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Theory of optical absorption of electronic impurity states coupled to optical phonons in high magnetic fields
Solid State Communications,
Vol. 8, pp. 1167—1171, 1970.
Pergamon Press.
Printed in Great Britain
THEORY OF OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES COUPLED TO OPTICAL PHONONS IN HIGH MAGNETIC FIELDS* R. F. Wallis* and A.A. Maradudin* University of California, Irvine, California 92664 and I.P. Ipatova A.F. loffe Physico-Technical Institute, Leningrad K—21, U.S.S.R. and R. Kaplan Naval Research Laboratory, Washington, D.C. 20390 (Received 30 March 1970 by F. Burns tein)
A theory is presented of the optical absorption associated with current carriers which are bound to impurities in semi-conductors in high magnetic fields and which interact with longitudinal optical phonons. The optical absorption coefficient is expressed in terms of a two-particle Green’s function using the Kubo formula. Using the Fröhlich interaction between electrons and longitudinal optical phonons, the Green’s function is evaluated for zero temperature with the aid of a decoupling approximation. The line positions and line intensities as functions of magnetic field are calculated for five lines in n—InSb. The results are compared with available experimental data. 1.
INTRODUCTION
replaced by a line at a somewhat higher frequency. When the magnetic field is such that this “impurity” cyclotron resonance (ICR) frequency is close to the long-wavelength longitudinal optical phonon frequency, ICR line splits 1 A the second-order perturinto a number of lines. bation theory of the phenomenon has been given1 which is valid only in the region of weak interaction. In the present paper a theory appropriate to the region of strong interaction is presented
IF A SEMICONDUCTOR such as n-type indium antimonide is placed in a magnetic field, electric dipole transitions corresponding to cyclotron resonance In sufficiently sufficientlyoccur. high magnetic fields,pure the material electronsat freeze out at the impurity sites at very low ternperatures, and the cyclotron resonance line is * Work supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under Grant No. AFOSR 68— 1448A. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.
for zero temperature. 2. HAMILTONIAN We are concerned with an electron in an external magnetic field moving in the potential of a periodic lattice plus an impurity atom and
Technical Report No. 70—2. 1167
1168
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
interacting with both electromagnetic waves and
H
0
longitudinal optical (LO) phonons. Utilizing the effective mass approximation for a spherical, parabolic band, we can write the Hamiltonian as the sum of three terms: H
=
H0 +HER
+HEP
hW(q)[b(q)b(q) HER
FIEP
~
=
n,
=
(2)
The electron—photon interaction is treated semiclassically in the dipole approximation using the Hamiltonian
3.
(q)
=
HER = — er . E(t) (3) where E(i) is the electric vector of the electroma gnetic field. The electron—phonon interaction is specified by the Fröhlich form of the Hamiltonian given by
D(q) e ~ ~ [h ~—q) + b(q)], ~
(7)
<
n~D(q)et~~n~> .
(9) (10)
=
i Lim di e~t<
M~>>~
(11)
where <<
M~(t);M
~
M~(t) =
=
—
~
iO(t)<[M~(t), Mj> (12) a~(t)a~’(t),
(13)
0(1) is the step function and the angular brackets denote an ensemble average. Equation (11) can be rewritten in the form ~ ~ Mmrn ‘i.~ =
-
~
nn’
where r0
Lim (—‘0
I
dt e ~
mm’
G
(t)
(14)
.J
where G~[,~1’(t)
The eigenfunctions and energy eigenvalues for the impurity electrons in a high magnetic field are taken to be those calculated by Wallis and Bowlden.2 In terms of these eigenstates, the electronic parts of the Hamiltonian can be second-quantized to give
—e< n~r~n’>
A semiclassical treatment of the interaction with the radiation yields the following Kubo formula for the dielectric susceptibility
(4) (5)
= [~/21n *~(q)]I, S is the volume in 3, and a is the polaron coupling conunits stant. ofIn r0 the following, we shall neglect the dependence of ~q) on q.
=
DIELECTRIC SUSCEPTIBILITY
1,>>~
=
M,~,y.E(t)a,~a~’
~
~
~
~(q) is the frequency of the LO phonon of wave ve~tor q, and b~(q)and b(q) are the creation and annihilation operators of this phonon.
