- Email: [email protected]
The nature of negative linear expansion in layer crystals C, Bn, GaS, GaSe and InSe
Solid State Communications, Vol. 53, No. 11, pp. 967-971, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
THE NATURE OF NEGATIVE LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe G.L. Belenkii, E. Yu. Salaev, R.A. Suleimanov, N.A. Abdullaev and V.Ya. Shteinshraiber Institute of Physics, Academy of Sciences of Azerbaidjan SSR, prosp. Narimanova 33, Baku 370143, U.S.S.R.
(Received 30 August 1984 by F. BassanO Experimental investigations of thermal expansion parallel and perpendicular to the layers plane of layer crystals GaS, GaSe and InSe are described. The obtained results are analized together with known thermal expansion data on graphite, C, and boron nitride, BN. Theoretical calculations of linear expansion coefficients are carried out on the basis of the model of highly anisotropic crystal. It is shown, that the negative thermal expansion in the layer plane, typical of layer crystals, is due to "bending" waves, acoustic waves propagating in the layer plane and polarized perpendicular to this plane (TA± mode) 1. INTRODUCTION THE CHARACTERISTIC FEATURE of thermal expansion in layer crystals Graphite, C, and Boron Nitride, BN, is the negative sign of the linear expansion coefficient in the layer plane, otll , in the wide temperature range, 30-600K [1,2], Fig. 1. It will be shown (see below) that the negative values of Otll are observed in layer crystals GaS, GaSe and InSe as well. The nature of n e g a t i v e oql in C and BN was explained in [3]. It is supposed, that considerable expansion in the direction perpendicular to the layer plane leads to contraction of layers. In 1952 the theory of thermal properties of highly anisotropic crystals was presented by Lifshits [4]. This theory predicted the negative thermal expansion in the layer plane due to "bending" waves, acoustic waves in highly anisotropic crystal, propagating in the layer plane and having polarisation vector perpendicular to this plane. This TA± mode was observed in layer crystals, however, no attempts were made to explain the negative art in layer crystals using the idea of "bending" waves. This paper presents the results of experimental investigations of thermal expansion in layer crystals GaS, GaSe and InSe. The obtained data are analised together with the known thermal expansion data on C and BN. It is shown, that the thermal expansion of layer crystals may be understood on the basis of Lifshits theory. 2. ELASTIC PROPERTIES AND PHONON SPECTRA OF LAYER CRYSTALS
crystal to crystal. Graphite and boron nitride have the most simple monoatomic layer structure. The layers stack to each other forming three-dimensional crystal structure with two layers in the unit cell. Semiconductors of AaB~ group GaS, GaSe, InSe etc., have a more complex, four-fold structure of a layer. The unit cell of GaS contain atoms from two layers, and that in GaSe and InSe include two or more layers. In Table 1, the elastic constants of some layer crystals having hexagonal structure are given. The elastic constants CH and C~2 charactedse the intralayer bonding, whereas C13, Caa and C~ are the "interlayer" elastic constants. In highly anisotropic hexagonal crystal CIa, Caa, C44 "¢ C11, C12. As it is seen from Table 1, graphite is the most anisotropic crystal with Ctl/C44 = 250. In PBI2 this ratio is equal to 4. The layer structure causes specific phonon spectra: low velocities of acoustic waves, low lying optical modes with weak dispersion in Kz direction in the Brillouin zone. There is a special dispersion law for TA1 acoustic wave: 60 = o/j +/3/j 2 , where G-reduced wave vector, a ~ x / ' ~ and/3 ~ x / ~ H • The ratio [3/ais the characteristic of anisotropy and is equal to 10 in graphite, 2 in GaS and 0 in GaSe and PbI2. Dispersion laws for acoustic waves in highly anisetropic hexagonal crystal were obtained in [4]. Assumhag, that Ca3, C44 '~ CH, C~2 the following dispersion relations may be written using the well known expressions of elasticity theory [8] : Transverse in-plane mode:
Crystals with layer structure are eharacterised by strong (covalent) bonding within the layers and weak (probably, of van-der-Waals type) bonding between the layers. The structure of individual layer differs from
In plane mode:
967
968
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe crystal can be calculated from:
Table 1. Elastic constants o f some layer crystals, 1011 dyn cm -2 .
