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Modulated reflectance at oblique incidence
Solid State Communications, Vol. 5, pp. 973- 976, 1967. Pergamon Press Ltd. Printed in Great Britain
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE J.E. Fischer andB.O. Seraphin Michelson Laboratory, China Lake, California 93555, U.S.A. (Received 8 November 1967 by H. Suhi)
This note demonstrates that observation of modulated reflectance at oblique rather than normal incidence increases the analytical potential of the method; the component polarized parallel to the plane of incidence exhibits an ‘angular line shape’ near the Brewster angle characteristic of the type of critical point involved in the transition.
MODULATED reflectance serves as a major tool in a search for critical points of the energy band structure. This note demonstrates that observation of modulated reflectance at oblique rather than normal incidence increases the analytical potential of the method.
can be used in the recognition of ~c~and c~, if the angle of incidence replaces the spectral coordinate as the variable parameter of the experiment: Scanning through various angles of incidence near the Brewster angle at fixed phonon energy and modulation displays an ‘angular’ line shape of (~R/R), which portrays the characteristic features of ~ and ~ Starting from the general form of Fresnel’ s equation (the standard approximations prove inadequate in the derivative near the Brewster angle), differentiation with respect to ~1 and 2 gives the coefficients ~.
—
—
Modulated reflectance correlates to band structure analysis through interpretation of A~ and t~ 2in 1~R/R
a~t
+~t~C2
.
(1)
distinguished The theory by the choice of the various of the modulation effects, as parameters electric field, stress, temperature predicts specific line shapes for ~ ~ and t~ 2 at critical points of a certain type and location in the Brillouin zone. Their identification seems feasible once these line shapes can be recognized in the measured spectral dependence of AR/R. This recognition, however, is difficult and has in no case been accomplished beyond doubt. —
2-3v2-s2) 2(u 2 r ~2 2, ) [.(u-s) +v ] i (u+s.i +v 2 2 2 2vs . (3u -v -s ) (u2+v2) [(u-s ) 2+v2 ] [ (u+s )
—
B = r~
-
2 a
=
P
At normal incidence the coefficients a and B in equation (1) are simply functions of the optical constants.’ At oblique incidence they are functions of the angle of incidence ~Poas well. For light polarized parallel to the plane of incidence, this angular dependence Is peculiar. It
=
a + 2ut r~ (u2+v2)
-
2~ 2 2 (u +v
)
2
(u -3v -t
2
)
(4)
[(u-t)2+v2)[(u+t)2+v2)
[(u-t)
(3u2-v2.-t2) 2 2 2 2, +v ][(u+t) +v
where we employ the following definitions: 973
(2)
2us +v
‘
(5)
974
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE
Vol. 5, No. 12
Ht 72.O6eV 0.12
,/,2.iOeV $p /2
H I
_____________________
0.010
H—
0.008
0.08
k—
0.04
~--
‘V-H
H
/ ,
—~
—
~:~ H
.-2.O6eV 0.006
—
0
4b
a
°°° C
oo:~
~
~
ANGLE OF INCIDENCE, DEG.
08 ~
~
ANGLE OF INCIDENCE, DEG.
FIG. 1 Normal and parallel components of the coefficients, equations (2) (5) (see text), as a function of angle of incidence. Calculated for germanium at 2. 06, 2. 10 and 2. 14 eV, using the optical constants of Ref. 2. -
s
=
n 0
t
=
cost’0 sln$
n °
2 u —v
.
2
2 =
Uv
tan~
°
n0
—
=
-
(6)
° .
2
Sifl
E2/2
The incident medium, refracting but not absorbing, is described by n0, the material under study by c, and 2. In solving explicitly for u and v, the signs of square roots are determined by requiring c2 0 and R~,R~ 1. Thus, in equations (2) (5), u ~ 0 and v 0. -
Figure 1 plots the coefficients (2) (5) vs. angle of incidence for the 2. 1 eV region in Ge. 2 The normal components a~and ~ -
decrease monotonically with increasing cpa,, indicating that (AR/R),, also decreases. The parallel components, on the other hand, exhibit extrema in a small angular region about the Brewster angle ~B (74. 40 in our example). These extrema reflect singularities in equations (4) and (5) at u = t which are damped by absorption. The smaller the absorption (small v), the sharper are the extrema. The near singularities in the parallel coefficients amplify the reflectance modulation considerably over the normal incidence values. In the case of Fig. 1 this amplification factor is more than one order of magnitude. It varies through the different spectral regions according to the admixture of a~and in the differential equation (1). In ~-dominated spectral regions v is large, the singularity in equations (4) and (5) is damped and the angular line shape broadened.