D(q)
(6)
where state n,c,~, is the energy eigenvalue of the electron
-
where p is the electron momentum, ~ is the vector potential of the constant external magnetic field, V(~)is the potential energy due to the impurity atom, m* is the electron effective mass,
HEP
+
q
electrons with the electromagnetic radiation and LO phonons, respectively. For H 0 we have 2/2m* -i~ V~) + C H0 S (~ + ~,40)
½1
ç, a~a~ +
=
(1)
In equation (1), H0 is the Hamiltonian of the non-interacting electronstheand phonons, while 1~1ER and HEP specify interactions of the
~~(q)[b~(q)b(q)+ q -
Vol. 8, No. 15
a~(t)a~(t);a~ am~>>r (15) is the retarded Green’s function. =
<<
The two-particle Green’s function in equation (15) satisfies a Bethe—Salpeter equation
Vol. 8, No. 15
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
which has the character of a transport equation. The systematic analysis of the Bethe—Salpeter equation is difficult. However, since we are not interested in the full spectrum, but only the resonant situation when the ICR line couples strongly to the LO phonons, it is sufficient for our purposes to use the equation of motion and decoupling approximation discussed by Zubarev.3 It may be noted that Korovin and Pavlov4 have discussed similar problems by restricting attention to the one-particle Green’s function.
4. CALCULATIONS In the present paper, we shall be concerned with the positions of the optical lines as functions of magnetic field. These line positions are specified by the poles of the Green’s function. We must therefore solve the following equation E
,~ in
=
9i +
h~(t)<[a~an’,a,~,am’]>
<<[a~(t)a~~(t),H~ a~a~i>>~
(16)
After evaluating the commutators, we obtain for the equation of motion
ui +
~
(t) =
at
h~(t)~n’m&3m’(Jn
(c~ — ~)G~’(t)
+
~
—
f~’)I
~
rq U —
~
2
+
E~ — — jVn~r(q)~
E(q)
r
(~1)
tiw and E(q)
=
hw(q). In terms of
=
the notation of Wallis and Bowlden2 the state n is (000) and n’ is (010). We have solved
m
au,~,,‘(t)
F
En,— E,~ +
=
where E We follow Zubarev3 and consider the equation of motion
1169
vrn(q)H~’~(t)
(17)
equation (21) for magnetic fields in the vicinity of 30kG by assuming that the LO phonon energy is independent of q, E(q) = FLO. The states r were taken to be(0I05, (020), (001), and (000). The energy separations e~— ç in the denominators of equation (21) were taken to have the values appropriate to 30kG. The small variation of these energy differences with magnetic field was neglected. The impurity state wave functions of Wallis and Bowiden were used to evaluate the electron—phonon matrix elements V~~~(q). The variational parameters were chosen to have the values appropriate for 3OkC and
r U
where ~ state n,
their variation with magnetic field was neglected. The squared magnitudes of V~r(q) integrated over q have the following forms:
is the mean occupation number of ~j = ±, and =
<>r (18)
(a) n’
(010), r
=
(OtO)
=
~q I~’r(q)
2
-
=
<>r (19)
The equations of motion for H’~°(z) involve higher order Green’s functions which are expressed in terms of the Green’s function G~1~ (j) through a decoupling approximation. After Fourier transforming this equation, solving mm’ U, and substituting into the Fourier for Ua 1~q ~w), transform of equation (17), we retain only diagonalG~~’(w.-ie) terms with aatresonant for 0°K: character and obtain (20) G~”(w
-
ic)
=
c
1
K
+
___________ 3K4 K2)3
4(1
~
=
(010), r
V,~(q)~2 =
q
(000)
=
2C(ss’/K)
-
(c) n’
=
(010), r
11 k
qr
-
2(1
—
K2)~
(23)
2)
9
I
(q) +
12
2K2
C [ss
=
‘~
3K2
-
1_K2)i
~(w+i)+
K(K)}
K2
—
(001)
=
2( (Ii /217) 16n’m~5m~’(fn fn’)~ V,, , (q) 2 ~ — ~‘~~h(w-iE)+ ~-~w(q)~
L(K)j
2
(22) (b) n’
~1
=
3K 2 2(1 - K2)
-
/K
~(1 —
L(K)~
j
I
K
(24)
1170
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
(d) n’
(010), r
=
q Vi,, (q) -
k
1
5K 2
—
+
4(1_k2)
between 1.5 and 4. 2°K. One sees that for the
(020)
=
2
15K 4
lines at the lower energies, the agreement between theory andphoton experiment is quite reason-
2K)
C (ss’
=
/
1SK
L(K))
8(1—K~2 — 16(1K2)~2
(25) where C
(a/8~) ELO 2(r
=
2),
0/a~)y~‘(1 — K s and s’ are the variational parameters for the states n and n’ respectively, K2 = (s2 + s ‘2 )/2, a~ is the effective Bohr radius, 7 is ~ measured in units of twice the effective Rydberg, and L(K)
log ([1
=
+
Vol. 8. No. 15
(l_~)i]/[l
-
(1_K2)u]).
