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Analysis of derivative spectrum of indirect exciton absorption in silicon
Solid State Communications,
Vol. 12, pp.1 137—1140, 1973.
Printed in Great Britain
Pergamon Press.
ANALYSIS OF DERIVATIVE SPECTRUM OF INDIRECT EXCITON ABSORPTION IN SILICON 1. Nishino, M. Takeda and Y. Hamakawa Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan (Received 6 March 1973 by H. Kawwnura)
It is shown that an analysis of the wavelength modulated absorption spectrum enables us to estimate the transition matrix elements in indirect absorption. The transition matrix elements in silicon are determined to be 0.ilO~’Afor indirect absorption with the TO phonon, O.0367h/A for the LO phonon and 0.0178 h/A for the TA phonon.
IT HAS been proposed that transition matrix elements in indirect absorption can be easily estimated by analysis of the wavelength modulated absorption spectrum.1 In this paper we report the results of measurements on silicon. From the analysis of data we have estimated associated with TO, LO and TA phonons, in addition the transition matrix elements in indirect absorption to the exciton energy gap and the exciton Rydberg of
SILICON
TA~.I
0.005
~.&,
2
_____
silicon. -O E 0.2
In modulation spectroscopy, the wavelength modu-
_____________
TOn.i
lated absorption measurement is particularly useful to investigate indirect exciton absorption.2 That is, the square root dependence on energy of indirect exciton absorption coefficient is transformed into sharp peak structures at the threshold energies in the wavelength derivative spectrum. l’his type of measurement for the study of indirect excitons in silicon has already been carried out, and the sharp peak structures expected have been observed by several workers.35 However, none of them has tried a quantitative analysis of the wavelength derivative spectrum and determined the transition matrix elements involved in the indirect absorption process. Furthermore, such an anlysis has not also been reported in other indirect gap semiconductors except silicon.
A
4
o.i
0
_________________________________ .~,
o
~
clW-E,X.l~WP(meV)
Figure 1 shows the wavelength derivative absorption spectrum of indirect excitons in silicon taken at I 8°K. The structures in the figure are due to the formations 1137
FIG. I. The wavelength derivative absorption spectra of silicon at 1 .8°Kin the region of TO, LO and TA phonon-assi.sted indirect exciton absorption. The experimental curves (solid line) are compared with the best fit to the F(x) function (dashed line) for the n = I exciton states with the emission of TO and TA phonons.
1138
INDIRECT EXCITON ABSORPTION IN Si
of then = 1 exciton ground states and then = 2 excited states accompanied by the emission of TO, LO and TA phonons, respectively. Wavelength modulation was performed using a quartz plate vibrated by a bimorph ceramic transducer inside the exit slit of a monochromator. The amplitude of wavelength modulation ~ was determined from a vibrating angle of a quartz plate, and the angle was measured accurately by using reflection of He—Ne laser beam. The energy resolution of the optical system at the slit widths used in this experiment is well within 0.2meV.
Vol.
12,
No. 11
the LO phonon and 0.21 ±0.01meV for the TA phonon, respectively. These values are compared with 0.3meV obtained in absolute absorption measurements by Dean er a!.9 030 ±0.04meV for
___________________________
SILICON
20
17.75
i.8 tc
TOn.i
E .~
10 LOn-i
Indirect exciton absorption coefficient with the emission of phonon is given by6 A
2.0
(1)
1.96
e
3 1_0
(~—Eex) 1/2
~-
0
0.479
—0-
TA,,., -r
0
0-
__________________
0.10 *f,w (meV) 0.05
where E 2 + fr~,,A is a constant including transition E~is energy gap, R exciton 0,, matrix = Eg elements. R/n Rydberg and ~~phonon energy. Considering lifetime broadening, the derivative of equation (I) becomes7
—ê-
0.15
,
—
FIG. 2. The relation between the measured A/l~.~ at .8°Kand the amplitude of ~lx~ for the n = I exciton states of TO, LU and TA phonon-assisted transitions in silicon.