Cu C Gas GaSe InSe PbI2
100 15.5 10.3 7.3 2.0
C~2 20 3.5 2.9 2.7 0.6
C13 1.5 1.3 1.2 3.0 0.18
Ca3 3.6 3.8 3.4 3.6 1.46
Vol. 53, No. 11
C,~
aj. = a T "
0.4 1.0 0.9 1.2 0.54
[5] [6] [6] [6] [7]
•
exp ( h 6 0 i ( k ) / k r ) - 1d3k k
(3') 0 { 1 all = -- 0~T" ~
(ff b,B60i(k)/Op } ~ J ! J e x p ( h 6 0 i ( k ) / k T ) - 1 dak
(3") Considering a highly anisotropic crystal and the contribution of acoustic waves only, equations (3') and (3") take the form:
Out-of-plane mode; TA l mode, when Kz = 0.: 2 2 +C2K2~ 2 + 2 = ¢2 (K~ + K~)11
2+
22
(1) In (1) C~ = (Czl -- Ct2)12p, C~ = C h i p , C 2 = 2 2 2 2czc2/(cl + c]), 712 = C ~ / p c 2 , ~2 = C33/pc 2 ' 7 = cav/Tr, p-density, v < 1, a-lattice parameter in the basal plane. The parameters r/and ~ are much smaller, than 1, due to high anisotropy of lattice. Quadratic dispersion law for 600 branch is obtained when bending forces are taken into account [9]. The parameter 7 characterises the bending rigidity of a layer and is determined by intralayer elastic constants. The acoustic wave 603 was called "bending" wave, because its propagation along the layers causes their bending [4]. For the first time the important role of "bending" waves was emphasized by Lifshits [4]. It can be shown, that the frequency distribution function g(60) corresponding to acoustic waves 6ol, 6o2 and coa has the form (as an example, for 3' = 0): 26°2
g(60) = (21r)2"
{~ - - ~c
1 _~ca ) + r/c~---~ + "
(2)
The third term in equation (2) is the contribution of "bending" mode 603. Because of high anisotropy, ~, < 1, the value of this contribution is much greater, than that of 601 and 602 modes. Thus, considering the thermal properties of highly anisotropic crystals, only the role of "bending" waves can be taken into account. It is seen from Table 1, that only graphite (and, probably, BN) can be considered as a highly anisotropic crystal. The crystals of A3B6 group do not have such a high anisotropy, and the contribution of acoustic waves 60z and 602 may be important in thermal properties of these crystals. 3. THEORY OF THERMAL EXPANSION OF A HIGHLY ANISOTROPIC CRYSTAL According to the well known thermodynamic relations, thermal expansion coefficients in uniaxial
c~±-
OT
k
•
h
+
kI°C330o +
.2-2 072 "To "o
2p603(k)" [exp (h60a(k)/kT)-- 1]
d3k} (4 ')
ho. a60]. d3e c~ll = -- ~
2p603 (k) " - ~ x - ~ o - ~ l / k r ) - -
1] "
k
(4") In (4') and (4") p-is the uniform areal pressure in the basal plane, o-uniaxial pressure, perpendicular to the basal plane. Besides the dispersion law for 6o3 mode, the derivatives a603/00 and 0603/0p a r e needed to calculate ai and Otll.The only available data are the results of experiments in which the influence of hydrostatic pressure on elastic constants of C, BN, GaS, GaSe and InSe was studied [10, 6]. Some simplification of integrals 4' and 4" is possible. It is shown in [10], that 0C33/OPhyd...~ 10, whereas OC44/OPhyd in graphite ~ 0.02, so one can neglect the first term in the numerator in 4'. Strong anisotropy of crystals allows to suggest, that the influence of interlayer distances on the value of intralayer elastic constant is insignificant. So the third term in the numerator in 4' can also be neglected. Thus, to calculate cxx only two fitting parameters 0Cs3/0o and v are needed. Calculating (0603/Op) the following main fact must be taken into account. The forces, which determine the frequency of out-of-plane vibrations in such a "membrane" change significantly in comparison with unstrained plane. New forces must be taken into account which are due to stretching of the layer under
Vol. 53, No. 11
