon

Table 3 Average Values of do/dex = no2 − 1/nex2 − 1 for the Compound CdSeXS1–X

Equations (61)

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1 = 1 + ( 4 π e 2 / w 2 ) υ i , k e ˆ · v υ i , k ; c , k + q v c , k + q ; υ i , k · e ˆ × [ f 0 υ i , k f c 0 , k + q ] e c , k + q e υ i , k ћ w ,
v c , k + q ; υ i , k = ( 1 / m ) ( c , k + q , s | p | υ i , k , s ) δ s s
e ˆ · v υ i , k ; c , k + q v c , k + q ; υ i , k · e ˆ = ( ) υ c , k + q ; υ i , k 2 , e ˆ C ,
e ˆ · v υ i , k ; c , k + q v c , k + q ; υ i , k · e ˆ = ( ) υ c , k + q ; υ i , k 2 , e ˆ C
1 ° = 1 + ( 4 π e 2 / w 2 ) υ i , k ( 1 / 2 s , s ( ) υ c , k ; υ i , k 2 ) × f υ i , k 0 f c , k 0 e c , k e υ i , k ћ w ( e ˆ c ) ,
e 1 ex = 1 + ( 4 π e 2 / w 2 ) υ i , k ( 1 / 2 s , s υ 2 c , k ; υ i , k ) × f 0 υ i , k f 0 c , k e c , k e υ i , k ћ w ( e ˆ c ) .
f 0 υ i , k f 0 c , k e c , k e υ i , k ћ w
d o ( i ) ( 0 , 0 ) ( G / 2 m n ) = 1 / 2 s , s ( ) υ c , k ; υ i , k 2 | k = 0 = 1 / 2 m 2 s , s | c , 0 , s | p · e ˆ | υ i , 0 , s | 2 ( e ˆ C ) ,
d ex ( i ) ( 0 , 0 ) ( G / 2 m n ) = 1 / 2 s , s ( ) υ c , k ; υ i , k 2 | k = 0 = 1 / 2 m 2 s , s | c , 0 , s | p · e ˆ | υ i , 0 , s | 2 ( e ˆ C ) .
1 G ( e c , k e υ i , k ћ w ) z 2 + λ r i 2 k 2 ,
z 2 = 1 ( ћ w / G ) ƛ r i 2 = ћ 2 2 m r i G , 1 / m r i = ( 1 / m n ) + ( 1 / m p i ) , i = 1 , 2 , 3 ,
N υ * = 1 / 3 π 2 ƛ r 1 3 , w υ 2 = 4 π N υ * e 2 / m n , w g = G / ћ .
n o 2 = 1 + 4 π e 2 / m n w g 2 1 2 υ i , k f υ i , k 0 f c , k 0 z 2 + ƛ r i 2 k 2 × d o ( i ) ( 0 , 0 ) = 1 + ( w υ 2 / w g 2 ) ( 3 / 2 ) υ i ƛ r 1 3 ƛ r i 3 × 0 y B y 2 d y z 2 + y 2 ( υ i , k f 0 c , k ) d o ( i ) ( 0 , 0 ) d Ω 4 π ,
n ex 2 = 1 + ( w υ 2 / w g 2 ) ( 3 / 2 ) υ i ƛ r 1 3 ƛ r i 3 0 y B y 2 d y z 2 + y 2 × ( f υ i , k 0 f c , k 0 ) d ex ( i ) ( 0 , 0 ) d Ω 4 π .
f υ i , k 0 = 1 , i = 1 , 2 , 3 f c , k 0 = 0 { Pure material ( No free carriers in conduction band ) ( All valence band states filled ) .
η o = ( 1 / 2 ) υ i ƛ r 1 3 ƛ r i 3 d o ( i ) ,
c o o = ( 3 / 2 ) η 0 ( w υ 2 / w g 2 ) ,
η ex = ( 1 / 2 ) υ i ƛ r 1 3 ƛ r i 3 d ex ( i ) ,
c o ex = ( 3 / 2 ) η ex ( w υ 2 / w g 2 )
d o ex ( i ) = d o ex ( i ) ( 0 , 0 ) d Ω 4 π , ƛ r 1 3 ƛ r i 3 = ( 1 + m n / m p 1 ) 3 / 2 ( 1 + m n / m p i ) 3 / 2 ,
n o 2 = 1 + 2 c o o [ y B z tan 1 ( y B / z ) ] ,
n ex 2 = 1 + 2 c ex o [ y B z tan 1 ( y B / z ) ] .
y B = y B ( a 0 , c 0 ) + ( a a 0 ) d y B d a | a = a 0 + ( C C 0 ) d y B dC | C = C 0 + .
y B = M 1 a ( Å ) + M 2 C ( Å ) + M 3 .