Vol. 5, No. 12 I
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE I
I
I
r_
-
2.01 ]
I
~RAB0UC EDGE 12
/76
— —
~
I I SADDLE POINT EDGE
•
. —
/~s~,~-78
—
1.6
—
60V~ \ -1.2
975
-
‘\
-
~
“
.~
/V
,,‘ \\
N. ‘I
~78~
//
—
...
-
~—
-
-0.4
-
-O.8~-
. ...
-
-1.6-
~72
I
-
2.06
aoe
I
I
aio
2.12 PHOTON ENERGY, eV
I 214
FIG. 2 Parallel component 01 the electroreflectance response for various angles of incidence. Calculated for a parabolic edge using the electro-optic functions of ~ef. 4. In o-dominated regions the angular line shape is very sharp, and amplification of the parallel reflectance modulation over the case of normal incidence may reach 100 or more. Thus observation at oblique incidence near the Brewster angle may be applicable to the study of weak modulation signals. Of greater significance than this amplification, however, is the manner in which the angular line shapes of ; and 6~are ‘staggered’ on the angular scale. As we increase c~through CPB, the differential response changes from ~e~ -dominated to ~c, -dominated. At 75°,(~R/R)2 will essentially duplicate the spectral dependence of t~2 (B2 ~ ar), whereas for ~o ~ 78° the line shape of t, c~emerges (a~ B2). Consequently, we can extract the sign and approximate magnibide of both ~ and ~ from a single sweep through the angle of incidence about CPB, without recourse to the Kramers-Kronig dispersion relation. Furthermore, the differences of the response functions ,~c, and M2 for critical points ~.
2.04
[j
2.06
I
~I
2.08 2.10 2.12 PHOTON ENERGY, eV
~ 2.14
FIG. 3 Parallel component of the electroreflectance response for various angles of incidence. Calculated for a saddle-point edge using the electro-optic functions of Ref. 4. of different type are preserved In the oblique incidence spectra, aiding in the interpretation of experimentally observed structure. We illustrate this significant feature by synthesizing the angular sweep for a simple case in electroreflectance. Consider (1) a parabolic edge in the center of the zone, and (2) a saddle point edge, also in the center of the zone, with the modulating electric field aligned parallel to the principal axis. 2Both and cases M are connected by the duality theorem 1 and ~2 in equation (1) are described by the electro-optic functions given in Ref. 4 for a Lorentzian broadening parameter r = 0. 035 eV. Each of the two types of critical points lead to a distinctive angular line shape for (t~R/R)2,as shown in Figs. 2 and 3. The parabolic case is characterized by a negative peak which grows to a maximum slightly before CPB is reached and then diminishes, all the while shifting to higher energies. At the same time a positive peak appears at lower energy, reaches a maximum slightly beyond ~PBand then diminishes, also shifting continuously toward higher energies. The angular dependence of the
976
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE
saddle point transition is markedly different: the major peak is always positive; it grows, reaches a maximum near °Band then diminishes, simultaneously shifting toward higher energies. Small negative satellites appear on the low-energy side of the main peak for ~ > ~B and on the high energy side for cp, < ~
~Rp 2.5
Vol. 5, No. 12
PHOTON ENERGY (eV) 2.4 2.3 22 2.1
2.0
10
7..
‘N y~.•’~ \..._75 ,
Ge
5
—8O•
1
:r,L~
300’ K
85’
/
We realize that our illustration is based on an oversimplified case. A more realistic analysis must match experimentally observed structure against a more complete theory of electroreflectance. ~ It is remarkable, however, how well the main features of this simple exercise agree with experiment: measurements of the 2.1/2.3 eV structure in germanium by Lakes (Fig. 4) show the amplification of the paral lel component near ~ by one order of magnitude, the growth and decay of positive and negative peaks and their respective ‘blue shifts’.
