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Peculiarities of thermal expansion of layered crystals
Solid State Communications, Vol. 44, No. 12, pp. 1613-1615, 1982. Printed in Great Britain.
0038-1098/82/481613-03 $03.00/0 Pergamon Press Ltd.
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS G.L. Belenkii, S.G. Abdullayeva, A.V. Solodukhin and R.A. Suleymanov (Institute of Physics, Academy of Sciences of Azerbaidjan SSR, Prospect Narimanova 33, 370143, Baku, USSR)
(Received 2 August 1982 by F. Bassani) Linear expansion coefficients parallel, ixll, and perpendicular, ix±, to layer plane of GaS, GaSe, and TIGaS2 are investigated in the temperature range 5 - 3 0 0 K. Negative values of ixll are observed in GaS and GaSe in the range of 3 0 - 5 0 K caused by negative Gruneisen parameters 'Yll in this interval. It is shown that negative ixll in layer crystal can be due to TA± acoustic branch contribution to ixllLAYER CRYSTALS consist of separate layers with strong (covalent) bonding between atoms within the layers. The bonding between the layers is weak, predominantly of Van der Waals type. Such crystal structure causes the specific shape of phonon branches in layer crystals [ 1]. The most important feature of acoustic waves determining the thermal properties of layer crystals at low termperatures is that acoustic waves propagating in the layer plane and polarized perpendicular to the layer plane (TA± mode) can have dispersion law of the form co = aK + BK 2, where k is the wave vector [2, 3]. These acoustic waves are called "bending waves" [3] (or "warping waves" using the terminology of [4]). IAfshits [3] was the first to point out the importance of "bending waves" describing thermal properties of layer and chain structures. According to [3 ] the specific heat of layer crystal is proportional to T 2 at low temperatures when the "bending waves" give the major contribution to the free energy. Note, that T 2 law may also be due to the contribution of acoustic waves with "ordinary" dispersion law, co "" k, if the interlayer interaction is neglected (the specific heat of two dimensional solid). The linear expansion coefficient in the direction parallel to the layer plane, ixll, has the negative value if "bending waves" give the major contribution to ixll. The negative value of Otllis due to "membrane effect" (or "tension effect" [4]), i.e. the increase of these frequencies with layers stretching [3]. The existence of "membrane effect" can be decisive in analysing the contribution of "bending waves" to thermal properties of layer crystals. "Bending waves" were first observed in graphite. The investigations of low energy neutron scattering in graphite [5] showed, that TA± mode has quadratic dispersion law. The linear expansion coefficient ixll has negative values in the wide temperature range 3 0 0 - 6 0 0 K [6], Fig. 2, and specific heat ~12. There are few investigations of ix in other layer crystals, and pecularities of a behaviour with temperature were
not observed, while the experimental investigations of neutron scattering in layer crystals indicate that there are TAt modes with quadratic dispersion law in some of them (for example, in GaS [7]). In this paper we present the results of investigations of linear expansion coefficients all and ix± in layer crystals GaS, GaSe and T1GaS2 at temperatures 5 - 3 0 0 K. The measurements were carried out by quartz dilatometer, described in [8]. The accuracy of measurements is approximately 0.I x 10-6K -1 in the whole temperature range. The lengths of the samples were 6 and 5 mm, respectively, measuring ixll and a± in GaS(GaSe) and 10 and 3 mm in T1GaS2. The experimental values for air and ix± for GaS, GaSe and T1GaS2 are given in Figs. 1-3. The most interesting detail is the ixll behaviour in GaS and GaSe, Figs. I and 2, at low temperatures: ixll is positive at T < 30K and T > 50K and negative in the temperature range 30 < T < 50K. At T = 10-20 K the averaged values of all in GaS are presented because of scattered character of experimental points, Fig. 1. The observed temperature behaviour of ixll in GaS and GaSe can be explained by the Lifshits theory [3]. The "bending waves" are assumed to give the major contribution to ixll in GaS(GaSe) in the temperature range 3 0 - 5 0 K. The investigations of specific heat in GaS(GaSe) in the same temperature interval showed that specific heat obeys T 2 law [9]. The positive values of ixll at T < 30 and T > 50K can be explained by the predominant contribution of phonons with "ordinary" dispersion law to ixll- This interpretation is based on the analysis of temperature behaviour of Gruneisen parameter, obtained from the relation [10]: V
ix, = ~ [ ( c ~
+ c12)ixl, + c~3ix,].