p-pressure. The dispersion law for w3 mode in “membrane” has the form: o: = p/p&i + k:) + C2t2kz [4,8]. So &.&ap = k; + kc/p, and one can retain only one term in numerator 4”, and only one fitting parameter v is necessary to calculate ~~11. The frequency of “bending” wave increases, when p-pressure stretches the plane, so the sign of derivate is positive. Thus, the “bending” waves lead to negative thermal expansion in the layer plane, (~11[ 141. If anisotropy of the crystal is not so high, the contribution of w1 and w2 waves must be taken into account. In this case the number of fitting parameters is increased. 4. EXPERIMENT
969
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, Gas, GaSe AND InSe
AND COMPARISON WITH THEORY
The linear expansion coefficients of C and BN measured in [ 1,2] in 30-700 K temperature region, Fig. 1. As it is seen from Fig. 1, both C and BN have large positive values of ol at 100 K and small negative (~11in a wide temperature range; (~11 becomes positive in Cat T-600K. The linear expansion coefficients cul and oil for Gas, GaSe and InSe were measured by using two methods, dilatometric and interferometric. The accuracy is lo-’ K-’ in dilatometric measurements [l l] and 51.10~’ K-’ in interferometric ones. The results are presented in Figs. 2-4. The temperature behaviour of ell in all crystals has the same character: all is negative in the narrow temperature interval, 30-50 K; at 50 K cyll
becomes positive. The low temperature measurements showed, that (~11reaches zero from the positive side; below 30 K (~1~ has the positive sign. InSe has the largest values of cul and (~11at high temperatures; then come GaSe and Gas. It turned out, that the theory of thermal expansion in a highly anisotropic crystal allows to explain experimental data obtained for C, BN, GaS, GaSe and InSe. In the case of C and BN using the values of fitting parameters v = 0.47(C), v = 0.35(BN) and (a&/au) = 16(for C and BN), al(T) and q(T) curves were calculated from 4’ and 4” on the basis of the above made assumptions. As it is seen from Fig. 1, theory is in good agreement with experiment. Thus, in accordance with theory [4] the negative value of crll in C and BN is due to “bending” waves. As it is shown above, the small anisotropy of Gas, GaSe and InSe leads to the necessity to take into account the contribution of w I and w2 modes. Using the same simplifications (i.e. neglecting the influence of intralayer distances on interlayer elastic constants and vice versa) as it was done before, the q(T) dependence for GaS was calculated from: h(k; + k;)d3k [ exp (hc+)/kT)
- l]
ac,, h*(k; + k;).----+1J-f2pw2 [exp (hw,/k~
’
- ljdjk
(5)
The sign of derivative aC,r /ap is negative because the stretching of the layers leads to decreasing of intralayer constants, so o2 leads to positive contribution in (err. Using the values of fitting parameters v = 0.6 and aCr, /ap = 16 good agreement between experiment and theory is achieved, Fig. 5. Thus, the negative value of all in GaS is again due to “bending” waves, but contrary to C and BN the role of w2 (and wl) modes becomes significant at relatively low temperatures (- 50 K, in comparison with 600 K in C). In fact, GaS (GaSe, InSe) has the interlayer elastic constants differ IO
GoS ? Y
. “a
a -I
T (K)
0 -I
Fig. 1. Thermal expansion coefficients for graphite and boron nitride [ 1,2] . Solid lines - theoretical calculations.
. .
.
.
AA
5
x
.
. .* I-.
.
A
l
..A/
I
I
100
I50
T(K)
Fig. 2. Thermal expansion
coefficients
for Gas.
I 200
970
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaS• AND InSe • ,
•
• "a~
• •
GaS•
•
Vol. 53, No. 11
can have another form if a~3/ap(o) are represented in the form: i)~3/Op(o) = O~3/Oa'Oa/Op(a) + ~6oa/ac" ac/ap((r) [12]:
,..¢
,'o_
5
•
•
•
•
au =
V[D
9'11--
3'. ; Cv-heat capacity, V-volume
•
I I00
%__• • •
0
•
I 2oo
i 150
T (K)
D
9"1"-- ---D-"9"11 ; D = ( C n + C t 2 ) 6'33 -- 2C]3
Fig. 3. Thermal expansion coefficients for GaSe. • • • • Aa± 15
dnSe
IC •
(111
I I00
o
I
| 2 O0
150
TIK)
Fig. 4. Thermal expansion coefficients for InSe.