The electron mass required in the calculations was taken to be 0.0165m as determined by Palik et al.5 for n-type InSb at 30kG, the static dielectric constant was taken to be 16, and the polaron coupling constant to be 0.02.
able. The slight disagreement at higher fields can be attributed to the neglect of non-parabolic effects and to the neglect of the dependence of the unperturbed energy differences of magnetic field. The major disagreement with experiment concerns the experimental line at 28.5meV and a second experimental line, not shown, at 33.1 meV. The first line might involve transitions to the continuum of the lowest Landau level, which are not included in the present study. Alternatively, multiphonon processes may be responsible for one or both of these lines. The procedure outlines in this paper can be extended to the calculation of the line intensities and line widths. We present here a simple theory of the integrated line intensities and defer the treatment of the line widths to a later paper. The Green’s function given in equation (20) can be rewritten as G~(x)
rrlnSb =
B~’{~ [x-E(q)-
~
+
~/~ (x~Es)~ (26)
32 30
equation (21), and ~ 28 Bmm,’ nfl ~26 a ~ 24
— —
-
—LO’ 020) —LO’(oiO)
——LO
-
z
0
22 I
a
20
~i2~)
=
6n~m6nm~Unfn’)
where x = ~(~— ie), E~~is s-th root of For the p-th line, we focus attention on the factor 1/(x — E~) which gives the singularity in equation (26). Taking the imaginary part as and intergrating over ~, we find that the integrated intensity of the p-th line is proportional to
25
30
MAGNETIC
35
40
45
FIELD (kG)
FIG. 1. Magnetic field dependence of the
frequencies of coupled modes of impurity electrons and longitudinal optical phonons. The points are experimental data of Kaplan and Wallis1 for n-InSb. The solid curves are the calculated line positions.
The results of our calculations for the line positions as functions of magnetic field are shown in Fig. 1. Also shown are the experimental points of Kaplan and Wallis for temperatures
l~,
=
U [E~
—
E(q)
—
Er +
Efi]
/ S U~ p (Ep
—
E 8). (27)
We have calculated the relative intensities of the five lines given in Fig. 1 using equation (27). The results plotted as functions of magnetic field are given in Fig. 2. The low field ICR line loses intensity as the field increases. The intermediate lines successively gain and lose intensity, while the high field ICR line increases continuously in intensity with increasing field. The theoretical results are in qualitative
Vol. 8. No. 15
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
agreement with the available experimental data.
1
n-I,, Sb
__________________________________
06
MAGNETIC
FIELD
(KG)
FIG. 2. Theoretical relative line intensities as functions of magnetic field. The lines are numbered in order of increasing frequency at a given magnetic field.
REFERENCES 1.
KAPLAN R. and WALLIS R.F., Phys. Rev. Lett. 20, 1499 (1968).
2.
WALLIS R.F. and BOWLDEN H.J., J. Phys. Chem. Solids 7, 78 (1958).
3.
ZUBAREV D.N., Usp. Fiz. Nauk 71, 320 (1960).
4.
KOROVIN LI. and PAVLOV S.T., Zh, Eksp. Teor. Fiz. 55, 349 (1968); Soviet Phys. JETP 28, 183 (1969).
5.
PALIK ED., PICUS G.S., TEITLER S. and WALLIS R.F., Phys. Rev. 122, 475 (1961).
flpe.~creBreHa TBO~Hfl ~oHa
cB~34H~c1k4 flpkl:tecR:i
no.inx
H
~flOTPB~PAR
~~BH~HX~
ri~
pacu.(en.qeHHR
n-InSb.
rpHRa
H
~~HHL(HH
OT
;1arHwTs.l~~x
YnoTpe~J1AP
~YHK~HAX
B38HM0,qBHcTBHH
npo,~oJibHaMH OflTHlecI4I4MH npH
uyne
.‘IarHHTHOrO
Pe~yi~ram: CpaBI45Ht~ C
.~aHHk~MH.