Using elastic constants, C~/~obtained at room tem. perature [7, 11 ] and specific heat, Co, measured in the wide temperature range T = 5 - 3 0 0 K [9] the ~'11 iS
1613
1614
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS
Vol. 44, No. 12
~.'I0,6 K -~
s
9 8
ZO
1
J .ss~
a 5 4 $ 2 I 0
I0
-' 40
80
~0
i ~lO
i ~00
,~0
~00
~.1(
Fig. 1. Temperature behaviour of linear expansion coefficients Oql and ~± in GaS.
,V.~ 2
Fig. 3. Temperature behaviour of linear expansion coefficients oql and ~± in T1GaS2.
x" xs
t
0
-!
2O
~ I g 0
x ~ "~4k•...~.
'
8o'
'
-,, g
Fig. 2. t~ll(T) dependence for GaS, GaSe and graphite. evaluated in the temperature range 5 - 3 0 0 K. It follows that ")'11has negative values at 30 < T < 50K. The ~± has the positive sign in the whole temperature range. From the definition of "/11 it follows that ~11 Eiq (-- a in ~iq/O In a), where a is the lattice parameter. Thus the sign of "/11 is determined by the sign of derivatives a6oi~/aa. If ')'11is negative then a~oiq/aa > o. The positive sign of acoiq/aa m e a n s that phonon frequencies increase while stretching the crystal along the layer plane. As it was already mentioned it is the property of "bending waves". In graphite ~'11 has negative values in the wide temperature range 3 0 - 6 0 0 K, i.e. in a considerably wider range than in GaS(GaSe). To explain this difference between graphite and GaS(GaSe) the following facts must be taken into account. Firstly, bonding between atoms within the layers is significantly stronger in graphite than in GaS(GaSe): Cla + C12 ~ 60 x 1011 dynecm -2 in graphite [12] and 21 x 1011 dynecm -2
(13 x 1011 dyne cm -2) in GaS(GaSe). The values of C33 are as follows: 2.0 x 1011 dyne cm -2 and 3.6 x l0 II (3.5 x 1011) dyne cm -2, respectively. Thus, the ratio (Cll + C12)/C3a is three times as great in graphite as in GaS(GaSe). Secondly, the layers in GaS(GaSe) have more complex crystal structure than in graphite. Each layer in GaS(GaSe) consists of four atomic sublayers S(Se)-Ga-Ga-S(Se), whereas in graphite layers are monoatomic. Thus it may be easier to "bend" the layer in graphite than in GaS(GaSe). The investigations of TA± mode dispersion law in GaS and in graphite show, that quadratic term in this law is more significant in graphite than in GaS (the ratio b/a in dispersion law 60 = aK + bK 2 is ~ 3 times larger in graphite [13]). If the above assumptions are true the increase of the number of atomic planes forming each layer, for example going from GaS (with 4 atomic planes, 8 atoms per unit cell) to TIGaS2 (with 7 atomic planes, 64 atoms per unit cell), leads to dispersion law for TA± branch having the "ordinary" form, ~o ~ k, at small values of k and as a consequence to temperature dependence of Otll determined by phonons with "ordinary" dispersion law (Fig. 3). It must be mentioned in conclusion, that temperature behaviour of linear expansion coefficients in layer crystals described above can lead to unusual behaviour of various parameters with temperature, such as values of direct and indirect gaps in GaS [14] or crystal field constants in GaSe [15]. These effects we hope to discuss in a separate paper.