GaS .v=0.6, d C J d p :16
]~
T ~e"
v
x
I
20
I
443
I
6O
I
80
I
I00
120
T (K)
Fig. 5. Linear expansion coefficient a n for GaS. Solid line - theoretical calculation. slightly from those in C, whereas the intralayer constants, which determine the contribution of co~ and o~2 modes in GaS (GaSe, INS•) are 10 times lower, than in in C. 5. DISCUSSION In [3] the negative all in C and BN is explained by the large value of a±. But this explanation is questionable. The formulas (3) for computation of a I and all
(6')
(6")
In (6) 3'11 = -- Xtk 0 In 6oih/a In a. f(~ik, T), 3'± = -- Zi,~ a In ~ i k / 0 In c. f(cotk, T); a, c- parameters of unit cell. 3ql,.L-WeU-known Grunaisen parameters. It is assumed in [3], that the whole negative value of all is due to the second term in (6'). The obtained value of elastic constant C13 in this case appears to be too large, 5.10 n dyn cm 2 (compare with C13 from Table 1). The values of two terms in (6) can be evaluated experimentally. For example, at T = 100 K all = (-- 0.9 -- 0.2). 10 -6 k - l . a.t = (17 + 0.8)" 10 -6 k -1 . So the negative of all in C is due primarily to the first term in equation (6'). The assumptions, that interlayer elastic constants depend slowly on intralayer distances and vice versa mean, that the second terms in (6) is negligible, and all is determined by the first term with negative Gruneisen p a r a m e t e r 9'11.The results of evaluation of two terms in equation (6) justify our assumption.~ Theory [4], discussed above, takes into account only acoustic waves. As it was mentioned above, lowlying optical modes are characteristic for layer crystals. The problem arises how to take into account these modes. It is necessary to bear in mind, that the unit cell of C, BN and GaS contains two layers. Integrating in equation (4) a value of kz = r¢/C, where Cparameter of unit cell in z direction, must be taken as the upper limit. This value of kzm,= leads to values of a± and all smaller, than obtained experimentally. To fit theory to experimental data the value o f k z m ~ = n/d was taken, where d-interlayer distance. This way allows to take into account out-of-plane optical modes.
6. CONCLUSION The "bending" waves are the main peculiarities of acoustic spectra of layer crystals. Their frequencies increases when pressure stretches the layers plane. In the framework of harmonic approximation this fact leads to negative linear expansion in the layer plane when "bending" waves play the main role in thermal properties of layer crystal.
Vol. 53, No. 11
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe REFERENCES
1. 2. 3. 4. 5, 6.
A.C. Bailey & B. Yates, Z Appl. Phys. 41,5088 (1970). B. Yates, M.S. Overy & O. Pirgon, Phil Mag. 32, 847 (1975). B.T. KeUy,PhilMag. 32, 859 (1975). I.M. Lifshits, JETP, 22,475, (1952). O.L Blakslee, P.G. Proctor, E.J. Seldin, G.B. Spence & T. Weng, J. Appl. Phys. 41, 3373 (1970). M. Gatulle, M. Fisher & A. Chevy,Phys. Stat. Sol. 119, 327 (1983).
971
W.M.Sears, M.L. Klein & J.A. Morrison,Phys. Rev. 19, 2305 (1979). 8. L.D. Landau * E.M. Lifshits, Theory of elasticity, Nauka, (1965). 9. A.M.Kosevich,Physical mechaniks of real crystals, Naukova dumka, (1981). 10. W.B. Gauster & I.J. Fritz, J. Appl. Phys. 45,3309 (1974) 11. G.L. Belenkii, S.G. Abdullaeva, A.V. Solodukhin & R.A. Suleimanov, Solid State Commun. 44, 1613 (1982). 12. T.H.K. Barron, J.C. Collins & G.K. White, Adv. Phys. 29,609 (1980). 7.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
THE NATURE OF NEGATIVE LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe G.L. Belenkii, E. Yu. Salaev, R.A. Suleimanov, N.A. Abdullaev and V.Ya. Shteinshraiber Institute of Physics, Academy of Sciences of Azerbaidjan SSR, prosp. Narimanova 33, Baku 370143, U.S.S.R.
