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Anisotropic Voigt effect in n-type silicon
Physica
63 (1973) 570-576
0
ANISOTROPIC
North-Holland
VOIGT EFFECT
G.P. SRIVASTAVA Department
Publishing
of Physics
Co.
IN n-TYPE SILICON
and P.C. KOTHARI
and Astrophysics,
University
of Delhi,
Delhi- 7, India
Received 22 March 1972
Synopsis Donovan and Webster have studied the free-carrier Voigt effect, valid for all frequencies and magnetic-field strengths, taking into account multiple reflections in n-type germanium semiconductors. This theory is now extended to the case of n-type silicon. The high-frequency magnetoconductivity tensors are derived for a nondegenerate system of electrons, considering lattice scattering only. Theoretical calculations for Voigt rotation are carried out for a typical nondegenerate specimen of n-type silicon having d.c. conductivity u0 = 1.8 x 10” e.s.u. (at 300 K).
1. Introduction. The theory of the free-carrier Voigt effect, valid for all frequencies and magnetic field strengths, taking into account multiple reflections, has been developed for cubic semiconductors with anisotropic effective masses by Donovan and Webster’). The behaviour of the Voigt rotation and ellipticity can be studied provided the high-frequency magnetoconductivity tensor components are known for particular orientations of the crystal and a particular direction of the magnetic field. Donovan and Webster have derived these tensor components for n-type germanium following the strong-field formulation of Abeles and Meiboom2). In the present paper, the high-frequency magnetoconductivity tensor components are calculated for n-type silicon semiconductors. These tensor components are valid for any orientation of the magnetic field and any crystal symmetry. The conductivity-tensor components are used for the theoretical investigation of the free-carrier Voigt effect. 2. Theory. A right-handed system of axes (1,2, 3) is chosen such that the incident radiation is propagated along the 1 direction, and the magnetic field is parallel to the 3 direction. The electric vector is initially inclined at an angle bE to the 3 direction. The cubic axes of the crystal are labelled as (x, y, z) and the propagation is considered specifically along the [IOO] or [I lo] direction. 4H is the angular displacement of the 3 axis with respect to the 2 axis. 570
ANISOTROPIC
VOIGT EFFECT IN n-TYPE SILICON
571
For electrons in the conduction band of silicon, the surfaces of constant energy in momentum space consist of three ellipsoids of revolution oriented along the [loo], [OlO] and [OOI] directions. The expression for the current density for one ellipsoid can be written asz*“)
where E is the energy and f. is the Fermi distribution function. The expression for 4, which is the high-frequency generalization of the expression given by Abeles and Meiboom, is2s3) ez
P,K
~2%
m,
m3
P-P
c (1 + iwt)
’ (p,Hx + P,H, + p,Hz)
1
&x
+ (two more terms obtained by cyclic permutation of the indices),
(2)
where
r is the relaxation time and E*,pi and Hi (i = x, y, z) are, respectively, the components of the time-dependent electric field, momentum and static magnetic field. For lattice scattering t = a (kT/E)‘, where a is a function of temperature. On the basis of a suitable model the current density may be obtained in the form
Ji = sij (w, *HH) 8~;
(&_I= x, Y, 4.
(3)
The conductivity-tensor components are obtained by solving eq. (1). Considering lattice scattering only, the current density for a single ellipsoid is obtained by an analysis similar to that given by Donovan and Webster3). In order to obtain the contribution to the current density from each individual ellipsoid the indices (x, y, z) in eq. (2) are replaced successively by the indices (x, y, z), (y, z, x) and (z, x, u), since the longitudinal axes of the ellipsoids in silicon are oriented along the direction of cubic edges. The total current density is obtained by summing up the contributions from all three ellipsoids. From expression (3), the general expressions for conductivity-tensor components, valid for any orientation of the static magnetic field and the electric vector, are obtained as
572
G. P. SRIVASTAVA
s,, =
-u
AND P. C. KOTHARI
’ H,H,, ; yj,
(K@I + I% + 183)Hz + u
j=l
Sij = Sji(-H),
(4)
where u = 4ne2a/(3r3mJ) and K = m3/ml, e is the electronic charge, Kj (w, H), lsj (W H) and y.t (m, H) are defined by eq. (12) of Donovan and Webster3). The cyclotron frequencies for the three ellipsoids are a, = (eK’/m,c) (H: + Hz + KHz)*,
(5)
sl?, = (eK*/m,c) (Hj? + Hz + KHZ)‘, D3 = (eK*/m,c) (Hz + Hz + KHC)‘,
where c is the velocity of light. We list below tensor components for two special cases which cover most of the popular orientations of the crystal and the directions of the magnetic field. 2.1. H confined to the (100) plane (Q, # 52, ZQ,). Let the magnetic field be confined to the (100) plane and & be the angle between the magnetic field (H) and the 2 axis (i.e., the [OOl]direction). Using the transformation matrix (see section 2.2 of Donovan and Webster3)) and proceeding as discussed earlier, we find that S,, = u (Ka, + 012+ Ka3), &2 = U 1% (Sin2 4~ + K cos’ 4H) + m2 + 01~(cos2 & + K sin2 &)I,
SJ3= u 01~ (cos24H + K sin2 4H) + a2 + 01~ (sin2 4* + K cos2 4H) [
S13 = -S31
= u (eaH/m,c) cos
4Hsin 4H(K - 1) (p3 - /II),
&3 = &2 = u cos 4~ sin 4~ (K - 1) (011- (x3), S,, = -S2,
= -u(eaH/m,c)
x [PI (sin2 &I + Kcos’
4d + B2+ p3(cos’4H + K sin2 4H)]. (6)
The cyclotron frequencies are 52, = (eK*/m,c) H (sin2
4B + K cos’ 4H)‘,
Q3 = (eK3/m3c) H (co?