Vol. 44, No. 12
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS REFERENCES
9.
L.N. Alieva, G.L. Belenkii, I.I. Reshina, E.Yu. Salaev & V.Y. Shteinshriber, Fis. T~erd. Tela 21, 126 (1979). 2. K. Komatsu, J. Phys. Soc. Japan 10, 471 (1955). 3. I.M. Lifshits, JETP 22, 471 (1952). 4. T.H.K. Barron,J. Appl. Phys. 41, 5044 (1970). 5. K.K. Mani & R. Ramani, Phys. Status Solidi (b) 61,659 (1974). 6. A.C. Bailey & B. Yates,J. Appl. Phys. 41, 5088 (1970). 7. B.M.Powell, S. Jandl, J.L. Brebner & F. Levy, J. Phys. CI0, 3059 (1977). 8. A.V. Solodukhin, Izmeritelnaja technika 2, 69 (1975).
10.
1.
11. 12. 13. 14. 15.
1615
K.K. Mamedov, M.A. Aldjanov, I.G. Kerimov & M.I. Mekhtiev, Fis. T~erd. Tela 20, 42 (1978). T.H.K. Barron, J.G. Collins & G.K. White, Adv. Phys. 29, 609 (1980). S.Adachi & C. Hamaguchi, Solid State Commun. 31,245 (1979). D.P. Riley, Proc. Phys. Soc. 57,486 (1945). A. Polian, M. Grimsditch & M. Gatulle, J. Phys. Lett. 43,405 (1982). G.L. Belenkii, E.Yu. Salaev, R.A. Suleymanov & E.I. Mirzoev, Phys. Status Solidi (a) 63, 97 (1981). S.S. Ischenko, V.I. Konovalov, S.M. Okulov, G.L. Belenkii & E.Yu. Salaev, Fiz. T~erd. Tela 21,287 (1979).
0038-1098/82/481613-03 $03.00/0 Pergamon Press Ltd.
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS G.L. Belenkii, S.G. Abdullayeva, A.V. Solodukhin and R.A. Suleymanov (Institute of Physics, Academy of Sciences of Azerbaidjan SSR, Prospect Narimanova 33, 370143, Baku, USSR)
(Received 2 August 1982 by F. Bassani) Linear expansion coefficients parallel, ixll, and perpendicular, ix±, to layer plane of GaS, GaSe, and TIGaS2 are investigated in the temperature range 5 - 3 0 0 K. Negative values of ixll are observed in GaS and GaSe in the range of 3 0 - 5 0 K caused by negative Gruneisen parameters 'Yll in this interval. It is shown that negative ixll in layer crystal can be due to TA± acoustic branch contribution to ixllLAYER CRYSTALS consist of separate layers with strong (covalent) bonding between atoms within the layers. The bonding between the layers is weak, predominantly of Van der Waals type. Such crystal structure causes the specific shape of phonon branches in layer crystals [ 1]. The most important feature of acoustic waves determining the thermal properties of layer crystals at low termperatures is that acoustic waves propagating in the layer plane and polarized perpendicular to the layer plane (TA± mode) can have dispersion law of the form co = aK + BK 2, where k is the wave vector [2, 3]. These acoustic waves are called "bending waves" [3] (or "warping waves" using the terminology of [4]). IAfshits [3] was the first to point out the importance of "bending waves" describing thermal properties of layer and chain structures. According to [3 ] the specific heat of layer crystal is proportional to T 2 at low temperatures when the "bending waves" give the major contribution to the free energy. Note, that T 2 law may also be due to the contribution of acoustic waves with "ordinary" dispersion law, co "" k, if the interlayer interaction is neglected (the specific heat of two dimensional solid). The linear expansion coefficient