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Optical gain and induced absorption from excitonic molecules in ZnO
Solid State Communications, Vol. 26, pp. 987—990. © Pergamon Press Ltd. 1978. Printed in Great Britain.
0038—1098/78/0622—0987
$02.00/0
OPTICAL GAIN AND INDUCED ABSORPTION FROM EXCITONIC MOLECULES IN ZnO J.M. Hvam Institute of Physics, Odense University, DK-5230 Odense M, Denmark (Received 10 February 1978 by M. Qirdona) Optical gain has been investigated ZnO 1 .8 K for different excitation 2. Gain in from theatrecombination of excitonic densities up to 10 MW cm molecules is found in two bands separated by 60 meV, corresponding to the final state being an exciton in state n = 1 (M-line) and an exciton in a continuum state (M’-line), respectively. At high excitation densities optical conversion of excitons into excitonic molecules is observed. The binding energy of the molecule is E~= 20.9 ±0.5 meV. IN THE Il—VI COMPOUNDS ZnO, CdS and CdSe a lot of controversy still exists, concerning the assignment of the new emission lines observed under intense excitation at low temperature. In all three materials an emission line, called theM-line, has been observed at a spectral position just below the bound exciton lines Fl, 2]. It has been attributed to the recombination of excitonic molecules (EM) [1—4],although alternative explanations, involving bound excitons interacting with acoustic phonons [51or free electrons [6], have been given. Very recently, it has been predicted that the carrier lattice coupling in polar materials provide stability to an electron—hole liquid (EHL) even in direct semiconductors [7] and, in accordance with that, optical gain from EHL has been reported in CdS [8, 9], CdSe [8] and ZnO [10]. In this paper we investigate the optical gain in the M-band region of ZnO under excitation densities covering three orders of magnitude, with a maximum of about 10MW cm2. Assuming an exciton penetration depth of 1 j.~mand a lifetime of 10~0sec, a conservative estimate of our maximum carrier density is about 1019 cm3, corresponding to close packing of excitons considered as spheres with radius 18 A [11]. From the gain spectra we are able to conclude that under these conditions EHL was not formed. On the contrary, our results give strong support to the formation of EM. The optical gain measurements were performed on thin (‘~‘2 ~zm)as-grown platelets of ZnO immersed in liquid He at 1.8K. The gain was measured by a new technique based on the method of amplified spontaneous emission [12]. By modulating the excitation length and applying synchronous detection and simple electronic processing we could record directly spectra with a straightforward relation to the induced gain/absorption. The technique and its applicability iS described in detail elsewhere [13]. The excitation source was a pulsed N 2-gas laser (Avco, C950). The
width of the line focus was smaller than 100 ~xmand the excitation length was modulated between 0.2 and 0.4 mm. The amplified spontaneous emission was analyzed by a double spectrometer followed by a photomultiplier, and the light pulses were averaged with boxcar technique using a gate width of 50 nsec. Figure 1 shows gain spectra for various excitation powers J. The spectra were recorded by modulating the excitation length between 1 and 2! and in this case the recorder signal F~gl)is given by [13] 1(21) F~g1)= K log = K log (et1 + 1) (1) I~j, where 1(1) is the amplified spontaneous emission with excitation length 1 [12], g is the gain coefficient and K is a calibration constant. The function F(gl) is shown in the insert of Fig. 1 with K = (log 2)’, thus F(gl) = 1 corresponds tog = 0. Note that the zero and unity of F(gl) are easily identified on the spectra recorded at low and medium excitation density (e.g. J = 0.1). For comparison with the gain spectra the spontaneous emission from