4H + K sin2 4H)‘.
8, = (eK’/m,c) H,
(7)
ANISOTROPIC
VOIGT EFFECT IN n-TYPE SILICON
573
2.2. H confined to the (li0) plane (Q1 # Qn, = Q3). Let the magnetic field be confined to the (110) plane and & be the angle between the magnetic field (H) and the 2 axis (i.e., the [OOl] direction). Using the transformation matrix (see section 2.1 of Donovan and Webster3)) and proceeding in the same way, we find that S,, = S,, = u {$x1 [sin* & + K(l + co? &)I + $a2 [K(sin* & + 2) + (1 + cos* &)I}, S33
=
24 {oL1 (cos*
&
+ (e4mIc)* s13
=
s32
=
h/J21
s23
=
s31
=
s,3(-H),
+
K
(71 + sin
sin* &) + 01~[K (1 + cos* &) + sin* &] 2Y*N,
#kcos4H(K-
S12 = u {&x1 [sin* & + K(cos* & -
1) [(a1
-
a,)+
(euH/m,~)(p,
--PI>,
l)]
+ fn2 [(Cos’ 4H - 1) + Ksin* f&J) - 24(euH/m,c) {/I1 (sin* 4H + Kcos2 &) + B2 [(l + cos* &) + Ksin* &]}, s2,
=
&2(-m.
(8)
The cyclotron frequencies are Q, = (eK3/m3c) H (sin* 4H + K cos* &)+,
(9)
Q2 = Q3 = (eK3/m3c) H [cos* & + + (K + 1) sin* &I’. When H lies along the [ill] direction (cos & = l/,/3) the three cyclotron frequencies reduce to a single value (isotropic case) Q, = Q, = Q3 = (eH/m,c) [K(K + 2)/3]*. The expression for complex propagation pendicular modes is
constants for the parallel and per-
where n,, and n,,,, the refractive indices for the two independent modes are given in the isotropic limit by ‘) nfr = &, and n& = 9. Thus knowing the magneto-
574
G. P. SRIVASTAVA
AND P. C. KOTHARI
conductivity tensor components and the complex propagation constant, the Voigt rotation 8 is calculated from the following expression of Donovan and Webster’). tan 20 = (J2 - K2)/2JKcos (l - 5).
(11)
The various symbols occurring above are defined in ref. 1. 3. Theoretical computation. Theoretical calculations reported here are carried out on an IBM 360 model 44. The numerical calculations relate to a typical specimen of n-type silicon, using the following data (applied at 300 K): d.c. conductivity c,, = 1.8 x loll e.s.u., carrier density N = 9.0 x 1014 cme3 (Irvin4), dielectric constant E = 11.6. The principal effective masses are : m, = m, = O.l9m,, m3 = 0.98m,.
The high-frequency magnetoconductivity tensor components are computed taking into account the lattice scattering only. The parameter a is calculated from the following expression of Donovan and Webste?): go = 4Ne2a/3x3m*, where m* = 0.26mo. In the calculation of the Voigt rotation the effect of multiple reflections?) within the sample are also taken into account. 4. Results and discussion. The variation of the Voigt rotation with a magnetic field oriented along the [OOl], [Oil] and [Ill] directions is shown in fig. 1 with w = 1.88 x 10” s-1 and specimen thickness x = 0.1 cm. The effect of multiple
MAGNETIC
FIELD
( kOe)+
Fig. 1. Variation of the Voigt rotation with magnetic field, with w = 1.88 x 10”~~‘,
x = 0.1 cm.