M 1 = 60.07 ± 0.52 ( Å 1 ) , M 2 = 42.00 ± 0.35 ( Å 1 ) , M 3 = 30.63 ± 0.19.
d o ( i ) ( q , k ) ( G / 2 m 2 ) = ( 1 / 2 m 2 ) s , s | ( υ i , k , s | p · e ˆ | c , k + q , s ) | 2 , d ex ( i ) ( q , k ) ( G / 2 m n ) = ( 1 / 2 m 2 ) s , s | ( υ i , k , s | p · e ˆ | c , k + q , s ) | 2 .
( U n , k , s | p | U c , k , s ) = ( a n k c c k + c n k a c k ) ( m P / ћ ) e ˆ 0 ( n = c , υ 2 , υ 3 ) , ( U υ 1 , k , | p | U c , k , ) = a c k ( m P / ћ ) e ˆ 1 , ( U υ 1 , k , | p | U c , k , ) = a c k ( m P / ћ ) e ˆ 1 .
| n , k , s ) = exp ( i k · r ) U n , k , s ,
e ˆ 1 = ( i ˆ + i j ˆ ) / 2 , e ˆ 0 = k / k , e ˆ 1 = ( i ˆ i j ˆ ) / 2 .
M 2 P 2 / ћ 2 = m 2 ( G / 2 m n ) = m 2 α 0 2 c 2 .
u = e c k / G , u 1 = e υ 1 k / G = ћ 2 k 2 / 2 m p 1 G , u 2 = e υ 2 k / G = ћ 2 k 2 / 2 m p 2 G , u 3 = e υ 3 k / G = ћ 2 k 2 / 2 m p 3 G , ( parabolic limit ) , W 2 = ( G 2 + e c k ) 2 = ћ 2 α 2 0 c 2 k 2 + ћ 2 α 0 2 c 2 k c 2 ( hyperbolic limit ) , k c = 1 / ƛ c = m n α 0 c / ћ .
a c k = [ ( 1 + u ) / ( 1 + 2 u ) ] 1 / 2 , c c k = [ u / ( 1 + 2 u ) ] 1 / 2 ; a υ 3 k = [ u / ( 1 + 2 u ) ] 1 / 2 , c υ 3 k = [ ( 1 + u ) / ( 1 + 2 u ) ] 1 / 2 ; a υ 2 k = u 1 / 2 ( 1 + u ) 1 / 2 { u ( 1 + u ) + [ 1 + R u ( 1 + u ) ] 2 } 1 / 2 , R = m n / m p 2 ; c υ 2 k = [ 1 + R u ( 1 + u ) ] { u ( 1 + u ) + [ 1 + R u ( 1 + u ) ] 2 } 1 / 2 .
k 2 = k c 2 β 2 ( u ) γ 2 ( u ) = 4 k c 2 u ( 1 + u ) , β ( u ) = 2 u 1 / 2 ( 1 + u ) 1 / 2 / ( 1 + 2 u ) , γ ( u ) = 1 + 2 u = [ 1 β 2 ( u ) ] 1 / 2 .
d o ( i ) ( 0 , k ) = ( a i k c c k + c i k a c k ) 2 ( k ˆ · e ˆ ) 2 , i = υ 2 , υ 3 , d o ( υ 1 ) ( 0 , k ) = ( a c k 2 / 2 ) [ ( e ˆ 1 · e ˆ ) 2 + ( e ˆ 1 · e ˆ ) 2 ] , d ex ( i ) ( 0 , k ) = ( a i k c c k + c i k a c k ) 2 ( k ˆ · e ˆ ) 2 , i = υ 2 , υ 3 , d ex ( υ 1 ) ( 0 , k ) = ( a c k 2 / 2 ) [ ( e ˆ 1 · e ˆ ) 2 + ( e ˆ 1 · e ˆ ) 2 ] .