(Received 30 August 1984 by F. BassanO Experimental investigations of thermal expansion parallel and perpendicular to the layers plane of layer crystals GaS, GaSe and InSe are described. The obtained results are analized together with known thermal expansion data on graphite, C, and boron nitride, BN. Theoretical calculations of linear expansion coefficients are carried out on the basis of the model of highly anisotropic crystal. It is shown, that the negative thermal expansion in the layer plane, typical of layer crystals, is due to "bending" waves, acoustic waves propagating in the layer plane and polarized perpendicular to this plane (TA± mode) 1. INTRODUCTION THE CHARACTERISTIC FEATURE of thermal expansion in layer crystals Graphite, C, and Boron Nitride, BN, is the negative sign of the linear expansion coefficient in the layer plane, otll , in the wide temperature range, 30-600K [1,2], Fig. 1. It will be shown (see below) that the negative values of Otll are observed in layer crystals GaS, GaSe and InSe as well. The nature of n e g a t i v e oql in C and BN was explained in [3]. It is supposed, that considerable expansion in the direction perpendicular to the layer plane leads to contraction of layers. In 1952 the theory of thermal properties of highly anisotropic crystals was presented by Lifshits [4]. This theory predicted the negative thermal expansion in the layer plane due to "bending" waves, acoustic waves in highly anisotropic crystal, propagating in the layer plane and having polarisation vector perpendicular to this plane. This TA± mode was observed in layer crystals, however, no attempts were made to explain the negative art in layer crystals using the idea of "bending" waves. This paper presents the results of experimental investigations of thermal expansion in layer crystals GaS, GaSe and InSe. The obtained data are analised together with the known thermal expansion data on C and BN. It is shown, that the thermal expansion of layer crystals may be understood on the basis of Lifshits theory. 2. ELASTIC PROPERTIES AND PHONON SPECTRA OF LAYER CRYSTALS
crystal to crystal. Graphite and boron nitride have the most simple monoatomic layer structure. The layers stack to each other forming three-dimensional crystal structure with two layers in the unit cell. Semiconductors of AaB~ group GaS, GaSe, InSe etc., have a more complex, four-fold structure of a layer. The unit cell of GaS contain atoms from two layers, and that in GaSe and InSe include two or more layers. In Table 1, the elastic constants of some layer crystals having hexagonal structure are given. The elastic constants CH and C~2 charactedse the intralayer bonding, whereas C13, Caa and C~ are the "interlayer" elastic constants. In highly anisotropic hexagonal crystal CIa, Caa, C44 "¢ C11, C12. As it is seen from Table 1, graphite is the most anisotropic crystal with Ctl/C44 = 250. In PBI2 this ratio is equal to 4. The layer structure causes specific phonon spectra: low velocities of acoustic waves, low lying optical modes with weak dispersion in Kz direction in the Brillouin zone. There is a special dispersion law for TA1 acoustic wave: 60 = o/j +/3/j 2 , where G-reduced wave vector, a ~ x / ' ~ and/3 ~ x / ~ H • The ratio [3/ais the characteristic of anisotropy and is equal to 10 in graphite, 2 in GaS and 0 in GaSe and PbI2. Dispersion laws for acoustic waves in highly anisetropic hexagonal crystal were obtained in [4]. Assumhag, that Ca3, C44 '~ CH, C~2 the following dispersion relations may be written using the well known expressions of elasticity theory [8] : Transverse in-plane mode:
Crystals with layer structure are eharacterised by strong (covalent) bonding within the layers and weak (probably, of van-der-Waals type) bonding between the layers. The structure of individual layer differs from
In plane mode:
967
968
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe crystal can be calculated from:
Table 1. Elastic constants o f some layer crystals, 1011 dyn cm -2 .