the excited surface of the sample is shown (broken curve). The spontaneous spectrum is dominated by the known exciton impurity lines12,16 and 19 [14] marked by the arrows. The gain spectra at low excitation density (J = 0.01—0.1) show absorption in the bound exciton region and a region with positive gain below 3.356 eV, i.e. in the previously reported M-band [2]. The gain profile is asymmetric and the line width (FWHM) is about 7.5 meV. Furthermore, we observe a new line, denoted M’, with positive gain. The width of the M~lineis about the same as that of the M-line and its spectral position at 3.297 eV is an exciton binding energy (60 meV) below the M-line. Increasing the excitation power above J = 0.1, the high energy slope of the gain profile (M-line) shifts toward lower energies. At the same time a new structure ~
appears in the gain/absorption curves peaking at 987
988
EXCITONIC MOLECULES IN ZnO
WAVELENGTH 3700
I
o
—
J=0.05 Mj l2~
1
I
Vol. 26, No. 12
(A) 3750 I I
I
7
-4 -2 P(2)~ P(1
0
2
4
gi
M’~
1619
z
~
335 3.30 PHOTON ENERGY (eV) Fig.2.1.The Optical relation gainbetween spectra (solid the recorder curves) signal for different and tbeexcitation gain coefficient, densities F(gl), J. J is= shown 10 corresponds in the insert. to about The broken 10MW cm is a spontaneous emission spectrum recorded at J = 1. curve 3.3563 eV. Increasing further the pumping the gain is growing in the region of exciton—exciton collision processes (P-band) [2]. In Fig. I we have subtracted this background from the M~linerevealing the latter to be slightly decreasing and broadening with increasing excitation. The transparency peak at 3.3563 eV clearly separates the induced absorption in the M-band from the band edge absorption, proving that no band shrinkage and EHL formation takes place, in contrast to previous claims [10]. Similar measurements on CdS [151
have recently led to the same conclusions, i.e. no condensation of excitons into an EHL for excitation densities up to about 10MW cm2. To explain the observed gain/absorption spectra in the M-band region consider the following reverse processes: EM(K) FE(R’) + hw; i.e. recombination of EM creating a photon (energy hw) and a free exciton (FE), or formation of EM by simultaneous annihilation of an exciton and a photon. If we take the optical matrix element to be wave vector independent and
Vol. 26, No. 12
EXCITONIC MOLECULES IN ZnO
M-line (E~—E~)has been fixed to coincide with the experimental peak, marked M in Fig. 1, i.e. E~ E~= 3.3563 eV ±0.5 meV. Hence, withEr = 3.3772 eV [18] —
/ 150
ZnO 1.8 K
‘,
I
\
I
we obtain a binding energy of EM,E~= 20.9 meV± 0.5 meV, which is close to previous estimates [2] based on luminescence spectra but exceeds the theoretically predicted value [19]. The position of the M~linesuggests a process similar to the one above, but with the final exciton left in a highly excited or completely dissociated state, i.e. EM eh + hw. This line is reported here for the first time and gives new evidence for the existence of EM in ZnO. The theoretical fit of the gain spectra in Fig. 2 implies effective temperatures common to EM and FE of 39 K (a) and 46K (b) which are reasonable values at the present excitation levels [2]. From the parameters 17 and ANM we deduce, that by increasing the excitation
—01 -
.
100 E X~ -~
THEORY
----
-+
50 (a)
I
M
O
E
I
0
5
10
,
15
1mev]
20
/ I
J 5 — -
density by a factor of 50, N~is increased by a factor of 11 and NM is almost unchanged, in fact slightly reduced. The variation in FE density agrees with the
1(b)
-50
3 36
989
3 35
3 34 ~ PHOTON ENERGY [eV)
Fig. 2. Comparison between experimental gain spectra and theoretical spectra, calculated from equation (2), for two different excitation densities (a)J = 0.1~(b) J = 5. The fitting parameters are (a) ANM = 3272 cm~ meV, 13 = 0.3 (meV)’, i~= 1.6; (b)ANM = 2985 cm’ meV, j3 = 0.25 (meV)’, ,~= 20.
