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L 0 - phonon instability in the free-carrier absorption in semiconductors under strong magnetic fields
003%1098/81/260345-05$02.00/O
Solid State Communications, Vo1.39, pp.345-349. Pergamon Press Ltd. 1981. Printed in Great Britain.
L
0
-
PHONON
INSTABILITY IN THE FREE-CARRIER STRONG MAGNETIC FIELDS O.A.C.
Nunes and Departamento Universidade
70.910
-
IN
SEMICONDUCTORS
UNDER
L.C.M. Mi randa de Fis i ca de Brasilia
Brasilia,
M. lnstituto
ABSORPTION
DF,
Brasil
and A.
Tenan de Fisica Universidade Estadual de Campinas 13.100 Campinas,thSP, Brasil (Received November 27 by R.C.C. Leite)
The LO-phonon damping in an electron-phonon system the presence of a laser beam and a d. c. magnetic field discussed. It is shown that near the laser-cyclotron resonance, and for the laser beam propagating perpendicularly to the magnetic field the Lq-phonon population in a relatively narrow range of K may grow with time.
The generation of nonequi librium LO-phonon population in semiconductors has been known for some time’-‘. It has been experimentally observed mainly in connection with Raman scattering studies under intense photoexcitation such that the incident photon energy ‘hw, is greater than the semiconductor energy gap. The steady-state nonequilibrium population of the LO-phonons is interpreted as a result of the relaxation of the photoexcited electrons to the bottom of the conduction band via successive emission of LO-phonons. Under the additional presence of a strong d.c. magnetic, field an enhancement of the LO-phonon Raman line has also been observedke7 when the LO-phonon frequency w equals an harmonic of the electron cyclo?ron frequency wc. Similarly to the zero field case, this strong departure of the LO-phonon population from its equilibrium value obtains when the injected photoelectrons cascade down the Landau levels emitting LO-phoIn contrast to these studies in nons. highly photoexcited semiconductors, very little has been published regarding LOphonon instability in connection with intraband 1 ight absorption experiments. In this paper we investigate an alternative mechanism for LO-phonon amplification in semiconductors namely, the LOphonon amplification induced by intraband absorption of electromagnetic radiation in the presence of an additional d. c. magnetic field. the In the absence of magnetic field, this problem d.c. has discussed by Gurevich and recent 1 1 been Parshin . Here, however, one includes the additional effects of a strong d.c. magnetic field, encouraged by the possibility of exploring the effects of the laser-cyclotron resonance. In fact,
in is
the resonance condition, where the laser frequency equals the electron cyclotron frequency , may be approached either by increasing the magnetic field strength or by using submillimeter lasers which are now becoming available. Hence, apart from its eventual application to intraband magneto-absorption experiments, we believe that the subject is of sufficient interest to warrant an independent investigation. We consider photon beam from incident quency uL.. conductor The magnetic vector+potential gauge effectiee electron
A
a a
circularly laser source on a n-type
polarized of fresemi-
In
a constant magnetic field. fielcJ is described by a A, given in the Landau Within the = (-Hy, 0.0). mass approximation the onestates are described by the
Landau wavefunctions’ n,px,pz>. where n is the Landau level quantum number. The corresponding one-electron energies are E,(p ) = (n + 1/2) hole+ hZp */2m, where H/me is the electronzcyclotron IA! =‘e fre The electron-LO-phonon interaction qcency. is assumed to be dominated by the longrange electrostatic interaction described by the FrHlich Hamiltonian”. The rate of change of the LO-phonon population Nk, due to the carrier-assisted process, “photon + carrier + phonon + “is described by the kinetic carrier, eauation
dNk -= &[+p-(a,@+ dt
+P+(a,a)
-
-P-(a,R)-
(a,$)]fa(l-fg) Where
345
fo
is
the
distribution
function
_p+
FREE-CARRIER
346
ABSORPTION
for electrons in the state CY = (n,px,pz). and the P’s represent the probability amplitude for an electronic transition where the superscript (subscript) indicates the emission (+) or the absorption (-) of a photon (phonon). The probabi 1 i ty ampl i tudes are calculated using second - or-der perturbation theory”712. In the following, we shall restrict ourselves only to the important situation where the laser beam propagates perpendicularThe reason ly to the magnetic field. for this is that one can show that this is the only configuration for which the P’s exhibit the laser-cyclotron (w = o? ) resonance. Hence, under these ci rkuns4 retaining only the resonant tances, terms and denoting by N the number of photons with wave vectob z and f requenone can show that Eh. reduces (I) CY w L, to
--dNk
=
.