H8CTI-1.SH,,tX
TeMneparypbJ
yp8BHeHkl~ ,~BHH4eNHA. flono~eHHR H
B.~I~SHCJ1BHkIM8E(
HOCI.ITB.1bPMH
npo,GOJ1bHSriH OnTHHBCKH:1H ~0NOH4MH.
Ky6o.
~neKTpoHa1H
BbI~lIlci1ae1 ~YHKLIHI0
c
e~nHcaH ~epe3 ,QByx
~opriy~iy
r-1e)M,~y
flOrflOL4eHHR
B flQflyflp0eO,~HHI4aX flpH BbICOKI’IX
B~aH:1oGeHcrBySou4HMH
HO3~HL(HHT nor2ou.(BHMP rpMHa
OflTHMBCI4OrO
flOflA
HMeSOLL(HMHCR
4oHoHar~H,
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flHHH~1 B
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1171
Vol. 8, pp. 1167—1171, 1970.
Pergamon Press.
Printed in Great Britain
THEORY OF OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES COUPLED TO OPTICAL PHONONS IN HIGH MAGNETIC FIELDS* R. F. Wallis* and A.A. Maradudin* University of California, Irvine, California 92664 and I.P. Ipatova A.F. loffe Physico-Technical Institute, Leningrad K—21, U.S.S.R. and R. Kaplan Naval Research Laboratory, Washington, D.C. 20390 (Received 30 March 1970 by F. Burns tein)
A theory is presented of the optical absorption associated with current carriers which are bound to impurities in semi-conductors in high magnetic fields and which interact with longitudinal optical phonons. The optical absorption coefficient is expressed in terms of a two-particle Green’s function using the Kubo formula. Using the Fröhlich interaction between electrons and longitudinal optical phonons, the Green’s function is evaluated for zero temperature with the aid of a decoupling approximation. The line positions and line intensities as functions of magnetic field are calculated for five lines in n—InSb. The results are compared with available experimental data. 1.
INTRODUCTION
replaced by a line at a somewhat higher frequency. When the magnetic field is such that this “impurity” cyclotron resonance (ICR) frequency is close to the long-wavelength longitudinal optical phonon frequency, ICR line splits 1 A the second-order perturinto a number of lines. bation theory of the phenomenon has been given1 which is valid only in the region of weak interaction. In the present paper a theory appropriate to the region of strong interaction is presented
IF A SEMICONDUCTOR such as n-type indium antimonide is placed in a magnetic field, electric dipole transitions corresponding to cyclotron resonance In sufficiently sufficientlyoccur. high magnetic fields,pure the material electronsat freeze out at the impurity sites at very low ternperatures, and the cyclotron resonance line is * Work supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under Grant No. AFOSR 68— 1448A. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.
for zero temperature. 2. HAMILTONIAN We are concerned with an electron in an external magnetic field moving in the potential of a periodic lattice plus an impurity atom and
Technical Report No. 70—2. 1167
1168
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
interacting with both electromagnetic waves and
H
0
longitudinal optical (LO) phonons. Utilizing the effective mass approximation for a spherical, parabolic band, we can write the Hamiltonian as the sum of three terms: H
=
H0 +HER
+HEP
hW(q)[b(q)b(q) HER
FIEP
~
=
n,
=
(2)
The electron—photon interaction is treated semiclassically in the dipole approximation using the Hamiltonian
3.
(q)
=
HER = — er . E(t) (3) where E(i) is the electric vector of the electroma gnetic field. The electron—phonon interaction is specified by the Fröhlich form of the Hamiltonian given by
D(q) e ~ ~ [h ~—q) + b(q)], ~
(7)
<
n~D(q)et~~n~> .
(9) (10)
=
i Lim di e~t<
M~>>~
(11)
where <<
M~(t);M
~
M~(t) =
=
—
~
iO(t)<[M~(t), Mj> (12) a~(t)a~’(t),
(13)
0(1) is the step function and the angular brackets denote an ensemble average. Equation (11) can be rewritten in the form ~ ~ Mmrn ‘i.~ =
-
~
nn’
where r0
Lim (—‘0
I
dt e ~
mm’
G
(t)
(14)
.J
where G~[,~1’(t)
The eigenfunctions and energy eigenvalues for the impurity electrons in a high magnetic field are taken to be those calculated by Wallis and Bowlden.2 In terms of these eigenstates, the electronic parts of the Hamiltonian can be second-quantized to give
—e< n~r~n’>
A semiclassical treatment of the interaction with the radiation yields the following Kubo formula for the dielectric susceptibility
(4) (5)
= [~/21n *~(q)]I, S is the volume in 3, and a is the polaron coupling conunits stant. ofIn r0 the following, we shall neglect the dependence of ~q) on q.
