- Email: [email protected]
Intensity of two-photon excitonic absorption in two-band and three-band models
Sofid State Communications,
Vol. 14, pp. 873-876, 1974.
Pergamon Press.
Printed in Great Britain
INTENSITY OF TWO-PHOTON EXCITONIC ABSORPTION IN TWO-BAND AND THREE-BAND MODELS E. Doni and G. Pastori Parravicini lstituto di Fisica deli'Universit~, Pisa, Italy and Gruppo Nazionale di Struttura della Materia del CNR, Pisa and R. Girlanda Istituto di Fisica dell'Universita, Messina, Italy and Gruppo Nazionale di Struttura della Materia del CNR, Messina and Pisa (Received 5 N o v e m b e r 1973)
The intensity of two.photon absorption to Wannier excitons is estimated in simplified models. Both two-band and three-band models simultaneously contribute to the optical two.photon spectrum, but it is shown that the two-band model is favoured in crystals with relevant electron-hole interaction. As an example, it is explained why two-photon absorption spectrum in CuCl exhibits ls, 2,o, 3p exciton states.
From second-order perturbation theory, e the absorption coefficient for photons of energy hco~ and polarization el, in the presence of N2 photons per unit volume of energy h ~ and polarization e 2 is given by
THE TWO-PHOTON absorption to Wannier excitons in crystals has been studied on the basis of time. dependent perturbation theory. The three-band model 1 and the two-band model 2 have been extensively used to interpret experimental spectra. We wish to remark explicitely that both models include processes that contribute to the optical absorption spectrum of actual crystals. In this paper we calculate for the first time the relative importance of these two contributions. It will be shown that the ratio of these two contributions depends on the dipols matrix elements between Bloch functions of interest; moreover it is found that a high dielectric constant favours the situation of the three-band model.
a(col) = Ct~ [(1 +PI2)
I2
e2~l'~-~Eo~'~l'(flPll)(llPI0)'el 6(Et -- e o -- h w l -- h.o2)
(1)
where C = (8nShe2N2)/(cm4nln2wl¢o2), [ 0 > is the ground state, I! > and I f > are intermediate and Final states of the crystal. In (1) P = ~ Pl, where Pl is the momentum operator for the/-th electron; the sum over f runs over all fmal degenerate states and the sum over ! runs over all intermediate states. P n interchanges et with e2 and ¢ol with ¢o2.
Here, we consider the case in which all the dipole interband matrix elements of interest are non vanishing. As an application of the theory, we discuss some features of the experimental twophoton absorption spectrum of CuCI. a.4 The case of symmetry-forbidden interband transitions can be treated with similar procedures and will be presented elsewhere. 5
We now apply (1) to a simplified model semi. conductor under the foUowing assumptions: (i) the 873
874
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
ground state is taken to be the Slater determinant ~o formed with all valence Bloch functions. (ii) The intermediate states are taken to be free electron-hole trial excited states ~c-,- v k obtained by replacing in 4'o the valence Bloch fun~ction ~0u](r,k) with the conduction function ~¢i(r, k). (iii) The final exciton states (with total wave vector l~x = 0) are written as:
l#! = ZkAt(k)Oc,k,vtk
a(wl) = CI~I(]+Pt2).,Z civj [e='(~At(k) X ~c,j,,v, klPt~c?,,,v/k,)(~&,,,v~k'lFl4~o)
•
ea] X
O) We notice that g (ePc#'.vjk'lPI ¢~o) = Mc~vj(k') = <¢,~(r, k')ipl Svi(r, k')>
(4) We also have: ~
1
fr(thxee'bemd)(6Jl) = C~f IFt(0)l 2 I(1 + P,2)e2"
Mc'ciMciv' ct
cj E c + Ac~ - h~°l
el12~
4- 2~ EG + Av~-- hoot vl - v ,