I
,-.‘~... ~
o
/
~
~f
‘~.
\
~
~N
1/
\“~‘....ti~~ “~ 7~7O~
._~
‘-‘I
“-....~‘ ...~
/ -__________________________ 450
500
550 WAVELENGTH m
650
600
‘
A consequent analysis of electroreflectance at oblique incidence must deal with the fact that the observed response represents7 a combination of longitudinal transverse electroreflectance; proper and accounting for this admixture must wait until the transverse version itself is better understood.
FIG. 4 Parallel component of the electroreflectance response of germanium for various angles of incidence near 2. 2 eV (after Lakes )~ Notice especially the ninefold increase in the measured AR 2 /R2 for 75° incidence as compared to near-normal incidence (represented by the solid curve). 6
References 1.
SERAPHIN B.O. and BOTTKA N., Phys. Rev. Lett.
2.
We used the optical constants reported by PHILLIP H.R. and TAFT E.A., 37 (1960).
3.
PHILLIPS J.C. and SERAPHIN B.O.,
4.
SERAPHIN B. 0. and BOTTKA N.,
5.
ASPNES D.E.,
Phys. Rev. 147,
15, 104 (1965).
Phys. Rev. Lett.
Phys. Rev. 145,
554 (1966);
ibid.
Phys. Rev.
120,
15, 107 (1965).
628 (1966). 153, 972 (1967);
AYMEHICH F. and BASSANI F., Nüovo Cim. 68B, 358 (1967); Phys. Rev. (in press); BOTTKA N. and ROESSLER U., Solid State Comm. 5, 939 (1967). 6.
LUKES F. and SERAPHIN B. 0. (to be published).
7.
REHN V. and KYSER D.S., Phys. Rev. Lett. 18, 848 (1967); FORMAN R.A., ASPNES D.E. and CARDONA M., Bull. Am. Soc.
12, 658 (1967).
Nous montrons que l’observation de la reflectance modulêe pour incidence oblique au lieu de normale augmente la puissance de la mêthode; on constate que le constituant polarisé parallel au plan d’incidence possède une ‘line shape angulaire’ autour de l’angle de Brewster, charactéristique du point critique compris dans la transition.
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE J.E. Fischer andB.O. Seraphin Michelson Laboratory, China Lake, California 93555, U.S.A. (Received 8 November 1967 by H. Suhi)
This note demonstrates that observation of modulated reflectance at oblique rather than normal incidence increases the analytical potential of the method; the component polarized parallel to the plane of incidence exhibits an ‘angular line shape’ near the Brewster angle characteristic of the type of critical point involved in the transition.
MODULATED reflectance serves as a major tool in a search for critical points of the energy band structure. This note demonstrates that observation of modulated reflectance at oblique rather than normal incidence increases the analytical potential of the method.
can be used in the recognition of ~c~and c~, if the angle of incidence replaces the spectral coordinate as the variable parameter of the experiment: Scanning through various angles of incidence near the Brewster angle at fixed phonon energy and modulation displays an ‘angular’ line shape of (~R/R), which portrays the characteristic features of ~ and ~ Starting from the general form of Fresnel’ s equation (the standard approximations prove inadequate in the derivative near the Brewster angle), differentiation with respect to ~1 and 2 gives the coefficients ~.
—
—
Modulated reflectance correlates to band structure analysis through interpretation of A~ and t~ 2in 1~R/R
a~t
+~t~C2
.
(1)
distinguished The theory by the choice of the various of the modulation effects, as parameters electric field, stress, temperature predicts specific line shapes for ~ ~ and t~ 2 at critical points of a certain type and location in the Brillouin zone. Their identification seems feasible once these line shapes can be recognized in the measured spectral dependence of AR/R. This recognition, however, is difficult and has in no case been accomplished beyond doubt. —
2-3v2-s2) 2(u 2 r ~2 2, ) [.(u-s) +v ] i (u+s.i +v 2 2 2 2vs . (3u -v -s ) (u2+v2) [(u-s ) 2+v2 ] [ (u+s )
—
B = r~
-
2 a
=
P
At normal incidence the coefficients a and B in equation (1) are simply functions of the optical constants.’ At oblique incidence they are functions of the angle of incidence ~Poas well. For light polarized parallel to the plane of incidence, this angular dependence Is peculiar. It
=
a + 2ut r~ (u2+v2)
-
2~ 2 2 (u +v
)
2
(u -3v -t
2
)
(4)
[(u-t)2+v2)[(u+t)2+v2)
[(u-t)
(3u2-v2.-t2) 2 2 2 2, +v ][(u+t) +v
where we employ the following definitions: 973
(2)
2us +v
‘
(5)
974
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE
Vol. 5, No. 12
Ht 72.O6eV 0.12
,/,2.iOeV $p /2
H I
_____________________
0.010
H—
0.008
0.08
k—
0.04
~--
‘V-H
H
/ ,
—~
—
~:~ H
.-2.O6eV 0.006
—
0
4b
a
°°° C
oo:~
~
~
ANGLE OF INCIDENCE, DEG.