ANISOTROPIC
VOIGT EFFECT
IN n-TYPE SILICON
575
3.5-
2.5 -
Fig. 2. Variation of the Voigt rotation with frequency, with H = 2 x lo4 Oe, x = 0.1 cm; a and c, H along [OOl]; b and d, H along [Oil]; c and d, with multiple reflections included.
8.0
70 t 6.0 t 3 0 S-oe g 2 4.06 g 50I=
THICKNESS
(mm)+
Fig. 3. Variation of the Voigt rotation with thickness x of the crystal, with w = 1.88 x 10’ 1 s-l, H = 2 x lo4 Oe.
576
G. P. SRIVASTAVA
AND P. C. KOTHARI
reflections within the sample is taken into account. It is seen that the rotation is highest and most pronounced for the [OOl] direction. The frequency dependence of the Voigt rotation is illustrated in fig. 2, with H = 2 x lo4 Oe and oriented along the [OOl] and [Ol l] directions, and x = 0.1 cm. The effect of multiple reflections within the specimen is shown by curves c and d. Fig. 3 shows the variation of the Voigt rotation as a function of the thickness x of the specimen for three different orientations of the magnetic field, oriented along the [OOl], [Oil] and [ill] directions, with (u = 1.88 x 10” s-l and H = 2 x lo4 Oe. The influence of multiple reflections is evident in the graph. Acknowledgements. The authors are thankful to Professor F.C.Auluck for his continued interest in the work. Thanks are also due to Arun Kumar for his helpful discussions.
REFERENCES 1) 2) 3) 4) 5) 6)
Webster,J. and Donovan, B., Brit. J. appl. Phys. 16 (1965) 25. Abeles, B. and Meiboom, S., Phys. Rev. 95 (1954) 31. Donovan, B. and Webster, J., Proc. Phys. Sot. 81 (1963) 90. Irvin, J.C., Bell. Syst. tech. J. 41 (1962) 387. Donovan, B. and Webster, J., Proc. Phys. Sot. 78 (1961) 120. Donovan, B. and Medcalf, T., Brit. J. appl. Phys. 15 (1964) 1139.
63 (1973) 570-576
0
ANISOTROPIC
North-Holland
VOIGT EFFECT
G.P. SRIVASTAVA Department
Publishing
of Physics
Co.
IN n-TYPE SILICON
and P.C. KOTHARI
and Astrophysics,
University
of Delhi,
Delhi- 7, India
Received 22 March 1972
Synopsis Donovan and Webster have studied the free-carrier Voigt effect, valid for all frequencies and magnetic-field strengths, taking into account multiple reflections in n-type germanium semiconductors. This theory is now extended to the case of n-type silicon. The high-frequency magnetoconductivity tensors are derived for a nondegenerate system of electrons, considering lattice scattering only. Theoretical calculations for Voigt rotation are carried out for a typical nondegenerate specimen of n-type silicon having d.c. conductivity u0 = 1.8 x 10” e.s.u. (at 300 K).
1. Introduction. The theory of the free-carrier Voigt effect, valid for all frequencies and magnetic field strengths, taking into account multiple reflections, has been developed for cubic semiconductors with anisotropic effective masses by Donovan and Webster’). The behaviour of the Voigt rotation and ellipticity can be studied provided the high-frequency magnetoconductivity tensor components are known for particular orientations of the crystal and a particular direction of the magnetic field. Donovan and Webster have derived these tensor components for n-type germanium following the strong-field formulation of Abeles and Meiboom2). In the present paper, the high-frequency magnetoconductivity tensor components are calculated for n-type silicon semiconductors. These tensor components are valid for any orientation of the magnetic field and any crystal symmetry. The conductivity-tensor components are used for the theoretical investigation of the free-carrier Voigt effect. 2. Theory. A right-handed system of axes (1,2, 3) is chosen such that the incident radiation is propagated along the 1 direction, and the magnetic field is parallel to the 3 direction. The electric vector is initially inclined at an angle bE to the 3 direction. The cubic axes of the crystal are labelled as (x, y, z) and the propagation is considered specifically along the [IOO] or [I lo] direction. 4H is the angular displacement of the 3 axis with respect to the 2 axis. 570
ANISOTROPIC
VOIGT EFFECT IN n-TYPE SILICON
571
For electrons in the conduction band of silicon, the surfaces of constant energy in momentum space consist of three ellipsoids of revolution oriented along the [loo], [OlO] and [OOI] directions. The expression for the current density for one ellipsoid can be written asz*“)
where E is the energy and f. is the Fermi distribution function. The expression for 4, which is the high-frequency generalization of the expression given by Abeles and Meiboom, is2s3) ez
P,K
~2%
m,
m3
P-P
c (1 + iwt)
’ (p,Hx + P,H, + p,Hz)
1
&x
+ (two more terms obtained by cyclic permutation of the indices),
(2)
where
r is the relaxation time and E*,pi and Hi (i = x, y, z) are, respectively, the components of the time-dependent electric field, momentum and static magnetic field. For lattice scattering t = a (kT/E)‘, where a is a function of temperature. On the basis of a suitable model the current density may be obtained in the form
Ji = sij (w, *HH) 8~;
(&_I= x, Y, 4.