a - ~ - h~o~ - h~oa
(7)
where R is the exciton binding energy, E a = Ec,(O) Ev,(0), Aci = Eci(0) --Ec,(0), A,j = Ev,(0) --Ev/(O). In deriving (7), we have taken advantage of the fact that At(k ) are strongly peaked in k.space, so that we have neglected the k-dependence in the energy denominators. From (7), we have that the three-band model leads to s-envelope excitons with [Ft(0)l 2 =
In the two-band model, 2 (6) holds, and (3) becomes: h2m 2 ,,("~°-~'"~)(,~,) = c 7 ~IVF:(O):I(1 + e , ~ ) x
e2"M%vVFf(O)'e,128(E R hco,--hco2)(8) (Ec - h~x)IVFt(0)I I
~ - n-~ -
The two-band model thus leads to p-envelope excitons with IVFt(0)I 2 = (n 2 - 1)/3rmSa~x .al
=
80k,~t'Mi,v.(k ) if c i = c~ and ~. #: v] (Sa) 5k.k'Mciv,(k) if v~ = h and ci #: cl (5b) ~,k'
In the three-band model a (Sa) and (5b) hold, and (3) becomes
l /rm3 a ~ .L:n
X
× [Eci(O ) _ Evj(O) _ h ~ 1 ]-1 [2 ~ ( E t _ Eo -- hw, -- hw2)
(4>~,k,v:,lPI ¢~)cik',vjk' )
suppose that all Mciv/(k ) of interest are non vanishing at k = 0, so that we can neglect their k-dependence.
(2)
where ¢'e,k,v,kare free electron-hole excited states formed with the top valence and bottom conduction bands. In the Wannier approach, the Fourier transforms Ft(r ) of A r ( k ) satisfy the effective mass equations. "t's The assumptions (i) and (iii) are standard. s The assumption (ii) is here adopted because using free electron-hole states instead of exciton states is justified whenever exciton binding energies are small with respect to band energy gaps; this allows a remarkable simplification in calculation. Equation (1) becomes:
Vol. 14, No. 9
From (7) and (8) we can write the intensity ratio for two-photon transitions to ns and np excitons (with n >/2) as:
~(two-band)(cOl ) /,12 __ 1 1 IA(tW°'band)12 Q(tkree.band)(o.~l ) = 3n------T - e2 IA(tla~ee.band)[2
[E~,(k)-E~,(k)] if c i = ct and v j = v t
(5c)
otherwise
(5d)
where (5c) is obtained from Slater's rulesa and the k • p approach. ~° In the following, we consider excitons at the edge of isotropic critical point Mo at k = 0; equation (5c) then becomes
((bc,k.v,klPl~e,k,,v#,> = ~51t,k'mhk. /a where p is the reduced effectivemass. Moreover we
(6)
(9)
where A (tw°'band) and A Cthree'band) are obtained by inspection from (8) and (7), respectively. For n ~ 2 excitons, we have transitions to ns and np excitons in the three- and two-band models respectively, the three-band contribution being favoured in crystals with high dielectric constant e. In the limit e ~ .o (neglect exciton effects) we recover the standard interband two-photon results, x2 Incidentally, we notice that the three- and two-band contributions for excitons share close similarity with inter-inter and inter-intra contributions to two.photon absorption to Landau levels,Is the external magnetic field playing a role similar to the electron-hole interaction.
Vol. 14, No. 9
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
Recently, very detailed two-photon absorption measurements became available for CuCI. a It has been found that the two-photon absorption spectrum of CuCl exhibits ls and np (n ;~ 2) exciton states. The above theory is well suited to interpreting the simultaneous appearance of exciton states associate with both three, and two-band models. The calculated band structure of CuCl is schematically reported in Fig. l(a); a4 for simplicity spin-orbit effects have been neglected because irrelevant in this calculation. In Fig. l(b) the relevant exciton states of the crystal are schematically shown. The independent matrix dements l~lvj(0 ) of interest are given in Table 1.