in the direction parallel to the layer plane, ixll, has the negative value if "bending waves" give the major contribution to ixll. The negative value of Otllis due to "membrane effect" (or "tension effect" [4]), i.e. the increase of these frequencies with layers stretching [3]. The existence of "membrane effect" can be decisive in analysing the contribution of "bending waves" to thermal properties of layer crystals. "Bending waves" were first observed in graphite. The investigations of low energy neutron scattering in graphite [5] showed, that TA± mode has quadratic dispersion law. The linear expansion coefficient ixll has negative values in the wide temperature range 3 0 0 - 6 0 0 K [6], Fig. 2, and specific heat ~12. There are few investigations of ix in other layer crystals, and pecularities of a behaviour with temperature were
not observed, while the experimental investigations of neutron scattering in layer crystals indicate that there are TAt modes with quadratic dispersion law in some of them (for example, in GaS [7]). In this paper we present the results of investigations of linear expansion coefficients all and ix± in layer crystals GaS, GaSe and T1GaS2 at temperatures 5 - 3 0 0 K. The measurements were carried out by quartz dilatometer, described in [8]. The accuracy of measurements is approximately 0.I x 10-6K -1 in the whole temperature range. The lengths of the samples were 6 and 5 mm, respectively, measuring ixll and a± in GaS(GaSe) and 10 and 3 mm in T1GaS2. The experimental values for air and ix± for GaS, GaSe and T1GaS2 are given in Figs. 1-3. The most interesting detail is the ixll behaviour in GaS and GaSe, Figs. I and 2, at low temperatures: ixll is positive at T < 30K and T > 50K and negative in the temperature range 30 < T < 50K. At T = 10-20 K the averaged values of all in GaS are presented because of scattered character of experimental points, Fig. 1. The observed temperature behaviour of ixll in GaS and GaSe can be explained by the Lifshits theory [3]. The "bending waves" are assumed to give the major contribution to ixll in GaS(GaSe) in the temperature range 3 0 - 5 0 K. The investigations of specific heat in GaS(GaSe) in the same temperature interval showed that specific heat obeys T 2 law [9]. The positive values of ixll at T < 30 and T > 50K can be explained by the predominant contribution of phonons with "ordinary" dispersion law to ixll- This interpretation is based on the analysis of temperature behaviour of Gruneisen parameter, obtained from the relation [10]: V
ix, = ~ [ ( c ~
+ c12)ixl, + c~3ix,].
Using elastic constants, C~/~obtained at room tem. perature [7, 11 ] and specific heat, Co, measured in the wide temperature range T = 5 - 3 0 0 K [9] the ~'11 iS
1613
1614
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS
Vol. 44, No. 12
~.'I0,6 K -~
s
9 8
ZO
1
J .ss~
a 5 4 $ 2 I 0
I0
-' 40
80
~0
i ~lO
i ~00
,~0
~00
~.1(
Fig. 1. Temperature behaviour of linear expansion coefficients Oql and ~± in GaS.
,V.~ 2
Fig. 3. Temperature behaviour of linear expansion coefficients oql and ~± in T1GaS2.
x" xs
t
0
-!
2O
~ I g 0
x ~ "~4k•...~.