(3)
The conductivity-tensor components are obtained by solving eq. (1). Considering lattice scattering only, the current density for a single ellipsoid is obtained by an analysis similar to that given by Donovan and Webster3). In order to obtain the contribution to the current density from each individual ellipsoid the indices (x, y, z) in eq. (2) are replaced successively by the indices (x, y, z), (y, z, x) and (z, x, u), since the longitudinal axes of the ellipsoids in silicon are oriented along the direction of cubic edges. The total current density is obtained by summing up the contributions from all three ellipsoids. From expression (3), the general expressions for conductivity-tensor components, valid for any orientation of the static magnetic field and the electric vector, are obtained as
572
G. P. SRIVASTAVA
s,, =
-u
AND P. C. KOTHARI
’ H,H,, ; yj,
(K@I + I% + 183)Hz + u
j=l
Sij = Sji(-H),
(4)
where u = 4ne2a/(3r3mJ) and K = m3/ml, e is the electronic charge, Kj (w, H), lsj (W H) and y.t (m, H) are defined by eq. (12) of Donovan and Webster3). The cyclotron frequencies for the three ellipsoids are a, = (eK’/m,c) (H: + Hz + KHz)*,
(5)
sl?, = (eK*/m,c) (Hj? + Hz + KHZ)‘, D3 = (eK*/m,c) (Hz + Hz + KHC)‘,
where c is the velocity of light. We list below tensor components for two special cases which cover most of the popular orientations of the crystal and the directions of the magnetic field. 2.1. H confined to the (100) plane (Q, # 52, ZQ,). Let the magnetic field be confined to the (100) plane and & be the angle between the magnetic field (H) and the 2 axis (i.e., the [OOl]direction). Using the transformation matrix (see section 2.2 of Donovan and Webster3)) and proceeding as discussed earlier, we find that S,, = u (Ka, + 012+ Ka3), &2 = U 1% (Sin2 4~ + K cos’ 4H) + m2 + 01~(cos2 & + K sin2 &)I,
SJ3= u 01~ (cos24H + K sin2 4H) + a2 + 01~ (sin2 4* + K cos2 4H) [
S13 = -S31
= u (eaH/m,c) cos
4Hsin 4H(K - 1) (p3 - /II),
&3 = &2 = u cos 4~ sin 4~ (K - 1) (011- (x3), S,, = -S2,
= -u(eaH/m,c)
x [PI (sin2 &I + Kcos’
4d + B2+ p3(cos’4H + K sin2 4H)]. (6)
The cyclotron frequencies are 52, = (eK*/m,c) H (sin2
4B + K cos’ 4H)‘,
Q3 = (eK3/m3c) H (co?
4H + K sin2 4H)‘.
8, = (eK’/m,c) H,
(7)
ANISOTROPIC
VOIGT EFFECT IN n-TYPE SILICON
573
2.2. H confined to the (li0) plane (Q1 # Qn, = Q3). Let the magnetic field be confined to the (110) plane and & be the angle between the magnetic field (H) and the 2 axis (i.e., the [OOl] direction). Using the transformation matrix (see section 2.1 of Donovan and Webster3)) and proceeding in the same way, we find that S,, = S,, = u {$x1 [sin* & + K(l + co? &)I + $a2 [K(sin* & + 2) + (1 + cos* &)I}, S33
=
24 {oL1 (cos*
&
+ (e4mIc)* s13
=
s32
=
h/J21
s23
=
s31
=
s,3(-H),
+
K
(71 + sin
sin* &) + 01~[K (1 + cos* &) + sin* &] 2Y*N,
#kcos4H(K-
S12 = u {&x1 [sin* & + K(cos* & -
1) [(a1
-
a,)+
(euH/m,~)(p,
--PI>,
l)]
+ fn2 [(Cos’ 4H - 1) + Ksin* f&J) - 24(euH/m,c) {/I1 (sin* 4H + Kcos2 &) + B2 [(l + cos* &) + Ksin* &]}, s2,
=
&2(-m.