~-fn+l(p,-kZ)]~
I . . . . I= (n+l)NL(Nk+l)fn(pZ) (E (p ) + I~uL-E~+~ n
(pz-kZ)-&k)+n(NL+l)
z
the
distribution
can
dNk --..-=y dt
yk
=
N
k
2niVk12
spontaneous
’ a2 No
‘Li
z
l;hm’ (wL-WC) ’
-fn(pz)
1 - f,+,
[
6
(hmL-il~~-h,,,
$+k,)-j
6
$-En+,
6
(Nk+l)
(En(pZ)-~~L-En_l
NLNkfn(pz)
(p,+kZ)
(p,-kZ)
fn
;
$3
c p,
(pZ+kz)
1
6
(E&p,)
- ,+oL
-
Yk
(pz)
performing yK :
the
=c+
pi e’ .) -
(3) l/2
Here and
(p,)
En_l(pZ+kz)
hwK)
Vk= VL=
i (2n
(27r
e*
-1
-p2*
*
‘dpZ
integrations
c
(EF
8
aiImthezo
gets
-1
woKL2(e
m -
-
0
-
En
~0~) FC
m
(n+l)Cp
1*
n=O
(EF-hwL+hwo-En(p,)
(p,)
-
8
(EF-hwL-htio-En(p2)fl
)
electron - LO - phonon and electron-photon vertices, respectively, a2 = hk*/mw , and V is the crystal volume; the ot k er q”uantities have their usual meaning. Now, assuming that the carriers form a completely degenerate electron gas, such that their distribution functions become step functions, it follows from Eq. (3) that if w > w > w. only the 1 ight L L absorbing processes (namely, the first and third I ines of Eq. (3) are non-vanish This, in turn, entai Is that one ing. may expect instability in the LO-phonon population at the laser cyclotron resonance condition only when the laser frequency is greater than the LO-phonon fre quency. In this case, assuming further>> I, that N the kinetic equation for L
)
(6) wherep=m(w
“2
one
_I”2
hwo/Vk2)
e’h/E,VWL)
fi
+ m
+ F
2k2/kZ/hwLm(wL-wc)
1-i- f,_l
+&.~kZ)
0
t ion
P
V ‘3rrwc 2Tih
and for
b-fn+l(pZ+kz)]6(En(pz)+
+ %IJJ~)- % (NL+l)NK
p,’
(hw,-+~:‘-‘~++~w
-co
-hw,)-(n+l)
+
(n+l)
n,pxlPZ
(pz-kz)]
tfJP,)l;-f,+,
c-t (pZ-KZ)]
written
+y’
k
is the irrelevant term and
where y’ emission
prescri
fn_l
be
as
fn(Pz)
I; -
LO-phonon
Vol. 39, No. 2
Equation (4) tells us that if yk is posi grows wi?h tive the LO-phonon population < 0 there is whereas for y time, Of courseka net growt (amp1 idamping. fication) of the LO-phonon population is only achievable provided yk is qreater than the phonon decay rate 11~ due to the lattice anharmonicity. Substituting in Eq. (5) the f’s by the corresponding step functions. transforming the summations over p and pz into integrals through the usGal
dt where:
IN SEMICONDUCTORS
P2
=
I m(wL
-
L wc
-w +
41:)
wo)/
hiKZI
and
+C;jkZI
To proceed further, we consider in the following the ultraquantum limit which the carrier concentration is such that only the n = 0 Landaullevel is fully occupied, namely, E =hw,; for an InSb sample in a magnetig ?ield of the order of 100 KOe this condition requires a carrier concentration of the order of 6.85 x 1016 cmm3. In this case, perform ing the summation over n in Eq. (6) witTT the help of the S-functions, the condition for yk> 0 can now be stablished. One gets: A.
For
wL>
wc>
wo,
2 k2/kZ13k3
mL (w,
- wc)’
Vol. 39, No. 2
FREE-CARRIER ABSORFTION IN SEMICONDUCTORS
6 f2, a
347
FREE-CARRIER
348
ABSORPTION
Vol. 39, No. 2
IN SEMICONDUCTORS
provided
- wc)j <
!kz!