A —
2(2r)”2 It.., F(x)
(2)
where F(x) is a well-known universal function in modu. lation spectroscopy and x = (It..~ Eex)Ir. And also in the derivation of equation (2) k~in A fIt.. is assumed to be constant because of the small energy region in which fine structures are observed. Thus, if one obtains a broadening parameter r from the analysis of the observed spectrum, one can determine the value of A proportional to the square of the transition matrix elements involved in the indirect absorption process. —
The spectra in Fig. I were analyzed according to equation (2). The parameters obtained by the best fit between experiment and theory are shown in Table 1. The exciton threshold energies of the n = 1 and n = 2 states for TO, LO and TA phonon-assisted indirect transitions are in good agreement with the results of Sialdee and Nahory4 within experimental error. The exciton Rydberg is 14.3meV for each phonon, and is also consistent with their result4 and the theoretical value.8 The intensity ratio of the n = I and fl = 2 exciton states was found to be 6—8: 1 and is very close to the ratio in a hydrogenic series of Wannier exciton. From the line shape fitting to F(x) function the broadening parameter r for the n = 1 exciton was determined to be 0.31 ±0.02meV for the TO phonon,
The prefactorA/l~.~ in equation (1) can be easily determined from the line shape and amplitude analysis of the wavelength derivative absorption spectrum, using the relation of equation (2). Figure 2 shows the relation between the measured A/It.. and the ampli. tude of wavelength modulation ~ for the n = 1 states of TO. LU and TA phonon-assisted indirect exciton absorption. The error in the evaluation of ~ is small within 5 per cent. The values of A/It.., are 17.75eV112cm’ for the TO phonon, and 0.479 for the TA phonon. These values are compared with 15.2 and 0.445,~and 18.08 and 0.50410 found by absolute absorption m~asurements.As can be seen in Fig. 2, the measured A/ho. does not change with ~hw, and therefore the relation of equation (2) has been justified. The matrix element M.~,of indirect absorption in silicon was estimated from the A values obtained here. M~,.is a transition matrix element which includes both electron—photon and electron—phonon matrix elements. The transition matrix elements of indirect absorption in silicon have not been determined experimentally except for the estimation by Lao er a!.1’ They obtained the matrix element for TO phononassisted transitions from the analysis of electroabsorp. tion data by Frova eta!.12 The value obtained was 0.0588h/A from the fit to their exciton electroabsorption theory and 0.l87h/A from one-electron theory.
Vol. 12, No. Ii
INDIRECT EXCITON ABSORPTION IN Si
1139
Table 1. Indirect exciton parameters of silicon determined by the analysis ofthe wavelength derivativespectra for indirect exciton absorption with the emission of TO, LO and TA phonons at 1.8°K
TO LO TA
r(mev)
E~(eV)
R(meV)
n=l n= 2
1.2127±0.0002 1.2234 ±0.0003
143±05
031±0.02
0.110
n= 1
1.2106 ±00003
143
030 ±0.04
0.0367
0.21
0.0178
n=2
n= 1
n2
1.2213±0.0003 1.1 733 ±0.0002
1.1840±0.0002
The corresponding transition matrix element obtained in our experiment is 0.11 Oh/A as shown in Table 1, where we used the same values of effective masses for electron and hole as those used by them. It is interesting to note that this value is just between the above two values which were obtained from the analysis of electroabsorption data. The transition matrix elements for LO and TA phonon-assisted transitions are 0.0367 h/A and 0.0178 h/A, respectively. Thus, transition matrix elements for indirect absorption can be more easily estimated by the wavelength derivative absorption measurements than by the analysis of electroabsorption measurements. As shown in Fig. 1, the derivative spectra for the n = I states of indirect exciton absorption with TO and TA phonons agree fairly well with the form of the F(x) function in the region close to each threshold, but deviate from F(x) above the n = 1 exciton region.
-
143
+
06
M~,(h/A)
—
±04
±0.01
This feature was also seen in the spectra measured by Shaklee and Nahory.4 This deviation cannot be cor-
rected by the consideration of It.., in A/It.., which is assumed to be constant in the derivation of equation (2), because it makes this deviation larger. Thus, it is difficult to explain indirect exciton absorption of silicon by a simple square root dependence on photon energy in the region between the ground and first excited states of indirect excitons. One should consider other effects; for example, the energy dependence of the broadening parameter 1’, non-parabolic effects of the bands involved and the complicated contribution of two more intermediate states in the transition process. One may consider that this deviation is related to the anomalous negative dip found in low field electroabsorption spectrum of silicon by Evangelisti eta!. 13.14 But it seems to be unreasonable because the negative dip was found only in the TO phonon-assisted indirect absorption edge.
REFERENCES
1.