-I®" _~_
..... 1 lll a~ ~[,ii~E~iii 1 i?i ii'iiii
875
Table 1. Matrix elements of interest for CuCl in units of 10"~°ergV2gr t/2. -i(rflp~lrf/(x)>
= 1.2" - - 6 •7 ¢~,c~ 1/2 t"
--i (r~lp~,l r~s(X))=
--i (r[$(x)lp, lrr~(z)) = 0 . 4 6 . - i (rl's~(x)l p , I r [ / ( z ) >
=
--i(F[lp~lF[~x)) = 4 . 2 " * Calculated by SONG K.S., Solid State Commun. 12, 865 (1973). t fe,e, denotes the oscillator strength between states F~ and F c15. Our calculation in the tight-binding approach.
.........:i in the calculation we have taken into account the actual experimental geometrical situation, a the actual values of t~ t and w2 a and the essential degeneracies of crystal states. The value of ~ has been taken to be
....6C'"r~i ~i°:i!iii~!i
,
6. is We obtain a(two'band)
n 2 -- ]
~(three-band) ~' 7
.....
- 0.16 ~
, iiiiiiii
FIG. 1. (a) Schematic band structure of CuC1 near k = 0 (after reference 14). The irreducible representations of Bloch states at k = 0 are also indicated. (b) Schematic diagram of relevant exciton states. Processes relevant in three-band model are indicated with letters a and b; two-band model process is indicated with letter c. Process involving F~'2 valence band ate forbidden because of symmetry. Because of the difficulty in calculating fc,c of Table 1, we notice that I'[ and I'[s are mainly formed with 4s and 4p atomic orbitals of Cu, respectively. In the atom the computed oscillator strength is 0.97, ~5 and we take this value as an upper limit for fc~e~. We can now estimate (9) using the numerical values of Table 1 and the computed energies at k = 0; 14
5.
(10)
The ratio given by (I 0) is in reasonable agreement with experimental data, a'16 which give for n = 2 ¢t(tw°'~ad)/a(tnree'band) ~- 12. We wish also to notice that an accurate calculation of fc~c2 is likely to give a value less than the assumed one, so increasing the ratio (10). In conclusion we have given for the first time a suitable model for the calculation of two-photon exciton absorption and we have shown that both two-band and three-band model ate needed for a complete interpretation of two-photon spectra, as in the case of CuC1. Finally, we wish to remark that twoand three-band models give different polarization dependence, iv and experiments of this kind would be useful in separating two- and three-band contributions. 5
Acknowledgements - The authors are grateful to Prof. F. Bassani for helpful discussions.
876
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
Vol. 14, No. 9
REFERENCES
1.
LOUDON R., Prec. Phys. Soc. Lend., 80,952 (1962).
2.
MAHAN G.D.,Phys. Rev. 170, 825 (1968).
3.
BIVASA., MARANGE C., GRUN J.B. and SCHWAB C., Optics Commu~ 6, 142 (1972).
4.
FROEHLICH D., STAGINNUS B. and SCHOENHERR E.,Phys. Rev. Lett. 19, 1(~32(1967); FROEHLICH D., MOHLER E. and WlESNER P., Phys. Rev. Lett. 26, 554 (1971).
5.
DONI E., GIRLANDA R. and PASTOR/PARRAVICINI G., to be published.
6.
GOEPPERT.MAYER M.,Ann. Phys. 9,273(1931).
7.
ONODERA Y. and TOYOZAWA Y., J. Phys. Soc. Japan 22, 833 (1967).
8.
See, for example, DIMMOCK J.O., in Semiconductors and Semimetals, (edited by WILLARDSON R~(. and BEER A.C., Vol. 3, p. 259, Academic Press, New York (1967); BASSANI F. and PASTORI PARRAVICINI G., Electronic States and Optical Transitions in Solids, Pergamon Press, Oxford (in press).
9.
SLATERJ.C.,Phys. Rev. 34, 1293(1929);38, 1109(1931).
10.
KITTEL C., Quantum Theory of Solids, p. 185. Wiley, New York (1963).
11.
ELLIOTTR.J.,Phys. Rev. 108, 1384(1957).
12.