Cu C Gas GaSe InSe PbI2
100 15.5 10.3 7.3 2.0
C~2 20 3.5 2.9 2.7 0.6
C13 1.5 1.3 1.2 3.0 0.18
Ca3 3.6 3.8 3.4 3.6 1.46
Vol. 53, No. 11
C,~
aj. = a T "
0.4 1.0 0.9 1.2 0.54
[5] [6] [6] [6] [7]
•
exp ( h 6 0 i ( k ) / k r ) - 1d3k k
(3') 0 { 1 all = -- 0~T" ~
(ff b,B60i(k)/Op } ~ J ! J e x p ( h 6 0 i ( k ) / k T ) - 1 dak
(3") Considering a highly anisotropic crystal and the contribution of acoustic waves only, equations (3') and (3") take the form:
Out-of-plane mode; TA l mode, when Kz = 0.: 2 2 +C2K2~ 2 + 2 = ¢2 (K~ + K~)11
2+
22
(1) In (1) C~ = (Czl -- Ct2)12p, C~ = C h i p , C 2 = 2 2 2 2czc2/(cl + c]), 712 = C ~ / p c 2 , ~2 = C33/pc 2 ' 7 = cav/Tr, p-density, v < 1, a-lattice parameter in the basal plane. The parameters r/and ~ are much smaller, than 1, due to high anisotropy of lattice. Quadratic dispersion law for 600 branch is obtained when bending forces are taken into account [9]. The parameter 7 characterises the bending rigidity of a layer and is determined by intralayer elastic constants. The acoustic wave 603 was called "bending" wave, because its propagation along the layers causes their bending [4]. For the first time the important role of "bending" waves was emphasized by Lifshits [4]. It can be shown, that the frequency distribution function g(60) corresponding to acoustic waves 6ol, 6o2 and coa has the form (as an example, for 3' = 0): 26°2
g(60) = (21r)2"
{~ - - ~c
1 _~ca ) + r/c~---~ + "
(2)
The third term in equation (2) is the contribution of "bending" mode 603. Because of high anisotropy, ~, < 1, the value of this contribution is much greater, than that of 601 and 602 modes. Thus, considering the thermal properties of highly anisotropic crystals, only the role of "bending" waves can be taken into account. It is seen from Table 1, that only graphite (and, probably, BN) can be considered as a highly anisotropic crystal. The crystals of A3B6 group do not have such a high anisotropy, and the contribution of acoustic waves 60z and 602 may be important in thermal properties of these crystals. 3. THEORY OF THERMAL EXPANSION OF A HIGHLY ANISOTROPIC CRYSTAL According to the well known thermodynamic relations, thermal expansion coefficients in uniaxial
c~±-
OT
k
•
h
+
kI°C330o +
.2-2 072 "To "o
2p603(k)" [exp (h60a(k)/kT)-- 1]
d3k} (4 ')
ho. a60]. d3e c~ll = -- ~
2p603 (k) " - ~ x - ~ o - ~ l / k r ) - -
1] "
k
(4") In (4') and (4") p-is the uniform areal pressure in the basal plane, o-uniaxial pressure, perpendicular to the basal plane. Besides the dispersion law for 6o3 mode, the derivatives a603/00 and 0603/0p a r e needed to calculate ai and Otll.The only available data are the results of experiments in which the influence of hydrostatic pressure on elastic constants of C, BN, GaS, GaSe and InSe was studied [10, 6]. Some simplification of integrals 4' and 4" is possible. It is shown in [10], that 0C33/OPhyd...~ 10, whereas OC44/OPhyd in graphite ~ 0.02, so one can neglect the first term in the numerator in 4'. Strong anisotropy of crystals allows to suggest, that the influence of interlayer distances on the value of intralayer elastic constant is insignificant. So the third term in the numerator in 4' can also be neglected. Thus, to calculate cxx only two fitting parameters 0Cs3/0o and v are needed. Calculating (0603/Op) the following main fact must be taken into account. The forces, which determine the frequency of out-of-plane vibrations in such a "membrane" change significantly in comparison with unstrained plane. New forces must be taken into account which are due to stretching of the layer under
Vol. 53, No. 11