expectation N~ JQ with 0.5
from simple rate equations for EM and FE, thermahzed within their respective bands. One could imagine that the implicit assumption of small signal gain [13],not affecting the balance neglect damping effects, the emission line shape for EM between NM and N~,does not hold. But even so, this recombination becomes an inverse Boltzmann [161. could not explain the conversion from gain to absorpConsidering also the reverse process, the resulting gain! tion in the M-band, since optical stimulation at most absorption profile takes the form [17] will tend to equalize NM and N~. g(E) = AN ~33/2(l 2e~E (2~ A possible explanation is to assume a Bose conM 17e~Ei~ densation (BC) of EM, acting as a drain for the where E = E~—E~ hw (E~is the FE energy at K = 0 thermalized EM entering into equation (2). This and EJ~is the EM binding energy), ~ = 23’2(Nex!NM) assumption is consistent with the appearance of the (Nex and NM are densities of FE and EM, respectively), peak at 3.3563 eV, corresponding to transitions of EM = 1 !kTe~ and A is a constant containing the matrix at K = 0. This peak, the width of which was resolution element (considered K-independent). In equation (2) it limited, showed threshold character and in some cases has been assumed that EM and FE are thermally distriit was more pronounced than in Fig. 1. Against the buted within their respective bands at an effective Bose condensation speaks the relatively high temperatemperature Teff which may exceed the lattice temperature of EM, determined from the theoretical fit. On the ture. In Fig. 2 we have compared two experimental other hand, if we consider EM as a weakly interacting spectra (J = 0.1 and J = 5), converted into a linear gas of bosons, the critical density N& for BC of EM is scale of gain coefficient, with the theory as expressed given by [20] by equation (2). The parameter ANM has been used to (MkT normalize the theoretical curves to the experimental Nivi = 6h3 (3) spectra, whereas 13 has been chosen to make the line widths match. Finally, ~ has been fitted to obtain the where Mis the translational mass of EM (M = I .74,n 0 best overall agreement, wMch is quite satisfactory. It in ZnO [21]). should be noted that the high energy cut-off of the At Teff = 45 K we obtain from equation (3): —
/
—
990
EXCITONIC MOLECULES IN ZnO Vol. 26, No. 12 Nf~= 4.4 x 10’s cm3, which still is below our estimated sitions. The induced absorption band associated with maximum excitation density. a gain peak (or at least a transmission window) on the In conclusion, the automatic recording of gain high energy side contradicts a condensation of FE into spectra employed in this work has made possible a more EHL. It is consistent, however, with a Bose condendetailed investigation of induced gain/absorption in ZnO sation of EM, which is further supported by the than previously reported. The induced absorption in the observed kinetics of the densities of FE and EM. M-band give evidence to the optical conversion of FE We suggest that further investigations of the kind into EM [16] in ZnO, and confirms at the same time performed here will prove valuable in the clarification of the previously reported M-line to be due to EM tranthe high excitation effects in direct gap semiconductors. REFERENCES 1.
SHIONOYA S., SAITO H., HANAMURA E. & AKIMOTO 0., Solid State Commun. 12, 223 (1973).
2. 3.
HVAM J.M., Phys. Status Solidi (b) 63, 511(1974). VOIGT J. & MAUERSBERGER G., Phys. Status Solidi (b) 60, 679 (1973).
4.
SEGAWA Y. & NAMBA S., Solid State Commun. 17, 489 (1975).
5.
DITE A.F., REVENKO V.!., TIMOFEEV V.B. & ALTUKHOV P.D.,JETPLett. 18, 341 (1973).
6.
KLINGSHIRN C., Solid State Commun. 15, 883 (1974).
7.
BENI G. & RICE T.M.,Phys. Rev. Lett. 37, 874 (1976).
8. 9. 10.
LEHENY R.F. & SHAH J.,Phys. Rev. Lett. 37, 871 (1976). MULLER G.O. WEBER H.H., LYSENKO V.G., REVENKO V.1. & TIMOFEEV V.B., Solid State Commun. 21,217(1977). SKETTRUP T., Solid State Commun. 23, 741 (1977).
11.
SKETTRUP T. & BALSLEV I. Phys. Status Solidi 40,93 (1970).
12.
SHAKLEE K.L. & LEHENY R.F.,Appl. Phys. Lett. 18, 475 (1971).
13.
HVAM J.M.,J. Appl. Phys. (to be published). The application of the technique is limited to the range of small signal gain. Gain saturation was avoided by keeping the excitation length small (1< 0.4mm).
14.
REYNOLDS D.C., LITTON C.W. & COLLINS T.C.,Phys. Rev. 140, A1726 (1965).
15.
HVAM J.M., Solid State Commun. (to be published).
16.
HANAMURA E., J. Phys. Soc. Japan 39, 1516 (1975).
17. 18.
KUSHIDA T. & MORIYA T.,Phys. Status Solidi 72, 385 (1975). SKETTRUP T. & BALSLEV I.,Phys. Rev. B3, 1457 (1971).
19. 20.
BRINKMAN W.F., RICE T.M. & BELL B.,Phys. Rev. B8, 1570 (1973). LANDAU L. & LIFSHITZ E., StatisticalPhysics, 2nd edn, p. 159. Pergamon Press, New York (1970).