<
20.0
$[j I
-
(I*IL_
wc)
For
wc>
6.
O (7b)
/ &IL>
2n e4kL’ k’jk
z
I3
w
cl
,
cEGih3
-.
150
‘o’)m~io’
w (~1 Lc
-
(8a)
(ii ) L
t” 9
provided
_
100
1a0
h;F -
+(wc
t
(w c
+
i*iL)
- wL) 1 < / kZ : < ,+;
1
lilo
/
(8b) 5.0
where
k
=
k ’
- 1”
F
1
from Eqs. (7) and (8) that condition for amp1 ification In other words, near the depends on k. the laser-cyclotron resonance condition, LO-phonon popula$ion, in a relatively narrow range of k, may become unstable, ie; there is a selective mechanism for In Table the excitation of LO phonons. the interval t , we show each value of wc/wL, LO-phonon amplification for an InSb sample with a carrier concentration such that E = 3bc/2. The following values for InSb have been the phyfsical parameters of used: m = 0.01 m E =18. F = 14.06 = 3.8 x 1013 “;-‘4” Here Le aote that w tf?e conditions of laser-c clotron resoand of E =3 tl oic/Z nance w can be met = w t nSb Sample illLminared by a CO2 for an laser (XL= IO pm) yro_vided H = 118 KOe and n = 5 x IO’ cm . In Fig. 1, we obeing show the smplification rate y / k YO a coefficient given by It
the
YO=
follows threshold
(N
)n
e4
K L’(Fi’
-
eo-‘)m
~il~,~k~,k
,3
L’V L
h3wL, as a function of the cyclotron frequency for an InSb sample i 1 luminated by a CO2 We have arbitrarily chosen, laser. without loss of generality, a geometry such that k = I .41 k . Actually, the criterion for the on:et of the instabiliis somewhat more involved: a net ty growth of the LO-phonon population is
Ic Fig. I (yk/yo) frequency illuminated
only than the
:
LO - phonon as a function (tic/~!,) for by CO2
achievable provided the LO-phonon decay lattice anharmonicity Yk
-
Fk
amp1 ification of the cyclotron an InSb sample laser.
‘k rate
,
is
greater n k due
1.
t: ;: 7.
to
ie,
b 0
(9)
The above condition enta Is that there is a threshold laser intens ty for the actual LO-phonon amplification. To get a numeri cal estimate of the crit cal intensity we consider an InSb samp e illuminated by laser havin a carrier concentration a CO -9 of 6225 x 1016cm in a magnetic phonon losses rjk are typically of the order of 0.01 wo, Eq. (9) gives us a critical intenstty of the order of IO’ w/cm’. These predictions could be experimentally observ for instance, in connection with free ed. s in -carrier absorpt/on experimen which a second strong pumping field is simul taneously present with the prob ng field. The LO-phonon instability wou in this d, case, be manifested as an inc ease in the absorption of the probing fie d as the pumping field approaches the aser-cvclotron resonance condition.
REFERENCES
2.
rate
and LEITE, R.C.C., Phys. Rev. Letters 22, 1304 (1969) SHAH, J. SHAH, J., LEITE, R.C.C. and SCOTT, J.F., Solid State Commun. 8, I089 (1970) MATTOS, J.C.V. and LEITE, R.C.C., Solid State Commun. 12, 465 (1973) VELLA - COLEIRO G.P., Phys. Rev. Letters 23, 697 (l969r and ZILBERBERG V.V., Sov. Phys. Solid State 11, 1465 (1970). GENKIN, G.M., LUZZI, R., Progr. Theor. Phys. 2, 13 (1972) FROTA-PESSOA, s. and LUZZI , R., Phys. Rev. 813, 5420 (1976)
FREE-CARRIER ABSORPTION IN SEMICONDUCTORS
Vol. 39, No. 2
8. 9. IO.
I 1. 12.
GUREVICH, LANDAU,
V. L.D.
Oxford. 1958) FROLICi, H., GOMES,M.A.F. COUTINHO, S.
L. and PARSHIN, and LIFSCHITZ, P.
Adv. and and
349
D. A. Zh. Eksp. Teor. Fiz. 2, 1589 (1977). E.M., "Quantum Mechanics" (Pergamon Press,
474. Phys. 2, 325 (1954). MIRANDA, L.C.M., Phys. MIRANDA, L.C.M., Progr.
Rev. B&, 3788 (1975). Theor. Phys. 51, 1570
(1975).