'
8o'
'
-,, g
Fig. 2. t~ll(T) dependence for GaS, GaSe and graphite. evaluated in the temperature range 5 - 3 0 0 K. It follows that ")'11has negative values at 30 < T < 50K. The ~± has the positive sign in the whole temperature range. From the definition of "/11 it follows that ~11 Eiq (-- a in ~iq/O In a), where a is the lattice parameter. Thus the sign of "/11 is determined by the sign of derivatives a6oi~/aa. If ')'11is negative then a~oiq/aa > o. The positive sign of acoiq/aa m e a n s that phonon frequencies increase while stretching the crystal along the layer plane. As it was already mentioned it is the property of "bending waves". In graphite ~'11 has negative values in the wide temperature range 3 0 - 6 0 0 K, i.e. in a considerably wider range than in GaS(GaSe). To explain this difference between graphite and GaS(GaSe) the following facts must be taken into account. Firstly, bonding between atoms within the layers is significantly stronger in graphite than in GaS(GaSe): Cla + C12 ~ 60 x 1011 dynecm -2 in graphite [12] and 21 x 1011 dynecm -2
(13 x 1011 dyne cm -2) in GaS(GaSe). The values of C33 are as follows: 2.0 x 1011 dyne cm -2 and 3.6 x l0 II (3.5 x 1011) dyne cm -2, respectively. Thus, the ratio (Cll + C12)/C3a is three times as great in graphite as in GaS(GaSe). Secondly, the layers in GaS(GaSe) have more complex crystal structure than in graphite. Each layer in GaS(GaSe) consists of four atomic sublayers S(Se)-Ga-Ga-S(Se), whereas in graphite layers are monoatomic. Thus it may be easier to "bend" the layer in graphite than in GaS(GaSe). The investigations of TA± mode dispersion law in GaS and in graphite show, that quadratic term in this law is more significant in graphite than in GaS (the ratio b/a in dispersion law 60 = aK + bK 2 is ~ 3 times larger in graphite [13]). If the above assumptions are true the increase of the number of atomic planes forming each layer, for example going from GaS (with 4 atomic planes, 8 atoms per unit cell) to TIGaS2 (with 7 atomic planes, 64 atoms per unit cell), leads to dispersion law for TA± branch having the "ordinary" form, ~o ~ k, at small values of k and as a consequence to temperature dependence of Otll determined by phonons with "ordinary" dispersion law (Fig. 3). It must be mentioned in conclusion, that temperature behaviour of linear expansion coefficients in layer crystals described above can lead to unusual behaviour of various parameters with temperature, such as values of direct and indirect gaps in GaS [14] or crystal field constants in GaSe [15]. These effects we hope to discuss in a separate paper.
Vol. 44, No. 12
PECULIARITIES OF THERMAL EXPANSION OF LAYERED CRYSTALS REFERENCES
9.
L.N. Alieva, G.L. Belenkii, I.I. Reshina, E.Yu. Salaev & V.Y. Shteinshriber, Fis. T~erd. Tela 21, 126 (1979). 2. K. Komatsu, J. Phys. Soc. Japan 10, 471 (1955). 3. I.M. Lifshits, JETP 22, 471 (1952). 4. T.H.K. Barron,J. Appl. Phys. 41, 5044 (1970). 5. K.K. Mani & R. Ramani, Phys. Status Solidi (b) 61,659 (1974). 6. A.C. Bailey & B. Yates,J. Appl. Phys. 41, 5088 (1970). 7. B.M.Powell, S. Jandl, J.L. Brebner & F. Levy, J. Phys. CI0, 3059 (1977). 8. A.V. Solodukhin, Izmeritelnaja technika 2, 69 (1975).
10.
1.
11. 12. 13. 14. 15.
1615
K.K. Mamedov, M.A. Aldjanov, I.G. Kerimov & M.I. Mekhtiev, Fis. T~erd. Tela 20, 42 (1978). T.H.K. Barron, J.G. Collins & G.K. White, Adv. Phys. 29, 609 (1980). S.Adachi & C. Hamaguchi, Solid State Commun. 31,245 (1979). D.P. Riley, Proc. Phys. Soc. 57,486 (1945). A. Polian, M. Grimsditch & M. Gatulle, J. Phys. Lett. 43,405 (1982). G.L. Belenkii, E.Yu. Salaev, R.A. Suleymanov & E.I. Mirzoev, Phys. Status Solidi (a) 63, 97 (1981). S.S. Ischenko, V.I. Konovalov, S.M. Okulov, G.L. Belenkii & E.Yu. Salaev, Fiz. T~erd. Tela 21,287 (1979).