BRAUNSTEIN R.,Phys. Rev. 125,475 (1962); BASSANI F. and HASSAN A.R., Optics Commun. 1,371 (1970); HASSAN A.R.,Nuovo Cint B 70, 21 (1970).
13.
BASSANIF. and GIRLANDA R., Optics Commun. 1, 359 (1979); GIRLANDA R., Nuovo C/n~ B 6, 53 (1971).
14. 15.
S~NG K.S.~J. Phys. (Parisj 28~195 (1967);J.Phys. Cherr~ S~lid$ 2 8 " 2 ~ 3 (~967);KHANM.A.~S~lid State CommurL 11,587 (1972); CALABRESE E. and FOWLERW.B.,Phys. Stares Solidi (b) 57, 135 (1973). MCGINNG.,J. Phys. Chem. 50, 1404 (1969).
16.
RINGEISSEN J. and NIKITINE S.,J. Phys. (ParisJ 28, C3-48 (1967).
17.
INOUE M. and TOYOZAWA Y., J. Phys. Soc. Japan 20, 363 (1965); BADER T.R. and GOLD A.,Phys. Rev. 171,997 (1968); DENISOV M.M. and MAKAROV V.P., J. Phys. C: Solid State Phys. 5,265 (1972).
C nOMO~b~ y r r p ~ F a s x Mo~e~e~ ~a~a oueHKa HHTeHCHBHOCTH ~ByX~OTOHHOPO nor~omea~s K 3ECHTOHaM Baabe. ~ByX- H Tp~x3OHHa£ Mo~e~H O~HOBpeMeHHO C~OC06CTBy~T ~By×~OTOHHOMy OHTHqOCKOMy cneKTpy, HO noKaBaao, qTO B K~HCTa~ax CO 3Haq~ITe~bHNM 9~eKTpOHHO-~I~pOHH~M B3aHMo~eRCTBHeM ~B~X'3OHHa~
MO~e~b ~zaeTca 6once n o ~ x o ~ e ~ . B KaqecTBe npHMepa npHBO~MTCn o6~acaeHne aa~Mqxa a ~ByX~X)TOHHOMcnexTpe Kp, c T a ~ a CuCI ~KCHTOHX~×ypoBne~ lS, 2p, 3p.
Vol. 14, pp. 873-876, 1974.
Pergamon Press.
Printed in Great Britain
INTENSITY OF TWO-PHOTON EXCITONIC ABSORPTION IN TWO-BAND AND THREE-BAND MODELS E. Doni and G. Pastori Parravicini lstituto di Fisica deli'Universit~, Pisa, Italy and Gruppo Nazionale di Struttura della Materia del CNR, Pisa and R. Girlanda Istituto di Fisica dell'Universita, Messina, Italy and Gruppo Nazionale di Struttura della Materia del CNR, Messina and Pisa (Received 5 N o v e m b e r 1973)
The intensity of two.photon absorption to Wannier excitons is estimated in simplified models. Both two-band and three-band models simultaneously contribute to the optical two.photon spectrum, but it is shown that the two-band model is favoured in crystals with relevant electron-hole interaction. As an example, it is explained why two-photon absorption spectrum in CuCl exhibits ls, 2,o, 3p exciton states.
From second-order perturbation theory, e the absorption coefficient for photons of energy hco~ and polarization el, in the presence of N2 photons per unit volume of energy h ~ and polarization e 2 is given by
THE TWO-PHOTON absorption to Wannier excitons in crystals has been studied on the basis of time. dependent perturbation theory. The three-band model 1 and the two-band model 2 have been extensively used to interpret experimental spectra. We wish to remark explicitely that both models include processes that contribute to the optical absorption spectrum of actual crystals. In this paper we calculate for the first time the relative importance of these two contributions. It will be shown that the ratio of these two contributions depends on the dipols matrix elements between Bloch functions of interest; moreover it is found that a high dielectric constant favours the situation of the three-band model.
a(col) = Ct~ [(1 +PI2)
I2
e2~l'~-~Eo~'~l'(flPll)(llPI0)'el 6(Et -- e o -- h w l -- h.o2)
(1)
where C = (8nShe2N2)/(cm4nln2wl¢o2), [ 0 > is the ground state, I! > and I f > are intermediate and Final states of the crystal. In (1) P = ~ Pl, where Pl is the momentum operator for the/-th electron; the sum over f runs over all fmal degenerate states and the sum over ! runs over all intermediate states. P n interchanges et with e2 and ¢ol with ¢o2.