Solid State Communications, Vo1.39, pp.345-349. Pergamon Press Ltd. 1981. Printed in Great Britain.
L
0
-
PHONON
INSTABILITY IN THE FREE-CARRIER STRONG MAGNETIC FIELDS O.A.C.
Nunes and Departamento Universidade
70.910
-
IN
SEMICONDUCTORS
UNDER
L.C.M. Mi randa de Fis i ca de Brasilia
Brasilia,
M. lnstituto
ABSORPTION
DF,
Brasil
and A.
Tenan de Fisica Universidade Estadual de Campinas 13.100 Campinas,thSP, Brasil (Received November 27 by R.C.C. Leite)
The LO-phonon damping in an electron-phonon system the presence of a laser beam and a d. c. magnetic field discussed. It is shown that near the laser-cyclotron resonance, and for the laser beam propagating perpendicularly to the magnetic field the Lq-phonon population in a relatively narrow range of K may grow with time.
The generation of nonequi librium LO-phonon population in semiconductors has been known for some time’-‘. It has been experimentally observed mainly in connection with Raman scattering studies under intense photoexcitation such that the incident photon energy ‘hw, is greater than the semiconductor energy gap. The steady-state nonequilibrium population of the LO-phonons is interpreted as a result of the relaxation of the photoexcited electrons to the bottom of the conduction band via successive emission of LO-phonons. Under the additional presence of a strong d.c. magnetic, field an enhancement of the LO-phonon Raman line has also been observedke7 when the LO-phonon frequency w equals an harmonic of the electron cyclo?ron frequency wc. Similarly to the zero field case, this strong departure of the LO-phonon population from its equilibrium value obtains when the injected photoelectrons cascade down the Landau levels emitting LO-phoIn contrast to these studies in nons. highly photoexcited semiconductors, very little has been published regarding LOphonon instability in connection with intraband 1 ight absorption experiments. In this paper we investigate an alternative mechanism for LO-phonon amplification in semiconductors namely, the LOphonon amplification induced by intraband absorption of electromagnetic radiation in the presence of an additional d. c. magnetic field. the In the absence of magnetic field, this problem d.c. has discussed by Gurevich and recent 1 1 been Parshin . Here, however, one includes the additional effects of a strong d.c. magnetic field, encouraged by the possibility of exploring the effects of the laser-cyclotron resonance. In fact,
in is
the resonance condition, where the laser frequency equals the electron cyclotron frequency , may be approached either by increasing the magnetic field strength or by using submillimeter lasers which are now becoming available. Hence, apart from its eventual application to intraband magneto-absorption experiments, we believe that the subject is of sufficient interest to warrant an independent investigation. We consider photon beam from incident quency uL.. conductor The magnetic vector+potential gauge effectiee electron
A
a a
circularly laser source on a n-type
polarized of fresemi-
In
a constant magnetic field. fielcJ is described by a A, given in the Landau Within the = (-Hy, 0.0). mass approximation the onestates are described by the
Landau wavefunctions’ n,px,pz>. where n is the Landau level quantum number. The corresponding one-electron energies are E,(p ) = (n + 1/2) hole+ hZp */2m, where H/me is the electronzcyclotron IA! =‘e fre The electron-LO-phonon interaction qcency. is assumed to be dominated by the longrange electrostatic interaction described by the FrHlich Hamiltonian”. The rate of change of the LO-phonon population Nk, due to the carrier-assisted process, “photon + carrier + phonon + “is described by the kinetic carrier, eauation
dNk -= &[+p-(a,@+ dt
+P+(a,a)
-
-P-(a,R)-
(a,$)]fa(l-fg) Where
345
fo
is
the
distribution
function
_p+
FREE-CARRIER
346
ABSORPTION
for electrons in the state CY = (n,px,pz). and the P’s represent the probability amplitude for an electronic transition where the superscript (subscript) indicates the emission (+) or the absorption (-) of a photon (phonon). The probabi 1 i ty ampl i tudes are calculated using second - or-der perturbation theory”712. In the following, we shall restrict ourselves only to the important situation where the laser beam propagates perpendicularThe reason ly to the magnetic field. for this is that one can show that this is the only configuration for which the P’s exhibit the laser-cyclotron (w = o? ) resonance. Hence, under these ci rkuns4 retaining only the resonant tances, terms and denoting by N the number of photons with wave vectob z and f requenone can show that Eh. reduces (I) CY w L, to
--dNk
=
.