(8)
The cyclotron frequencies are Q, = (eK3/m3c) H (sin* 4H + K cos* &)+,
(9)
Q2 = Q3 = (eK3/m3c) H [cos* & + + (K + 1) sin* &I’. When H lies along the [ill] direction (cos & = l/,/3) the three cyclotron frequencies reduce to a single value (isotropic case) Q, = Q, = Q3 = (eH/m,c) [K(K + 2)/3]*. The expression for complex propagation pendicular modes is
constants for the parallel and per-
where n,, and n,,,, the refractive indices for the two independent modes are given in the isotropic limit by ‘) nfr = &, and n& = 9. Thus knowing the magneto-
574
G. P. SRIVASTAVA
AND P. C. KOTHARI
conductivity tensor components and the complex propagation constant, the Voigt rotation 8 is calculated from the following expression of Donovan and Webster’). tan 20 = (J2 - K2)/2JKcos (l - 5).
(11)
The various symbols occurring above are defined in ref. 1. 3. Theoretical computation. Theoretical calculations reported here are carried out on an IBM 360 model 44. The numerical calculations relate to a typical specimen of n-type silicon, using the following data (applied at 300 K): d.c. conductivity c,, = 1.8 x loll e.s.u., carrier density N = 9.0 x 1014 cme3 (Irvin4), dielectric constant E = 11.6. The principal effective masses are : m, = m, = O.l9m,, m3 = 0.98m,.
The high-frequency magnetoconductivity tensor components are computed taking into account the lattice scattering only. The parameter a is calculated from the following expression of Donovan and Webste?): go = 4Ne2a/3x3m*, where m* = 0.26mo. In the calculation of the Voigt rotation the effect of multiple reflections?) within the sample are also taken into account. 4. Results and discussion. The variation of the Voigt rotation with a magnetic field oriented along the [OOl], [Oil] and [Ill] directions is shown in fig. 1 with w = 1.88 x 10” s-1 and specimen thickness x = 0.1 cm. The effect of multiple
MAGNETIC
FIELD
( kOe)+
Fig. 1. Variation of the Voigt rotation with magnetic field, with w = 1.88 x 10”~~‘,
x = 0.1 cm.
ANISOTROPIC
VOIGT EFFECT
IN n-TYPE SILICON
575
3.5-
2.5 -
Fig. 2. Variation of the Voigt rotation with frequency, with H = 2 x lo4 Oe, x = 0.1 cm; a and c, H along [OOl]; b and d, H along [Oil]; c and d, with multiple reflections included.
8.0
70 t 6.0 t 3 0 S-oe g 2 4.06 g 50I=
THICKNESS
(mm)+
Fig. 3. Variation of the Voigt rotation with thickness x of the crystal, with w = 1.88 x 10’ 1 s-l, H = 2 x lo4 Oe.
576
G. P. SRIVASTAVA
AND P. C. KOTHARI
reflections within the sample is taken into account. It is seen that the rotation is highest and most pronounced for the [OOl] direction. The frequency dependence of the Voigt rotation is illustrated in fig. 2, with H = 2 x lo4 Oe and oriented along the [OOl] and [Ol l] directions, and x = 0.1 cm. The effect of multiple reflections within the specimen is shown by curves c and d. Fig. 3 shows the variation of the Voigt rotation as a function of the thickness x of the specimen for three different orientations of the magnetic field, oriented along the [OOl], [Oil] and [ill] directions, with (u = 1.88 x 10” s-l and H = 2 x lo4 Oe. The influence of multiple reflections is evident in the graph. Acknowledgements. The authors are thankful to Professor F.C.Auluck for his continued interest in the work. Thanks are also due to Arun Kumar for his helpful discussions.
REFERENCES 1) 2) 3) 4) 5) 6)
Webster,J. and Donovan, B., Brit. J. appl. Phys. 16 (1965) 25. Abeles, B. and Meiboom, S., Phys. Rev. 95 (1954) 31. Donovan, B. and Webster, J., Proc. Phys. Sot. 81 (1963) 90. Irvin, J.C., Bell. Syst. tech. J. 41 (1962) 387. Donovan, B. and Webster, J., Proc. Phys. Sot. 78 (1961) 120. Donovan, B. and Medcalf, T., Brit. J. appl. Phys. 15 (1964) 1139.