Here, we consider the case in which all the dipole interband matrix elements of interest are non vanishing. As an application of the theory, we discuss some features of the experimental twophoton absorption spectrum of CuCI. a.4 The case of symmetry-forbidden interband transitions can be treated with similar procedures and will be presented elsewhere. 5
We now apply (1) to a simplified model semi. conductor under the foUowing assumptions: (i) the 873
874
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
ground state is taken to be the Slater determinant ~o formed with all valence Bloch functions. (ii) The intermediate states are taken to be free electron-hole trial excited states ~c-,- v k obtained by replacing in 4'o the valence Bloch fun~ction ~0u](r,k) with the conduction function ~¢i(r, k). (iii) The final exciton states (with total wave vector l~x = 0) are written as:
l#! = ZkAt(k)Oc,k,vtk
a(wl) = CI~I(]+Pt2).,Z civj [e='(~At(k) X ~c,j,,v, klPt~c?,,,v/k,)(~&,,,v~k'lFl4~o)
•
ea] X
O) We notice that g (ePc#'.vjk'lPI ¢~o) = Mc~vj(k') = <¢,~(r, k')ipl Svi(r, k')>
(4) We also have: ~
1
fr(thxee'bemd)(6Jl) = C~f IFt(0)l 2 I(1 + P,2)e2"
Mc'ciMciv' ct
cj E c + Ac~ - h~°l
el12~
4- 2~ EG + Av~-- hoot vl - v ,
a - ~ - h~o~ - h~oa
(7)
where R is the exciton binding energy, E a = Ec,(O) Ev,(0), Aci = Eci(0) --Ec,(0), A,j = Ev,(0) --Ev/(O). In deriving (7), we have taken advantage of the fact that At(k ) are strongly peaked in k.space, so that we have neglected the k-dependence in the energy denominators. From (7), we have that the three-band model leads to s-envelope excitons with [Ft(0)l 2 =
In the two-band model, 2 (6) holds, and (3) becomes: h2m 2 ,,("~°-~'"~)(,~,) = c 7 ~IVF:(O):I(1 + e , ~ ) x
e2"M%vVFf(O)'e,128(E R hco,--hco2)(8) (Ec - h~x)IVFt(0)I I
~ - n-~ -
The two-band model thus leads to p-envelope excitons with IVFt(0)I 2 = (n 2 - 1)/3rmSa~x .al
=
80k,~t'Mi,v.(k ) if c i = c~ and ~. #: v] (Sa) 5k.k'Mciv,(k) if v~ = h and ci #: cl (5b) ~,k'
In the three-band model a (Sa) and (5b) hold, and (3) becomes
l /rm3 a ~ .L:n
X
× [Eci(O ) _ Evj(O) _ h ~ 1 ]-1 [2 ~ ( E t _ Eo -- hw, -- hw2)
(4>~,k,v:,lPI ¢~)cik',vjk' )
suppose that all Mciv/(k ) of interest are non vanishing at k = 0, so that we can neglect their k-dependence.