21.
HuMMER K.,Phys. Status Solidi(b) 56, 249 (1973).
0038—1098/78/0622—0987
$02.00/0
OPTICAL GAIN AND INDUCED ABSORPTION FROM EXCITONIC MOLECULES IN ZnO J.M. Hvam Institute of Physics, Odense University, DK-5230 Odense M, Denmark (Received 10 February 1978 by M. Qirdona) Optical gain has been investigated ZnO 1 .8 K for different excitation 2. Gain in from theatrecombination of excitonic densities up to 10 MW cm molecules is found in two bands separated by 60 meV, corresponding to the final state being an exciton in state n = 1 (M-line) and an exciton in a continuum state (M’-line), respectively. At high excitation densities optical conversion of excitons into excitonic molecules is observed. The binding energy of the molecule is E~= 20.9 ±0.5 meV. IN THE Il—VI COMPOUNDS ZnO, CdS and CdSe a lot of controversy still exists, concerning the assignment of the new emission lines observed under intense excitation at low temperature. In all three materials an emission line, called theM-line, has been observed at a spectral position just below the bound exciton lines Fl, 2]. It has been attributed to the recombination of excitonic molecules (EM) [1—4],although alternative explanations, involving bound excitons interacting with acoustic phonons [51or free electrons [6], have been given. Very recently, it has been predicted that the carrier lattice coupling in polar materials provide stability to an electron—hole liquid (EHL) even in direct semiconductors [7] and, in accordance with that, optical gain from EHL has been reported in CdS [8, 9], CdSe [8] and ZnO [10]. In this paper we investigate the optical gain in the M-band region of ZnO under excitation densities covering three orders of magnitude, with a maximum of about 10MW cm2. Assuming an exciton penetration depth of 1 j.~mand a lifetime of 10~0sec, a conservative estimate of our maximum carrier density is about 1019 cm3, corresponding to close packing of excitons considered as spheres with radius 18 A [11]. From the gain spectra we are able to conclude that under these conditions EHL was not formed. On the contrary, our results give strong support to the formation of EM. The optical gain measurements were performed on thin (‘~‘2 ~zm)as-grown platelets of ZnO immersed in liquid He at 1.8K. The gain was measured by a new technique based on the method of amplified spontaneous emission [12]. By modulating the excitation length and applying synchronous detection and simple electronic processing we could record directly spectra with a straightforward relation to the induced gain/absorption. The technique and its applicability iS described in detail elsewhere [13]. The excitation source was a pulsed N 2-gas laser (Avco, C950). The
width of the line focus was smaller than 100 ~xmand the excitation length was modulated between 0.2 and 0.4 mm. The amplified spontaneous emission was analyzed by a double spectrometer followed by a photomultiplier, and the light pulses were averaged with boxcar technique using a gate width of 50 nsec. Figure 1 shows gain spectra for various excitation powers J. The spectra were recorded by modulating the excitation length between 1 and 2! and in this case the recorder signal F~gl)is given by [13] 1(21) F~g1)= K log = K log (et1 + 1) (1) I~j, where 1(1) is the amplified spontaneous emission with excitation length 1 [12], g is the gain coefficient and K is a calibration constant. The function F(gl) is shown in the insert of Fig. 1 with K = (log 2)’, thus F(gl) = 1 corresponds tog = 0. Note that the zero and unity of F(gl) are easily identified on the spectra recorded at low and medium excitation density (e.g. J = 0.1). For comparison with the gain spectra the spontaneous emission from the excited surface of the sample is shown (broken curve). The spontaneous spectrum