NISHINO T., TAKEDA M. and HAMAKAWA Y., Proc. 1st mt. Conf on Modulation Specrroscopy, Tucson (1972).
2.
CARDONA M.,Modulanon Spectroscopy, p. 105, Academic Press, New York (1969).
3.
BALSLEV I.,Phys. Rev. 143,639(1966).
SHAKLEE K.L. and NAHORY R.E.,Phys. Rev. Lett. 24,942(1970). 5. NISHINO T. and HAMAKAWA Y,Thys. Status Solidi(b) 50,345 (1972). 6. ELL1OT RJ.,Thys. Rev. 108, 1384 (1957). 7. BATZ B.,Semiconductors and Semimeta!s, Vol.9, p.316 (Edited by WILLARDSON R.K. and BEER A.C.), Academic Press, New York (1972). LIPARI N.O. and BALDERESCHI A.,Fhys. Rev. B3, 2497 (1971).
8. 9. DEAN PJ., YAFET 3. and HAYNES J.R.,Thys. Rev. 184,837(1969). 10. MACFARLANE G.G., MCLEAN T.P., QUARRINGTON i.E. and ROBERTS V.,Thys. Rev. 111, 1245 (1958). 11. LAO B.Y., DOW J.D. and WEINSTEIN F.C., Pays. Rev. B4, 4424 (1971).
1140 12.
13. 14.
INDIRECT EXCITON ABSORPTION IN Si
Vol. 12, No. 11
FROVA A., HANDLER P., GERMANO F.A. and ASPNES D.E.,Thys. Rev. 145, 575 (1966). EVANGELIST! F., FROVA A. and ZANINI M.,Solid Stare Commun. 9, 1467 (1971).
EVANGELISTI F., FROVA A., ZANINI M. and KANE E.O., Solid State Commun. 11,611(1972).
Es wird gezeigt, dass die Ubergängsmatrixelemente für indirekten Absorption aus der Analysis der Wellenlangen-modulations-Spektren bestimmt werden kOnnen. Für indirekten Absorption von Siizium unter TO Phonon wird das Ubergängsmatrixelement 0.1 lOb/A bestimmt, O.0367h/A für LO Phonon und 0.0178 h/A für TA Phonon.
Vol. 12, pp.1 137—1140, 1973.
Printed in Great Britain
Pergamon Press.
ANALYSIS OF DERIVATIVE SPECTRUM OF INDIRECT EXCITON ABSORPTION IN SILICON 1. Nishino, M. Takeda and Y. Hamakawa Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan (Received 6 March 1973 by H. Kawwnura)
It is shown that an analysis of the wavelength modulated absorption spectrum enables us to estimate the transition matrix elements in indirect absorption. The transition matrix elements in silicon are determined to be 0.ilO~’Afor indirect absorption with the TO phonon, O.0367h/A for the LO phonon and 0.0178 h/A for the TA phonon.
IT HAS been proposed that transition matrix elements in indirect absorption can be easily estimated by analysis of the wavelength modulated absorption spectrum.1 In this paper we report the results of measurements on silicon. From the analysis of data we have estimated associated with TO, LO and TA phonons, in addition the transition matrix elements in indirect absorption to the exciton energy gap and the exciton Rydberg of
SILICON
TA~.I
0.005
~.&,
2
_____
silicon. -O E 0.2
In modulation spectroscopy, the wavelength modu-
_____________
TOn.i
lated absorption measurement is particularly useful to investigate indirect exciton absorption.2 That is, the square root dependence on energy of indirect exciton absorption coefficient is transformed into sharp peak structures at the threshold energies in the wavelength derivative spectrum. l’his type of measurement for the study of indirect excitons in silicon has already been carried out, and the sharp peak structures expected have been observed by several workers.35 However, none of them has tried a quantitative analysis of the wavelength derivative spectrum and determined the transition matrix elements involved in the indirect absorption process. Furthermore, such an anlysis has not also been reported in other indirect gap semiconductors except silicon.
A
4
o.i
0
_________________________________ .~,
o
~
clW-E,X.l~WP(meV)
Figure 1 shows the wavelength derivative absorption spectrum of indirect excitons in silicon taken at I 8°K. The structures in the figure are due to the formations 1137
FIG. I. The wavelength derivative absorption spectra of silicon at 1 .8°Kin the region of TO, LO and TA phonon-assi.sted indirect exciton absorption. The experimental curves (solid line) are compared with the best fit to the F(x) function (dashed line) for the n = I exciton states with the emission of TO and TA phonons.