(2)
where ¢'e,k,v,kare free electron-hole excited states formed with the top valence and bottom conduction bands. In the Wannier approach, the Fourier transforms Ft(r ) of A r ( k ) satisfy the effective mass equations. "t's The assumptions (i) and (iii) are standard. s The assumption (ii) is here adopted because using free electron-hole states instead of exciton states is justified whenever exciton binding energies are small with respect to band energy gaps; this allows a remarkable simplification in calculation. Equation (1) becomes:
Vol. 14, No. 9
From (7) and (8) we can write the intensity ratio for two-photon transitions to ns and np excitons (with n >/2) as:
~(two-band)(cOl ) /,12 __ 1 1 IA(tW°'band)12 Q(tkree.band)(o.~l ) = 3n------T - e2 IA(tla~ee.band)[2
[E~,(k)-E~,(k)] if c i = ct and v j = v t
(5c)
otherwise
(5d)
where (5c) is obtained from Slater's rulesa and the k • p approach. ~° In the following, we consider excitons at the edge of isotropic critical point Mo at k = 0; equation (5c) then becomes
((bc,k.v,klPl~e,k,,v#,> = ~51t,k'mhk. /a where p is the reduced effectivemass. Moreover we
(6)
(9)
where A (tw°'band) and A Cthree'band) are obtained by inspection from (8) and (7), respectively. For n ~ 2 excitons, we have transitions to ns and np excitons in the three- and two-band models respectively, the three-band contribution being favoured in crystals with high dielectric constant e. In the limit e ~ .o (neglect exciton effects) we recover the standard interband two-photon results, x2 Incidentally, we notice that the three- and two-band contributions for excitons share close similarity with inter-inter and inter-intra contributions to two.photon absorption to Landau levels,Is the external magnetic field playing a role similar to the electron-hole interaction.
Vol. 14, No. 9
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
Recently, very detailed two-photon absorption measurements became available for CuCI. a It has been found that the two-photon absorption spectrum of CuCl exhibits ls and np (n ;~ 2) exciton states. The above theory is well suited to interpreting the simultaneous appearance of exciton states associate with both three, and two-band models. The calculated band structure of CuCl is schematically reported in Fig. l(a); a4 for simplicity spin-orbit effects have been neglected because irrelevant in this calculation. In Fig. l(b) the relevant exciton states of the crystal are schematically shown. The independent matrix dements l~lvj(0 ) of interest are given in Table 1.
-I®" _~_
..... 1 lll a~ ~[,ii~E~iii 1 i?i ii'iiii
875
Table 1. Matrix elements of interest for CuCl in units of 10"~°ergV2gr t/2. -i(rflp~lrf/(x)>
= 1.2" - - 6 •7 ¢~,c~ 1/2 t"
--i (r~lp~,l r~s(X))=
--i (r[$(x)lp, lrr~(z)) = 0 . 4 6 . - i (rl's~(x)l p , I r [ / ( z ) >
=
--i(F[lp~lF[~x)) = 4 . 2 " * Calculated by SONG K.S., Solid State Commun. 12, 865 (1973). t fe,e, denotes the oscillator strength between states F~ and F c15. Our calculation in the tight-binding approach.
.........:i in the calculation we have taken into account the actual experimental geometrical situation, a the actual values of t~ t and w2 a and the essential degeneracies of crystal states. The value of ~ has been taken to be
....6C'"r~i ~i°:i!iii~!i
,
6. is We obtain a(two'band)
n 2 -- ]
~(three-band) ~' 7
.....
- 0.16 ~
, iiiiiiii
FIG. 1. (a) Schematic band structure of CuC1 near k = 0 (after reference 14). The irreducible representations of Bloch states at k = 0 are also indicated. (b) Schematic diagram of relevant exciton states. Processes relevant in three-band model are indicated with letters a and b; two-band model process is indicated with letter c. Process involving F~'2 valence band ate forbidden because of symmetry. Because of the difficulty in calculating fc,c of Table 1, we notice that I'[ and I'[s are mainly formed with 4s and 4p atomic orbitals of Cu, respectively. In the atom the computed oscillator strength is 0.97, ~5 and we take this value as an upper limit for fc~e~. We can now estimate (9) using the numerical values of Table 1 and the computed energies at k = 0; 14
5.