=
DIELECTRIC SUSCEPTIBILITY
1,>>~
=
M,~,y.E(t)a,~a~’
~
~
~
~(q) is the frequency of the LO phonon of wave ve~tor q, and b~(q)and b(q) are the creation and annihilation operators of this phonon.
D(q)
(6)
where state n,c,~, is the energy eigenvalue of the electron
-
where p is the electron momentum, ~ is the vector potential of the constant external magnetic field, V(~)is the potential energy due to the impurity atom, m* is the electron effective mass,
HEP
+
q
electrons with the electromagnetic radiation and LO phonons, respectively. For H 0 we have 2/2m* -i~ V~) + C H0 S (~ + ~,40)
½1
ç, a~a~ +
=
(1)
In equation (1), H0 is the Hamiltonian of the non-interacting electronstheand phonons, while 1~1ER and HEP specify interactions of the
~~(q)[b~(q)b(q)+ q -
Vol. 8, No. 15
a~(t)a~(t);a~ am~>>r (15) is the retarded Green’s function. =
<<
The two-particle Green’s function in equation (15) satisfies a Bethe—Salpeter equation
Vol. 8, No. 15
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
which has the character of a transport equation. The systematic analysis of the Bethe—Salpeter equation is difficult. However, since we are not interested in the full spectrum, but only the resonant situation when the ICR line couples strongly to the LO phonons, it is sufficient for our purposes to use the equation of motion and decoupling approximation discussed by Zubarev.3 It may be noted that Korovin and Pavlov4 have discussed similar problems by restricting attention to the one-particle Green’s function.
4. CALCULATIONS In the present paper, we shall be concerned with the positions of the optical lines as functions of magnetic field. These line positions are specified by the poles of the Green’s function. We must therefore solve the following equation E
,~ in
=
9i +
h~(t)<[a~an’,a,~,am’]>
<<[a~(t)a~~(t),H~ a~a~i>>~
(16)
After evaluating the commutators, we obtain for the equation of motion
ui +
~
(t) =
at
h~(t)~n’m&3m’(Jn
(c~ — ~)G~’(t)
+
~
—
f~’)I
~
rq U —
~
2
+
E~ — — jVn~r(q)~
E(q)
r
(~1)
tiw and E(q)
=
hw(q). In terms of
=
the notation of Wallis and Bowlden2 the state n is (000) and n’ is (010). We have solved
m
au,~,,‘(t)
F
En,— E,~ +
=
where E We follow Zubarev3 and consider the equation of motion
1169
vrn(q)H~’~(t)
(17)
equation (21) for magnetic fields in the vicinity of 30kG by assuming that the LO phonon energy is independent of q, E(q) = FLO. The states r were taken to be(0I05, (020), (001), and (000). The energy separations e~— ç in the denominators of equation (21) were taken to have the values appropriate to 30kG. The small variation of these energy differences with magnetic field was neglected. The impurity state wave functions of Wallis and Bowiden were used to evaluate the electron—phonon matrix elements V~~~(q). The variational parameters were chosen to have the values appropriate for 3OkC and
r U
where ~ state n,
their variation with magnetic field was neglected. The squared magnitudes of V~r(q) integrated over q have the following forms:
is the mean occupation number of ~j = ±, and =
<
(a) n’
(010), r
=
(OtO)
=
~q I~’r(q)
2
-
=
<
The equations of motion for H’~°(z) involve higher order Green’s functions which are expressed in terms of the Green’s function G~1~ (j) through a decoupling approximation. After Fourier transforming this equation, solving mm’ U, and substituting into the Fourier for Ua 1~q ~w), transform of equation (17), we retain only diagonalG~~’(w.-ie) terms with aatresonant for 0°K: character and obtain (20) G~”(w
-
ic)
=
c
1
K
+
___________ 3K4 K2)3
4(1
~
=
(010), r
V,~(q)~2 =
q
(000)
=
2C(ss’/K)
-
(c) n’
=
(010), r
11 k
qr
-
2(1
—
K2)~
(23)
2)
9
I
(q) +
12
2K2
C [ss
=
‘~
3K2
-
1_K2)i
~(w+i)+
K(K)}
K2
—
(001)
=
2( (Ii /217) 16n’m~5m~’(fn fn’)~ V,, , (q) 2 ~ — ~‘~~h(w-iE)+ ~-~w(q)~
L(K)j
2
(22) (b) n’
~1
=
3K 2 2(1 - K2)
-
/K
~(1 —
L(K)~
j
I
K
(24)
1170
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
(d) n’
(010), r
=
q Vi,, (q) -
k
1
5K 2
—
+
4(1_k2)
between 1.5 and 4. 2°K. One sees that for the
(020)
=
2
15K 4
lines at the lower energies, the agreement between theory andphoton experiment is quite reason-
2K)
C (ss’
=
/
1SK
L(K))
8(1—K~2 — 16(1K2)~2
(25) where C
(a/8~) ELO 2(r
=
2),
0/a~)y~‘(1 — K s and s’ are the variational parameters for the states n and n’ respectively, K2 = (s2 + s ‘2 )/2, a~ is the effective Bohr radius, 7 is ~ measured in units of twice the effective Rydberg, and L(K)
log ([1
=
+
Vol. 8. No. 15
(l_~)i]/[l
-
(1_K2)u]).