1138
INDIRECT EXCITON ABSORPTION IN Si
of then = 1 exciton ground states and then = 2 excited states accompanied by the emission of TO, LO and TA phonons, respectively. Wavelength modulation was performed using a quartz plate vibrated by a bimorph ceramic transducer inside the exit slit of a monochromator. The amplitude of wavelength modulation ~ was determined from a vibrating angle of a quartz plate, and the angle was measured accurately by using reflection of He—Ne laser beam. The energy resolution of the optical system at the slit widths used in this experiment is well within 0.2meV.
Vol.
12,
No. 11
the LO phonon and 0.21 ±0.01meV for the TA phonon, respectively. These values are compared with 0.3meV obtained in absolute absorption measurements by Dean er a!.9 030 ±0.04meV for
___________________________
SILICON
20
17.75
i.8 tc
TOn.i
E .~
10 LOn-i
Indirect exciton absorption coefficient with the emission of phonon is given by6 A
2.0
(1)
1.96
e
3 1_0
(~—Eex) 1/2
~-
0
0.479
—0-
TA,,., -r
0
0-
__________________
0.10 *f,w (meV) 0.05
where E 2 + fr~,,A is a constant including transition E~is energy gap, R exciton 0,, matrix = Eg elements. R/n Rydberg and ~~phonon energy. Considering lifetime broadening, the derivative of equation (I) becomes7
—ê-
0.15
,
—
FIG. 2. The relation between the measured A/l~.~ at .8°Kand the amplitude of ~lx~ for the n = I exciton states of TO, LU and TA phonon-assisted transitions in silicon.
A —
2(2r)”2 It.., F(x)
(2)
where F(x) is a well-known universal function in modu. lation spectroscopy and x = (It..~ Eex)Ir. And also in the derivation of equation (2) k~in A fIt.. is assumed to be constant because of the small energy region in which fine structures are observed. Thus, if one obtains a broadening parameter r from the analysis of the observed spectrum, one can determine the value of A proportional to the square of the transition matrix elements involved in the indirect absorption process. —
The spectra in Fig. I were analyzed according to equation (2). The parameters obtained by the best fit between experiment and theory are shown in Table 1. The exciton threshold energies of the n = 1 and n = 2 states for TO, LO and TA phonon-assisted indirect transitions are in good agreement with the results of Sialdee and Nahory4 within experimental error. The exciton Rydberg is 14.3meV for each phonon, and is also consistent with their result4 and the theoretical value.8 The intensity ratio of the n = I and fl = 2 exciton states was found to be 6—8: 1 and is very close to the ratio in a hydrogenic series of Wannier exciton. From the line shape fitting to F(x) function the broadening parameter r for the n = 1 exciton was determined to be 0.31 ±0.02meV for the TO phonon,
The prefactorA/l~.~ in equation (1) can be easily determined from the line shape and amplitude analysis of the wavelength derivative absorption spectrum, using the relation of equation (2). Figure 2 shows the relation between the measured A/It.. and the ampli. tude of wavelength modulation ~ for the n = 1 states of TO. LU and TA phonon-assisted indirect exciton absorption. The error in the evaluation of ~ is small within 5 per cent. The values of A/It.., are 17.75eV112cm’ for the TO phonon, and 0.479 for the TA phonon. These values are compared with 15.2 and 0.445,~and 18.08 and 0.50410 found by absolute absorption m~asurements.As can be seen in Fig. 2, the measured A/ho. does not change with ~hw, and therefore the relation of equation (2) has been justified. The matrix element M.~,of indirect absorption in silicon was estimated from the A values obtained here. M~,.is a transition matrix element which includes both electron—photon and electron—phonon matrix elements. The transition matrix elements of indirect absorption in silicon have not been determined experimentally except for the estimation by Lao er a!.1’ They obtained the matrix element for TO phononassisted transitions from the analysis of electroabsorp. tion data by Frova eta!.12 The value obtained was 0.0588h/A from the fit to their exciton electroabsorption theory and 0.l87h/A from one-electron theory.