(10)
The ratio given by (I 0) is in reasonable agreement with experimental data, a'16 which give for n = 2 ¢t(tw°'~ad)/a(tnree'band) ~- 12. We wish also to notice that an accurate calculation of fc~c2 is likely to give a value less than the assumed one, so increasing the ratio (10). In conclusion we have given for the first time a suitable model for the calculation of two-photon exciton absorption and we have shown that both two-band and three-band model ate needed for a complete interpretation of two-photon spectra, as in the case of CuC1. Finally, we wish to remark that twoand three-band models give different polarization dependence, iv and experiments of this kind would be useful in separating two- and three-band contributions. 5
Acknowledgements - The authors are grateful to Prof. F. Bassani for helpful discussions.
876
ABSORPTION IN TWO-BAND AND THREE-BAND MODELS
Vol. 14, No. 9
REFERENCES
1.
LOUDON R., Prec. Phys. Soc. Lend., 80,952 (1962).
2.
MAHAN G.D.,Phys. Rev. 170, 825 (1968).
3.
BIVASA., MARANGE C., GRUN J.B. and SCHWAB C., Optics Commu~ 6, 142 (1972).
4.
FROEHLICH D., STAGINNUS B. and SCHOENHERR E.,Phys. Rev. Lett. 19, 1(~32(1967); FROEHLICH D., MOHLER E. and WlESNER P., Phys. Rev. Lett. 26, 554 (1971).
5.
DONI E., GIRLANDA R. and PASTOR/PARRAVICINI G., to be published.
6.
GOEPPERT.MAYER M.,Ann. Phys. 9,273(1931).
7.
ONODERA Y. and TOYOZAWA Y., J. Phys. Soc. Japan 22, 833 (1967).
8.
See, for example, DIMMOCK J.O., in Semiconductors and Semimetals, (edited by WILLARDSON R~(. and BEER A.C., Vol. 3, p. 259, Academic Press, New York (1967); BASSANI F. and PASTORI PARRAVICINI G., Electronic States and Optical Transitions in Solids, Pergamon Press, Oxford (in press).
9.
SLATERJ.C.,Phys. Rev. 34, 1293(1929);38, 1109(1931).
10.
KITTEL C., Quantum Theory of Solids, p. 185. Wiley, New York (1963).
11.
ELLIOTTR.J.,Phys. Rev. 108, 1384(1957).
12.
BRAUNSTEIN R.,Phys. Rev. 125,475 (1962); BASSANI F. and HASSAN A.R., Optics Commun. 1,371 (1970); HASSAN A.R.,Nuovo Cint B 70, 21 (1970).
13.
BASSANIF. and GIRLANDA R., Optics Commun. 1, 359 (1979); GIRLANDA R., Nuovo C/n~ B 6, 53 (1971).
14. 15.
S~NG K.S.~J. Phys. (Parisj 28~195 (1967);J.Phys. Cherr~ S~lid$ 2 8 " 2 ~ 3 (~967);KHANM.A.~S~lid State CommurL 11,587 (1972); CALABRESE E. and FOWLERW.B.,Phys. Stares Solidi (b) 57, 135 (1973). MCGINNG.,J. Phys. Chem. 50, 1404 (1969).
16.
RINGEISSEN J. and NIKITINE S.,J. Phys. (ParisJ 28, C3-48 (1967).
17.
INOUE M. and TOYOZAWA Y., J. Phys. Soc. Japan 20, 363 (1965); BADER T.R. and GOLD A.,Phys. Rev. 171,997 (1968); DENISOV M.M. and MAKAROV V.P., J. Phys. C: Solid State Phys. 5,265 (1972).
C nOMO~b~ y r r p ~ F a s x Mo~e~e~ ~a~a oueHKa HHTeHCHBHOCTH ~ByX~OTOHHOPO nor~omea~s K 3ECHTOHaM Baabe. ~ByX- H Tp~x3OHHa£ Mo~e~H O~HOBpeMeHHO C~OC06CTBy~T ~By×~OTOHHOMy OHTHqOCKOMy cneKTpy, HO noKaBaao, qTO B K~HCTa~ax CO 3Haq~ITe~bHNM 9~eKTpOHHO-~I~pOHH~M B3aHMo~eRCTBHeM ~B~X'3OHHa~
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