08 ~
~
ANGLE OF INCIDENCE, DEG.
FIG. 1 Normal and parallel components of the coefficients, equations (2) (5) (see text), as a function of angle of incidence. Calculated for germanium at 2. 06, 2. 10 and 2. 14 eV, using the optical constants of Ref. 2. -
s
=
n 0
t
=
cost’0 sln$
n °
2 u —v
.
2
2 =
Uv
tan~
°
n0
—
=
-
(6)
° .
2
Sifl
E2/2
The incident medium, refracting but not absorbing, is described by n0, the material under study by c, and 2. In solving explicitly for u and v, the signs of square roots are determined by requiring c2 0 and R~,R~ 1. Thus, in equations (2) (5), u ~ 0 and v 0. -
Figure 1 plots the coefficients (2) (5) vs. angle of incidence for the 2. 1 eV region in Ge. 2 The normal components a~and ~ -
decrease monotonically with increasing cpa,, indicating that (AR/R),, also decreases. The parallel components, on the other hand, exhibit extrema in a small angular region about the Brewster angle ~B (74. 40 in our example). These extrema reflect singularities in equations (4) and (5) at u = t which are damped by absorption. The smaller the absorption (small v), the sharper are the extrema. The near singularities in the parallel coefficients amplify the reflectance modulation considerably over the normal incidence values. In the case of Fig. 1 this amplification factor is more than one order of magnitude. It varies through the different spectral regions according to the admixture of a~and in the differential equation (1). In ~-dominated spectral regions v is large, the singularity in equations (4) and (5) is damped and the angular line shape broadened.
Vol. 5, No. 12 I
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE I
I
I
r_
-
2.01 ]
I
~RAB0UC EDGE 12
/76
— —
~
I I SADDLE POINT EDGE
•
. —
/~s~,~-78
—
1.6
—
60V~ \ -1.2
975
-
‘\
-
~
“
.~
/V
,,‘ \\
N. ‘I
~78~
//
—
...
-
~—
-
-0.4
-
-O.8~-
. ...
-
-1.6-
~72
I
-
2.06
aoe
I
I
aio
2.12 PHOTON ENERGY, eV
I 214
FIG. 2 Parallel component 01 the electroreflectance response for various angles of incidence. Calculated for a parabolic edge using the electro-optic functions of ~ef. 4. In o-dominated regions the angular line shape is very sharp, and amplification of the parallel reflectance modulation over the case of normal incidence may reach 100 or more. Thus observation at oblique incidence near the Brewster angle may be applicable to the study of weak modulation signals. Of greater significance than this amplification, however, is the manner in which the angular line shapes of ; and 6~are ‘staggered’ on the angular scale. As we increase c~through CPB, the differential response changes from ~e~ -dominated to ~c, -dominated. At 75°,(~R/R)2 will essentially duplicate the spectral dependence of t~2 (B2 ~ ar), whereas for ~o ~ 78° the line shape of t, c~emerges (a~ B2). Consequently, we can extract the sign and approximate magnibide of both ~ and ~ from a single sweep through the angle of incidence about CPB, without recourse to the Kramers-Kronig dispersion relation. Furthermore, the differences of the response functions ,~c, and M2 for critical points ~.