p-pressure. The dispersion law for w3 mode in “membrane” has the form: o: = p/p&i + k:) + C2t2kz [4,8]. So &.&ap = k; + kc/p, and one can retain only one term in numerator 4”, and only one fitting parameter v is necessary to calculate ~~11. The frequency of “bending” wave increases, when p-pressure stretches the plane, so the sign of derivate is positive. Thus, the “bending” waves lead to negative thermal expansion in the layer plane, (~11[ 141. If anisotropy of the crystal is not so high, the contribution of w1 and w2 waves must be taken into account. In this case the number of fitting parameters is increased. 4. EXPERIMENT
969
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, Gas, GaSe AND InSe
AND COMPARISON WITH THEORY
The linear expansion coefficients of C and BN measured in [ 1,2] in 30-700 K temperature region, Fig. 1. As it is seen from Fig. 1, both C and BN have large positive values of ol at 100 K and small negative (~11in a wide temperature range; (~11 becomes positive in Cat T-600K. The linear expansion coefficients cul and oil for Gas, GaSe and InSe were measured by using two methods, dilatometric and interferometric. The accuracy is lo-’ K-’ in dilatometric measurements [l l] and 51.10~’ K-’ in interferometric ones. The results are presented in Figs. 2-4. The temperature behaviour of ell in all crystals has the same character: all is negative in the narrow temperature interval, 30-50 K; at 50 K cyll
becomes positive. The low temperature measurements showed, that (~11reaches zero from the positive side; below 30 K (~1~ has the positive sign. InSe has the largest values of cul and (~11at high temperatures; then come GaSe and Gas. It turned out, that the theory of thermal expansion in a highly anisotropic crystal allows to explain experimental data obtained for C, BN, GaS, GaSe and InSe. In the case of C and BN using the values of fitting parameters v = 0.47(C), v = 0.35(BN) and (a&/au) = 16(for C and BN), al(T) and q(T) curves were calculated from 4’ and 4” on the basis of the above made assumptions. As it is seen from Fig. 1, theory is in good agreement with experiment. Thus, in accordance with theory [4] the negative value of crll in C and BN is due to “bending” waves. As it is shown above, the small anisotropy of Gas, GaSe and InSe leads to the necessity to take into account the contribution of w I and w2 modes. Using the same simplifications (i.e. neglecting the influence of intralayer distances on interlayer elastic constants and vice versa) as it was done before, the q(T) dependence for GaS was calculated from: h(k; + k;)d3k [ exp (hc+)/kT)
- l]
ac,, h*(k; + k;).----+1J-f2pw2 [exp (hw,/k~
’
- ljdjk
(5)
The sign of derivative aC,r /ap is negative because the stretching of the layers leads to decreasing of intralayer constants, so o2 leads to positive contribution in (err. Using the values of fitting parameters v = 0.6 and aCr, /ap = 16 good agreement between experiment and theory is achieved, Fig. 5. Thus, the negative value of all in GaS is again due to “bending” waves, but contrary to C and BN the role of w2 (and wl) modes becomes significant at relatively low temperatures (- 50 K, in comparison with 600 K in C). In fact, GaS (GaSe, InSe) has the interlayer elastic constants differ IO
GoS ? Y
. “a
a -I
T (K)
0 -I
Fig. 1. Thermal expansion coefficients for graphite and boron nitride [ 1,2] . Solid lines - theoretical calculations.
. .
.
.
AA
5
x
.
. .* I-.
.
A
l
..A/
I
I
100
I50
T(K)
Fig. 2. Thermal expansion
coefficients
for Gas.
I 200
970
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaS• AND InSe • ,
•
• "a~
• •
GaS•
•
Vol. 53, No. 11
can have another form if a~3/ap(o) are represented in the form: i)~3/Op(o) = O~3/Oa'Oa/Op(a) + ~6oa/ac" ac/ap((r) [12]:
,..¢
,'o_
5
•
•
•
•
au =
V[D
9'11--
3'. ; Cv-heat capacity, V-volume
•
I I00
%__• • •
0
•
I 2oo
i 150
T (K)
D
9"1"-- ---D-"9"11 ; D = ( C n + C t 2 ) 6'33 -- 2C]3
Fig. 3. Thermal expansion coefficients for GaSe. • • • • Aa± 15
dnSe
IC •
(111
I I00
o
I
| 2 O0
150
TIK)
Fig. 4. Thermal expansion coefficients for InSe.