~-fn+l(p,-kZ)]~
I . . . . I= (n+l)NL(Nk+l)fn(pZ) (E (p ) + I~uL-E~+~ n
(pz-kZ)-&k)+n(NL+l)
z
the
distribution
can
dNk --..-=y dt
yk
=
N
k
2niVk12
spontaneous
’ a2 No
‘Li
z
l;hm’ (wL-WC) ’
-fn(pz)
1 - f,+,
[
6
(hmL-il~~-h,,,
$+k,)-j
6
$-En+,
6
(Nk+l)
(En(pZ)-~~L-En_l
NLNkfn(pz)
(p,+kZ)
(p,-kZ)
fn
;
$3
c p,
(pZ+kz)
1
6
(E&p,)
- ,+oL
-
Yk
(pz)
performing yK :
the
=c+
pi e’ .) -
(3) l/2
Here and
(p,)
En_l(pZ+kz)
hwK)
Vk= VL=
i (2n
(27r
e*
-1
-p2*
*
‘dpZ
integrations
c
(EF
8
aiImthezo
gets
-1
woKL2(e
m -
-
0
-
En
~0~) FC
m
(n+l)Cp
1*
n=O
(EF-hwL+hwo-En(p,)
(p,)
-
8
(EF-hwL-htio-En(p2)fl
)
electron - LO - phonon and electron-photon vertices, respectively, a2 = hk*/mw , and V is the crystal volume; the ot k er q”uantities have their usual meaning. Now, assuming that the carriers form a completely degenerate electron gas, such that their distribution functions become step functions, it follows from Eq. (3) that if w > w > w. only the 1 ight L L absorbing processes (namely, the first and third I ines of Eq. (3) are non-vanish This, in turn, entai Is that one ing. may expect instability in the LO-phonon population at the laser cyclotron resonance condition only when the laser frequency is greater than the LO-phonon fre quency. In this case, assuming further>> I, that N the kinetic equation for L
)
(6) wherep=m(w
“2
one
_I”2
hwo/Vk2)
e’h/E,VWL)
fi
+ m
+ F
2k2/kZ/hwLm(wL-wc)
1-i- f,_l
+&.~kZ)
0
t ion
P
V ‘3rrwc 2Tih
and for
b-fn+l(pZ+kz)]6(En(pz)+
+ %IJJ~)- % (NL+l)NK
p,’
(hw,-+~:‘-‘~++~w
-co
-hw,)-(n+l)
+
(n+l)
n,pxlPZ
(pz-kz)]
tfJP,)l;-f,+,
c-t (pZ-KZ)]
written
+y’
k
is the irrelevant term and
where y’ emission
prescri
fn_l
be
as
fn(Pz)
I; -
LO-phonon
Vol. 39, No. 2
Equation (4) tells us that if yk is posi grows wi?h tive the LO-phonon population < 0 there is whereas for y time, Of courseka net growt (amp1 idamping. fication) of the LO-phonon population is only achievable provided yk is qreater than the phonon decay rate 11~ due to the lattice anharmonicity. Substituting in Eq. (5) the f’s by the corresponding step functions. transforming the summations over p and pz into integrals through the usGal
dt where:
IN SEMICONDUCTORS
P2
=
I m(wL
-
L wc
-w +
41:)
wo)/
hiKZI
and
+C;jkZI
To proceed further, we consider in the following the ultraquantum limit which the carrier concentration is such that only the n = 0 Landaullevel is fully occupied, namely, E =hw,; for an InSb sample in a magnetig ?ield of the order of 100 KOe this condition requires a carrier concentration of the order of 6.85 x 1016 cmm3. In this case, perform ing the summation over n in Eq. (6) witTT the help of the S-functions, the condition for yk> 0 can now be stablished. One gets: A.
For
wL>
wc>
wo,
2 k2/kZ13k3
mL (w,
- wc)’
Vol. 39, No. 2
FREE-CARRIER ABSORFTION IN SEMICONDUCTORS
6 f2, a
347
FREE-CARRIER
348
ABSORPTION
Vol. 39, No. 2
IN SEMICONDUCTORS
provided
- wc)j <
!kz!
<
20.0
$[j I
-
(I*IL_
wc)
For
wc>
6.