Vol. 12, No. Ii
INDIRECT EXCITON ABSORPTION IN Si
1139
Table 1. Indirect exciton parameters of silicon determined by the analysis ofthe wavelength derivativespectra for indirect exciton absorption with the emission of TO, LO and TA phonons at 1.8°K
TO LO TA
r(mev)
E~(eV)
R(meV)
n=l n= 2
1.2127±0.0002 1.2234 ±0.0003
143±05
031±0.02
0.110
n= 1
1.2106 ±00003
143
030 ±0.04
0.0367
0.21
0.0178
n=2
n= 1
n2
1.2213±0.0003 1.1 733 ±0.0002
1.1840±0.0002
The corresponding transition matrix element obtained in our experiment is 0.11 Oh/A as shown in Table 1, where we used the same values of effective masses for electron and hole as those used by them. It is interesting to note that this value is just between the above two values which were obtained from the analysis of electroabsorption data. The transition matrix elements for LO and TA phonon-assisted transitions are 0.0367 h/A and 0.0178 h/A, respectively. Thus, transition matrix elements for indirect absorption can be more easily estimated by the wavelength derivative absorption measurements than by the analysis of electroabsorption measurements. As shown in Fig. 1, the derivative spectra for the n = I states of indirect exciton absorption with TO and TA phonons agree fairly well with the form of the F(x) function in the region close to each threshold, but deviate from F(x) above the n = 1 exciton region.
-
143
+
06
M~,(h/A)
—
±04
±0.01
This feature was also seen in the spectra measured by Shaklee and Nahory.4 This deviation cannot be cor-
rected by the consideration of It.., in A/It.., which is assumed to be constant in the derivation of equation (2), because it makes this deviation larger. Thus, it is difficult to explain indirect exciton absorption of silicon by a simple square root dependence on photon energy in the region between the ground and first excited states of indirect excitons. One should consider other effects; for example, the energy dependence of the broadening parameter 1’, non-parabolic effects of the bands involved and the complicated contribution of two more intermediate states in the transition process. One may consider that this deviation is related to the anomalous negative dip found in low field electroabsorption spectrum of silicon by Evangelisti eta!. 13.14 But it seems to be unreasonable because the negative dip was found only in the TO phonon-assisted indirect absorption edge.
REFERENCES
1.
NISHINO T., TAKEDA M. and HAMAKAWA Y., Proc. 1st mt. Conf on Modulation Specrroscopy, Tucson (1972).
2.
CARDONA M.,Modulanon Spectroscopy, p. 105, Academic Press, New York (1969).
3.
BALSLEV I.,Phys. Rev. 143,639(1966).
SHAKLEE K.L. and NAHORY R.E.,Phys. Rev. Lett. 24,942(1970). 5. NISHINO T. and HAMAKAWA Y,Thys. Status Solidi(b) 50,345 (1972). 6. ELL1OT RJ.,Thys. Rev. 108, 1384 (1957). 7. BATZ B.,Semiconductors and Semimeta!s, Vol.9, p.316 (Edited by WILLARDSON R.K. and BEER A.C.), Academic Press, New York (1972). LIPARI N.O. and BALDERESCHI A.,Fhys. Rev. B3, 2497 (1971).
8. 9. DEAN PJ., YAFET 3. and HAYNES J.R.,Thys. Rev. 184,837(1969). 10. MACFARLANE G.G., MCLEAN T.P., QUARRINGTON i.E. and ROBERTS V.,Thys. Rev. 111, 1245 (1958). 11. LAO B.Y., DOW J.D. and WEINSTEIN F.C., Pays. Rev. B4, 4424 (1971).
1140 12.
13. 14.
INDIRECT EXCITON ABSORPTION IN Si
Vol. 12, No. 11
FROVA A., HANDLER P., GERMANO F.A. and ASPNES D.E.,Thys. Rev. 145, 575 (1966). EVANGELIST! F., FROVA A. and ZANINI M.,Solid Stare Commun. 9, 1467 (1971).
EVANGELISTI F., FROVA A., ZANINI M. and KANE E.O., Solid State Commun. 11,611(1972).
Es wird gezeigt, dass die Ubergängsmatrixelemente für indirekten Absorption aus der Analysis der Wellenlangen-modulations-Spektren bestimmt werden kOnnen. Für indirekten Absorption von Siizium unter TO Phonon wird das Ubergängsmatrixelement 0.1 lOb/A bestimmt, O.0367h/A für LO Phonon und 0.0178 h/A für TA Phonon.