is dominated by the known exciton impurity lines12,16 and 19 [14] marked by the arrows. The gain spectra at low excitation density (J = 0.01—0.1) show absorption in the bound exciton region and a region with positive gain below 3.356 eV, i.e. in the previously reported M-band [2]. The gain profile is asymmetric and the line width (FWHM) is about 7.5 meV. Furthermore, we observe a new line, denoted M’, with positive gain. The width of the M~lineis about the same as that of the M-line and its spectral position at 3.297 eV is an exciton binding energy (60 meV) below the M-line. Increasing the excitation power above J = 0.1, the high energy slope of the gain profile (M-line) shifts toward lower energies. At the same time a new structure ~
appears in the gain/absorption curves peaking at 987
988
EXCITONIC MOLECULES IN ZnO
WAVELENGTH 3700
I
o
—
J=0.05 Mj l2~
1
I
Vol. 26, No. 12
(A) 3750 I I
I
7
-4 -2 P(2)~ P(1
0
2
4
gi
M’~
1619
z
~
335 3.30 PHOTON ENERGY (eV) Fig.2.1.The Optical relation gainbetween spectra (solid the recorder curves) signal for different and tbeexcitation gain coefficient, densities F(gl), J. J is= shown 10 corresponds in the insert. to about The broken 10MW cm is a spontaneous emission spectrum recorded at J = 1. curve 3.3563 eV. Increasing further the pumping the gain is growing in the region of exciton—exciton collision processes (P-band) [2]. In Fig. I we have subtracted this background from the M~linerevealing the latter to be slightly decreasing and broadening with increasing excitation. The transparency peak at 3.3563 eV clearly separates the induced absorption in the M-band from the band edge absorption, proving that no band shrinkage and EHL formation takes place, in contrast to previous claims [10]. Similar measurements on CdS [151
have recently led to the same conclusions, i.e. no condensation of excitons into an EHL for excitation densities up to about 10MW cm2. To explain the observed gain/absorption spectra in the M-band region consider the following reverse processes: EM(K) FE(R’) + hw; i.e. recombination of EM creating a photon (energy hw) and a free exciton (FE), or formation of EM by simultaneous annihilation of an exciton and a photon. If we take the optical matrix element to be wave vector independent and
Vol. 26, No. 12
EXCITONIC MOLECULES IN ZnO
M-line (E~—E~)has been fixed to coincide with the experimental peak, marked M in Fig. 1, i.e. E~ E~= 3.3563 eV ±0.5 meV. Hence, withEr = 3.3772 eV [18] —
/ 150
ZnO 1.8 K
‘,
I
\
I
we obtain a binding energy of EM,E~= 20.9 meV± 0.5 meV, which is close to previous estimates [2] based on luminescence spectra but exceeds the theoretically predicted value [19]. The position of the M~linesuggests a process similar to the one above, but with the final exciton left in a highly excited or completely dissociated state, i.e. EM eh + hw. This line is reported here for the first time and gives new evidence for the existence of EM in ZnO. The theoretical fit of the gain spectra in Fig. 2 implies effective temperatures common to EM and FE of 39 K (a) and 46K (b) which are reasonable values at the present excitation levels [2]. From the parameters 17 and ANM we deduce, that by increasing the excitation
—01 -
.
100 E X~ -~
THEORY
----
-+
50 (a)
I
M
O
E
I
0
5
10
,
15
1mev]
20
/ I
J 5 — -
density by a factor of 50, N~is increased by a factor of 11 and NM is almost unchanged, in fact slightly reduced. The variation in FE density agrees with the
1(b)
-50
3 36
989
3 35
3 34 ~ PHOTON ENERGY [eV)
Fig. 2. Comparison between experimental gain spectra and theoretical spectra, calculated from equation (2), for two different excitation densities (a)J = 0.1~(b) J = 5. The fitting parameters are (a) ANM = 3272 cm~ meV, 13 = 0.3 (meV)’, i~= 1.6; (b)ANM = 2985 cm’ meV, j3 = 0.25 (meV)’, ,~= 20.