O (7b)
/ &IL>
2n e4kL’ k’jk
z
I3
w
cl
,
cEGih3
-.
150
‘o’)m~io’
w (~1 Lc
-
(8a)
(ii ) L
t” 9
provided
_
100
1a0
h;F -
+(wc
t
(w c
+
i*iL)
- wL) 1 < / kZ : < ,+;
1
lilo
/
(8b) 5.0
where
k
=
k ’
- 1”
F
1
from Eqs. (7) and (8) that condition for amp1 ification In other words, near the depends on k. the laser-cyclotron resonance condition, LO-phonon popula$ion, in a relatively narrow range of k, may become unstable, ie; there is a selective mechanism for In Table the excitation of LO phonons. the interval t , we show each value of wc/wL, LO-phonon amplification for an InSb sample with a carrier concentration such that E = 3bc/2. The following values for InSb have been the phyfsical parameters of used: m = 0.01 m E =18. F = 14.06 = 3.8 x 1013 “;-‘4” Here Le aote that w tf?e conditions of laser-c clotron resoand of E =3 tl oic/Z nance w can be met = w t nSb Sample illLminared by a CO2 for an laser (XL= IO pm) yro_vided H = 118 KOe and n = 5 x IO’ cm . In Fig. 1, we obeing show the smplification rate y / k YO a coefficient given by It
the
YO=
follows threshold
(N
)n
e4
K L’(Fi’
-
eo-‘)m
~il~,~k~,k
,3
L’V L
h3wL, as a function of the cyclotron frequency for an InSb sample i 1 luminated by a CO2 We have arbitrarily chosen, laser. without loss of generality, a geometry such that k = I .41 k . Actually, the criterion for the on:et of the instabiliis somewhat more involved: a net ty growth of the LO-phonon population is
Ic Fig. I (yk/yo) frequency illuminated
only than the
:
LO - phonon as a function (tic/~!,) for by CO2
achievable provided the LO-phonon decay lattice anharmonicity Yk
-
Fk
amp1 ification of the cyclotron an InSb sample laser.
‘k rate
,
is
greater n k due
1.
t: ;: 7.
to
ie,
b 0
(9)
The above condition enta Is that there is a threshold laser intens ty for the actual LO-phonon amplification. To get a numeri cal estimate of the crit cal intensity we consider an InSb samp e illuminated by laser havin a carrier concentration a CO -9 of 6225 x 1016cm in a magnetic phonon losses rjk are typically of the order of 0.01 wo, Eq. (9) gives us a critical intenstty of the order of IO’ w/cm’. These predictions could be experimentally observ for instance, in connection with free ed. s in -carrier absorpt/on experimen which a second strong pumping field is simul taneously present with the prob ng field. The LO-phonon instability wou in this d, case, be manifested as an inc ease in the absorption of the probing fie d as the pumping field approaches the aser-cvclotron resonance condition.
REFERENCES
2.
rate
and LEITE, R.C.C., Phys. Rev. Letters 22, 1304 (1969) SHAH, J. SHAH, J., LEITE, R.C.C. and SCOTT, J.F., Solid State Commun. 8, I089 (1970) MATTOS, J.C.V. and LEITE, R.C.C., Solid State Commun. 12, 465 (1973) VELLA - COLEIRO G.P., Phys. Rev. Letters 23, 697 (l969r and ZILBERBERG V.V., Sov. Phys. Solid State 11, 1465 (1970). GENKIN, G.M., LUZZI, R., Progr. Theor. Phys. 2, 13 (1972) FROTA-PESSOA, s. and LUZZI , R., Phys. Rev. 813, 5420 (1976)
FREE-CARRIER ABSORPTION IN SEMICONDUCTORS
Vol. 39, No. 2
8. 9. IO.
I 1. 12.
GUREVICH, LANDAU,
V. L.D.
Oxford. 1958) FROLICi, H., GOMES,M.A.F. COUTINHO, S.
L. and PARSHIN, and LIFSCHITZ, P.
Adv. and and
349
D. A. Zh. Eksp. Teor. Fiz. 2, 1589 (1977). E.M., "Quantum Mechanics" (Pergamon Press,
474. Phys. 2, 325 (1954). MIRANDA, L.C.M., Phys. MIRANDA, L.C.M., Progr.
Rev. B&, 3788 (1975). Theor. Phys. 51, 1570
(1975).