The electron mass required in the calculations was taken to be 0.0165m as determined by Palik et al.5 for n-type InSb at 30kG, the static dielectric constant was taken to be 16, and the polaron coupling constant to be 0.02.
able. The slight disagreement at higher fields can be attributed to the neglect of non-parabolic effects and to the neglect of the dependence of the unperturbed energy differences of magnetic field. The major disagreement with experiment concerns the experimental line at 28.5meV and a second experimental line, not shown, at 33.1 meV. The first line might involve transitions to the continuum of the lowest Landau level, which are not included in the present study. Alternatively, multiphonon processes may be responsible for one or both of these lines. The procedure outlines in this paper can be extended to the calculation of the line intensities and line widths. We present here a simple theory of the integrated line intensities and defer the treatment of the line widths to a later paper. The Green’s function given in equation (20) can be rewritten as G~(x)
rrlnSb =
B~’{~ [x-E(q)-
~
+
~/~ (x~Es)~ (26)
32 30
equation (21), and ~ 28 Bmm,’ nfl ~26 a ~ 24
— —
-
—LO’ 020) —LO’(oiO)
——LO
-
z
0
22 I
a
20
~i2~)
=
6n~m6nm~Unfn’)
where x = ~(~— ie), E~~is s-th root of For the p-th line, we focus attention on the factor 1/(x — E~) which gives the singularity in equation (26). Taking the imaginary part as and intergrating over ~, we find that the integrated intensity of the p-th line is proportional to
25
30
MAGNETIC
35
40
45
FIELD (kG)
FIG. 1. Magnetic field dependence of the
frequencies of coupled modes of impurity electrons and longitudinal optical phonons. The points are experimental data of Kaplan and Wallis1 for n-InSb. The solid curves are the calculated line positions.
The results of our calculations for the line positions as functions of magnetic field are shown in Fig. 1. Also shown are the experimental points of Kaplan and Wallis for temperatures
l~,
=
U [E~
—
E(q)
—
Er +
Efi]
/ S U~ p (Ep
—
E 8). (27)
We have calculated the relative intensities of the five lines given in Fig. 1 using equation (27). The results plotted as functions of magnetic field are given in Fig. 2. The low field ICR line loses intensity as the field increases. The intermediate lines successively gain and lose intensity, while the high field ICR line increases continuously in intensity with increasing field. The theoretical results are in qualitative
Vol. 8. No. 15
OPTICAL ABSORPTION OF ELECTRONIC IMPURITY STATES
agreement with the available experimental data.
1
n-I,, Sb
__________________________________
06
MAGNETIC
FIELD
(KG)
FIG. 2. Theoretical relative line intensities as functions of magnetic field. The lines are numbered in order of increasing frequency at a given magnetic field.
REFERENCES 1.
KAPLAN R. and WALLIS R.F., Phys. Rev. Lett. 20, 1499 (1968).
2.
WALLIS R.F. and BOWLDEN H.J., J. Phys. Chem. Solids 7, 78 (1958).
3.
ZUBAREV D.N., Usp. Fiz. Nauk 71, 320 (1960).
4.
KOROVIN LI. and PAVLOV S.T., Zh, Eksp. Teor. Fiz. 55, 349 (1968); Soviet Phys. JETP 28, 183 (1969).
5.
PALIK ED., PICUS G.S., TEITLER S. and WALLIS R.F., Phys. Rev. 122, 475 (1961).
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