expectation N~ JQ with 0.5
from simple rate equations for EM and FE, thermahzed within their respective bands. One could imagine that the implicit assumption of small signal gain [13],not affecting the balance neglect damping effects, the emission line shape for EM between NM and N~,does not hold. But even so, this recombination becomes an inverse Boltzmann [161. could not explain the conversion from gain to absorpConsidering also the reverse process, the resulting gain! tion in the M-band, since optical stimulation at most absorption profile takes the form [17] will tend to equalize NM and N~. g(E) = AN ~33/2(l 2e~E (2~ A possible explanation is to assume a Bose conM 17e~Ei~ densation (BC) of EM, acting as a drain for the where E = E~—E~ hw (E~is the FE energy at K = 0 thermalized EM entering into equation (2). This and EJ~is the EM binding energy), ~ = 23’2(Nex!NM) assumption is consistent with the appearance of the (Nex and NM are densities of FE and EM, respectively), peak at 3.3563 eV, corresponding to transitions of EM = 1 !kTe~ and A is a constant containing the matrix at K = 0. This peak, the width of which was resolution element (considered K-independent). In equation (2) it limited, showed threshold character and in some cases has been assumed that EM and FE are thermally distriit was more pronounced than in Fig. 1. Against the buted within their respective bands at an effective Bose condensation speaks the relatively high temperatemperature Teff which may exceed the lattice temperature of EM, determined from the theoretical fit. On the ture. In Fig. 2 we have compared two experimental other hand, if we consider EM as a weakly interacting spectra (J = 0.1 and J = 5), converted into a linear gas of bosons, the critical density N& for BC of EM is scale of gain coefficient, with the theory as expressed given by [20] by equation (2). The parameter ANM has been used to (MkT normalize the theoretical curves to the experimental Nivi = 6h3 (3) spectra, whereas 13 has been chosen to make the line widths match. Finally, ~ has been fitted to obtain the where Mis the translational mass of EM (M = I .74,n 0 best overall agreement, wMch is quite satisfactory. It in ZnO [21]). should be noted that the high energy cut-off of the At Teff = 45 K we obtain from equation (3): —
/
—
990
EXCITONIC MOLECULES IN ZnO Vol. 26, No. 12 Nf~= 4.4 x 10’s cm3, which still is below our estimated sitions. The induced absorption band associated with maximum excitation density. a gain peak (or at least a transmission window) on the In conclusion, the automatic recording of gain high energy side contradicts a condensation of FE into spectra employed in this work has made possible a more EHL. It is consistent, however, with a Bose condendetailed investigation of induced gain/absorption in ZnO sation of EM, which is further supported by the than previously reported. The induced absorption in the observed kinetics of the densities of FE and EM. M-band give evidence to the optical conversion of FE We suggest that further investigations of the kind into EM [16] in ZnO, and confirms at the same time performed here will prove valuable in the clarification of the previously reported M-line to be due to EM tranthe high excitation effects in direct gap semiconductors. REFERENCES 1.
SHIONOYA S., SAITO H., HANAMURA E. & AKIMOTO 0., Solid State Commun. 12, 223 (1973).
2. 3.
HVAM J.M., Phys. Status Solidi (b) 63, 511(1974). VOIGT J. & MAUERSBERGER G., Phys. Status Solidi (b) 60, 679 (1973).
4.
SEGAWA Y. & NAMBA S., Solid State Commun. 17, 489 (1975).
5.
DITE A.F., REVENKO V.!., TIMOFEEV V.B. & ALTUKHOV P.D.,JETPLett. 18, 341 (1973).
6.
KLINGSHIRN C., Solid State Commun. 15, 883 (1974).
7.
BENI G. & RICE T.M.,Phys. Rev. Lett. 37, 874 (1976).
8. 9. 10.
LEHENY R.F. & SHAH J.,Phys. Rev. Lett. 37, 871 (1976). MULLER G.O. WEBER H.H., LYSENKO V.G., REVENKO V.1. & TIMOFEEV V.B., Solid State Commun. 21,217(1977). SKETTRUP T., Solid State Commun. 23, 741 (1977).
11.
SKETTRUP T. & BALSLEV I. Phys. Status Solidi 40,93 (1970).
12.
SHAKLEE K.L. & LEHENY R.F.,Appl. Phys. Lett. 18, 475 (1971).
13.
HVAM J.M.,J. Appl. Phys. (to be published). The application of the technique is limited to the range of small signal gain. Gain saturation was avoided by keeping the excitation length small (1< 0.4mm).
14.
REYNOLDS D.C., LITTON C.W. & COLLINS T.C.,Phys. Rev. 140, A1726 (1965).
15.
HVAM J.M., Solid State Commun. (to be published).
16.
HANAMURA E., J. Phys. Soc. Japan 39, 1516 (1975).
17. 18.
KUSHIDA T. & MORIYA T.,Phys. Status Solidi 72, 385 (1975). SKETTRUP T. & BALSLEV I.,Phys. Rev. B3, 1457 (1971).
19. 20.
BRINKMAN W.F., RICE T.M. & BELL B.,Phys. Rev. B8, 1570 (1973). LANDAU L. & LIFSHITZ E., StatisticalPhysics, 2nd edn, p. 159. Pergamon Press, New York (1970).
21.
HuMMER K.,Phys. Status Solidi(b) 56, 249 (1973).