GaS .v=0.6, d C J d p :16
]~
T ~e"
v
x
I
20
I
443
I
6O
I
80
I
I00
120
T (K)
Fig. 5. Linear expansion coefficient a n for GaS. Solid line - theoretical calculation. slightly from those in C, whereas the intralayer constants, which determine the contribution of co~ and o~2 modes in GaS (GaSe, INS•) are 10 times lower, than in in C. 5. DISCUSSION In [3] the negative all in C and BN is explained by the large value of a±. But this explanation is questionable. The formulas (3) for computation of a I and all
(6')
(6")
In (6) 3'11 = -- Xtk 0 In 6oih/a In a. f(~ik, T), 3'± = -- Zi,~ a In ~ i k / 0 In c. f(cotk, T); a, c- parameters of unit cell. 3ql,.L-WeU-known Grunaisen parameters. It is assumed in [3], that the whole negative value of all is due to the second term in (6'). The obtained value of elastic constant C13 in this case appears to be too large, 5.10 n dyn cm 2 (compare with C13 from Table 1). The values of two terms in (6) can be evaluated experimentally. For example, at T = 100 K all = (-- 0.9 -- 0.2). 10 -6 k - l . a.t = (17 + 0.8)" 10 -6 k -1 . So the negative of all in C is due primarily to the first term in equation (6'). The assumptions, that interlayer elastic constants depend slowly on intralayer distances and vice versa mean, that the second terms in (6) is negligible, and all is determined by the first term with negative Gruneisen p a r a m e t e r 9'11.The results of evaluation of two terms in equation (6) justify our assumption.~ Theory [4], discussed above, takes into account only acoustic waves. As it was mentioned above, lowlying optical modes are characteristic for layer crystals. The problem arises how to take into account these modes. It is necessary to bear in mind, that the unit cell of C, BN and GaS contains two layers. Integrating in equation (4) a value of kz = r¢/C, where Cparameter of unit cell in z direction, must be taken as the upper limit. This value of kzm,= leads to values of a± and all smaller, than obtained experimentally. To fit theory to experimental data the value o f k z m ~ = n/d was taken, where d-interlayer distance. This way allows to take into account out-of-plane optical modes.
6. CONCLUSION The "bending" waves are the main peculiarities of acoustic spectra of layer crystals. Their frequencies increases when pressure stretches the layers plane. In the framework of harmonic approximation this fact leads to negative linear expansion in the layer plane when "bending" waves play the main role in thermal properties of layer crystal.
Vol. 53, No. 11
LINEAR EXPANSION IN LAYER CRYSTALS C, BN, GaS, GaSe AND InSe REFERENCES
1. 2. 3. 4. 5, 6.
A.C. Bailey & B. Yates, Z Appl. Phys. 41,5088 (1970). B. Yates, M.S. Overy & O. Pirgon, Phil Mag. 32, 847 (1975). B.T. KeUy,PhilMag. 32, 859 (1975). I.M. Lifshits, JETP, 22,475, (1952). O.L Blakslee, P.G. Proctor, E.J. Seldin, G.B. Spence & T. Weng, J. Appl. Phys. 41, 3373 (1970). M. Gatulle, M. Fisher & A. Chevy,Phys. Stat. Sol. 119, 327 (1983).
971
W.M.Sears, M.L. Klein & J.A. Morrison,Phys. Rev. 19, 2305 (1979). 8. L.D. Landau * E.M. Lifshits, Theory of elasticity, Nauka, (1965). 9. A.M.Kosevich,Physical mechaniks of real crystals, Naukova dumka, (1981). 10. W.B. Gauster & I.J. Fritz, J. Appl. Phys. 45,3309 (1974) 11. G.L. Belenkii, S.G. Abdullaeva, A.V. Solodukhin & R.A. Suleimanov, Solid State Commun. 44, 1613 (1982). 12. T.H.K. Barron, J.C. Collins & G.K. White, Adv. Phys. 29,609 (1980). 7.