2.04
[j
2.06
I
~I
2.08 2.10 2.12 PHOTON ENERGY, eV
~ 2.14
FIG. 3 Parallel component of the electroreflectance response for various angles of incidence. Calculated for a saddle-point edge using the electro-optic functions of Ref. 4. of different type are preserved In the oblique incidence spectra, aiding in the interpretation of experimentally observed structure. We illustrate this significant feature by synthesizing the angular sweep for a simple case in electroreflectance. Consider (1) a parabolic edge in the center of the zone, and (2) a saddle point edge, also in the center of the zone, with the modulating electric field aligned parallel to the principal axis. 2Both and cases M are connected by the duality theorem 1 and ~2 in equation (1) are described by the electro-optic functions given in Ref. 4 for a Lorentzian broadening parameter r = 0. 035 eV. Each of the two types of critical points lead to a distinctive angular line shape for (t~R/R)2,as shown in Figs. 2 and 3. The parabolic case is characterized by a negative peak which grows to a maximum slightly before CPB is reached and then diminishes, all the while shifting to higher energies. At the same time a positive peak appears at lower energy, reaches a maximum slightly beyond ~PBand then diminishes, also shifting continuously toward higher energies. The angular dependence of the
976
MODULATED REFLECTANCE AT OBLIQUE INCIDENCE
saddle point transition is markedly different: the major peak is always positive; it grows, reaches a maximum near °Band then diminishes, simultaneously shifting toward higher energies. Small negative satellites appear on the low-energy side of the main peak for ~ > ~B and on the high energy side for cp, < ~
~Rp 2.5
Vol. 5, No. 12
PHOTON ENERGY (eV) 2.4 2.3 22 2.1
2.0
10
7..
‘N y~.•’~ \..._75 ,
Ge
5
—8O•
1
:r,L~
300’ K
85’
/
We realize that our illustration is based on an oversimplified case. A more realistic analysis must match experimentally observed structure against a more complete theory of electroreflectance. ~ It is remarkable, however, how well the main features of this simple exercise agree with experiment: measurements of the 2.1/2.3 eV structure in germanium by Lakes (Fig. 4) show the amplification of the paral lel component near ~ by one order of magnitude, the growth and decay of positive and negative peaks and their respective ‘blue shifts’.
I
,-.‘~... ~
o
/
~
~f
‘~.
\
~
~N
1/
\“~‘....ti~~ “~ 7~7O~
._~
‘-‘I
“-....~‘ ...~
/ -__________________________ 450
500
550 WAVELENGTH m
650
600
‘
A consequent analysis of electroreflectance at oblique incidence must deal with the fact that the observed response represents7 a combination of longitudinal transverse electroreflectance; proper and accounting for this admixture must wait until the transverse version itself is better understood.
FIG. 4 Parallel component of the electroreflectance response of germanium for various angles of incidence near 2. 2 eV (after Lakes )~ Notice especially the ninefold increase in the measured AR 2 /R2 for 75° incidence as compared to near-normal incidence (represented by the solid curve). 6
References 1.
SERAPHIN B.O. and BOTTKA N., Phys. Rev. Lett.
2.
We used the optical constants reported by PHILLIP H.R. and TAFT E.A., 37 (1960).
3.
PHILLIPS J.C. and SERAPHIN B.O.,
4.
SERAPHIN B. 0. and BOTTKA N.,
5.
ASPNES D.E.,
Phys. Rev. 147,
15, 104 (1965).
Phys. Rev. Lett.
Phys. Rev. 145,
554 (1966);
ibid.
Phys. Rev.
120,
15, 107 (1965).
628 (1966). 153, 972 (1967);
AYMEHICH F. and BASSANI F., Nüovo Cim. 68B, 358 (1967); Phys. Rev. (in press); BOTTKA N. and ROESSLER U., Solid State Comm. 5, 939 (1967). 6.
LUKES F. and SERAPHIN B. 0. (to be published).
7.
REHN V. and KYSER D.S., Phys. Rev. Lett. 18, 848 (1967); FORMAN R.A., ASPNES D.E. and CARDONA M., Bull. Am. Soc.
12, 658 (1967).
Nous montrons que l’observation de la reflectance modulêe pour incidence oblique au lieu de normale augmente la puissance de la mêthode; on constate que le constituant polarisé parallel au plan d’incidence possède une ‘line shape angulaire’ autour de l’angle de Brewster, charactéristique du point critique compris dans la transition.