| e ˆ 1 · e ˆ | 2 = | e ˆ 1 · e ˆ | 2 = ( 1 / 2 ) ( cos 2 ϕ + sin 2 ϕ cos 2 θ ) , | e ˆ 1 · e ˆ | 2 = | e ˆ 1 · e ˆ | 2 = ( 1 / 2 ) sin 2 θ , | k ˆ · e ˆ | 2 = sin 2 θ sin 2 ϕ , | k ˆ · e ˆ | 2 = cos 2 θ ,
d o ( υ i ) ( k ) = ( 1 / 4 π ) Ω d o ( υ i ) ( 0 , k ) d Ω , i = 1 , 2 , 3 ,
d ex ( υ i ) ( k ) = ( 1 / 4 π ) Ω d ex ( υ i ) ( 0 , k ) d Ω , i = 1 , 2 , 3 ,
d o ( υ i ) ( k ) = ( 1 / 3 ) a c k 2 ( 1 / 3 ) ,
d o ( υ 2 ) ( k ) = ( 1 / 3 ) a c k 2 F ( u ) ( 1 / 3 ) ,
d o ( υ 3 ) ( k ) = ( 1 / 3 ) [ 1 / γ 2 ( u ) ] ( 1 / 3 ) , u 0 k 0 ,
d ex ( υ i ) ( k ) = d o ( υ i ) ( k ) , i = 1 , 2 , 3 , cubic zinc blende ,
F ( u ) = [ 1 + Ru ( 1 + u ) u ] 2 u ( 1 + u ) + [ 1 + Ru ( 1 + u ) ] 2 1 , u 0 k 0 .
d o ( υ 1 ) ( 0 ) = d o ( υ 2 ) ( 0 ) = d o ( υ 3 ) ( 0 ) = { 1 / 3 , cubic zinc blende d o , hexagonal wurtzite ;
d ex ( υ 1 ) ( 0 ) = d ex ( υ 2 ) ( 0 ) = d ex ( υ 3 ) ( 0 ) = { 1 / 3 , cubic zinc blende d ex , hexagonal wurtzite .
[ n o 2 n ex 2 ] = 1 + 4 π e 2 / m n w g 2 1 2 i = 1 3 2 ( 2 π ) 3 k 2 d k { 4 π d o ( υ i ) ( k ) 4 π d e x ( υ i ) ( k ) } × f υ i , k 0 f c , k 0 z 2 + ƛ r i 2 k 2 = 1 + 4 π e 2 / m n w g 2 1 2 [ 4 π d o 4 π d ex ] 2 ( 2 π ) 3 × k 2 d k [ ( 1 + u ) ( 1 + 2 u ) ( f υ 1 0 f c k 0 z 2 + ƛ 1 2 k 2 ) + F ( u ) ( 1 + u ) ( 1 + 2 u ) × ( f υ 2 0 f c k 0 z 2 + ƛ r 2 2 k 2 ) + 1 ( 1 + 2 u ) 2 ( f υ 3 k 0 f c k 0 z 2 + ƛ r 3 2 k 2 ) ] ,
a c k 2 = ( 1 + u ) ( 1 + 2 u ) 1 = a c k 2 | k = 0 u = 0
f i = [ 1 + ( m n / m p 1 ) ] 3 / 2 [ 1 + ( m n / m p i ) ] 3 / 2 { 1 , m p i m p 1 0 , m p i 0 .
n o 2 = 1 + 2 c o o [ y B z tan 1 ( y B / z ) ] ,
n ex 2 = 1 + 2 c o ex [ y B z tan 1 ( y B / z ) ] ,
c o o = ( 3 / 2 ) η o ( w υ 2 / w g 2 ) ,
c o ex = ( 3 / 2 ) η ex ( w υ 2 / w g 2 ) ,
η o = ( d o / 2 ) i ( 1 + m n / m p 1 ) 3 / 2 ( 1 + m n / m p i ) 3 / 2 ,
η ex = ( d ex / 2 ) i ( 1 + m n / m p 1 ) 3 / 2 ( 1 + m n / m p i ) 3 / 2 ,
d o d ex = n o 2 1 n ex 2 1 .
n 2 = 1 + [ i ( 1 + m n / m p 1 ) 3 / 2 ( 1 + m n / m p i ) 3 / 2 ] × ( w υ 2 / w g 2 ) [ y B z tan 1 ( y B / z ) ] ,
y B = m ( a 0 a ) , II-VI cubic zinc blende compounds , m = ( 0.346 ± 0.020 ) Å 1 , a 0 = ( 17.37 ± 0.88 ) Å .
y B = M 1 a + M 2 C + M 3 , II-VI hexagonal wurtzite compounds , M 1 = 60.07 ± 0.52 ( Å 1 ) , M 2 = 42.00 ± 0.35 ( Å 1 ) , M 3 = 30.63 ± 0.19 ,
d o / d ex = n o 2 1 n ex 2 1 .
d o / d ex 0.98 , 0 x 1 ,
0.453 d o 0.237 , 0 x 1.
n ex n o [ ( d ex / d o ) 1 ] ( n o 2 1 ) 2 n o ,

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