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Temperature dependent acoustical characterization of alkaline earth monochalcogenides in B1 and B2 phase
ARTICLE IN PRESS Physica B 405 (2010) 77–84
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Temperature dependent acoustical characterization of alkaline earth monochalcogenides in B1 and B2 phase Rishi Pal Singh, Manish Pratap Singh, Prakash Chandra Srivastava, Rajendra Kumar Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India
a r t i c l e in fo
abstract
Article history: Received 28 January 2009 Received in revised form 2 August 2009 Accepted 5 August 2009
The Born model of ionic solids has been applied to study the temperature dependence of acoustical dissipation due to phonon–phonon interaction, thermo-elastic mechanism and dislocation damping due to screw and edge dislocations in alkaline earth chalcogenides viz. calcium and barium chalcogenides [CaX and BaX; X ¼ S, Se and Te] in FCC as well as BCC phases, in the wide temperature range (50–500 K); for longitudinal and shear modes of propagation along /1 0 0S, /11 0S and /111S directions and for shear modes with different polarizations. Electrostatic and short-range repulsive potentials have been used for evaluating second and third order elastic constants at required temperature, which in turn have been used to obtain acoustical dissipation and related parameters at different temperatures. Hardness parameter, lattice constants and experimental values of second order elastic constants have been used as input parameters. Results have been discussed and temperature variation of acoustical dissipation has been found to exhibit exponential behaviour. & 2009 Elsevier B.V. All rights reserved.
PACS: 62.20Dc 62.65+K 43.35Cg Keywords: Alkaline earth monochalcogenides Elastic constant Phonon–phonon interaction Phonon viscosity Nonlinearity coupling constants
1. Introduction Alkaline earth monochalcogenides form very important closed shell ionic systems crystallizing in NaCl type (B1) structure with six fold coordination and CsCl (B2) type structure with eight fold coordination at ambient conditions [1–3]. There is no d electron in the valence band. Alkaline earth chalcogenides are currently under intense investigations driven by their applications in light emitting diodes (LEDs) and laser diodes (LDs). It is expected that these compounds may provide new II–IV candidates for the fabrication of various electrical and optical devices. Among the wide band gap II–IV semiconductors, calcium chalcogenides CaX and barium chalcogenides BaX (X ¼ S, Se and Te) and their alloys are interesting in connection with optoelectronic applications in blue light wavelength regime. High temperature investigations of alkaline earth chalcogenides have recently attracted interest due to their technological usefulness and interesting physical properties. These compounds are technologically important materials having many applications ranging from catalysis to microelec-
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tronics. They also have applications in the area of luminescent devices. Recently, structural, elastic, and electronic properties of these chalcogenides have been investigated extensively [1–5], but, acoustical studies have not been made in these chalcogenides. So, we have chosen this important class of compounds for studying temperature variation of acoustical dissipation. Acoustic ¨ attenuation, phonon viscosity, Gruneisen parameters, nonlinearity parameters and thermal relaxation time, which are of immense importance to explain the microstructure and related physical properties of these compounds, have been obtained over a wide temperature range. Experimental studies of FCC and BCC type compounds reveal that dislocation mechanism yields a negligible contribution to acoustic attenuation in well annealed NaCl (B1) type and CsCl (B2) type crystals and contribution to thermoelastic loss too is negligibly small [6]. Thus the alkaline earth chalcogenides are ideally suited for the study of acoustic attenuation due to phonon–phonon (p–p) interaction mechanism [7]. In the present investigation, using simple potentials (electrostatic and repulsive potentials), second and third order elastic constants (SOEM and TOEM) have been obtained at different temperatures, which in turn have been used to obtain acoustical dissipation and related parameters. The calculation of temperature dependence of acoustical behaviour have been presented for longitudinal and
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shear waves along different directions of propagation viz. [1 0 0], [11 0] and [111] in the temperature range 50–500 K for CaX and BaX (X ¼ S, Se and Te) in FCC and BCC phases in the Akhiezer regime [8] using Mason’s approach [9–12].
Table 3 ¨ Square average Gruneisen parameter for longitudinal wave /gj2 SL , average, square i ¨ Gruneisen parameter for longitudinal wave /gji S2L j i
/g
S2S
and shear wave /gji S2S ,
.
Compounds
SL /gj2 i
/gji S2L
/gji S2S
/gji S2S
DL
DS
DS
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
14.31 12.78 1.45 1.62
1.84 2.37 0.03 0.13
0.08 0.40 0.14 1.04
– 31.73 – 2.74
119.74 103.36 13.10 14.55
0.75 3.60 0.90 9.43
– 285.57 – 24.67
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
13.36 11.90 1.19 1.32
1.77 2.29 0.03 0.07
0.08 0.33 0.05 0.48
– 29.56 – 2.39
112.51 97.19 10.57 11.67
0.75 2.99 0.53 4.32
– 266.04 – 21.53
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
12.60 11.21 1.50 1.89
1.71 2.20 1.13 1.56
0.08 0.29 0.26 3.74
– 27.89 – 1.41
106.85 92.38 9.33 11.16
0.74 2.60 2.38 33.69
– 2.51 – 12.75
2. Theoretical approach CaS B1
2.1. Theory of elastic moduli
B2
According to Brugger’s definition [13], nth order elastic constant is defined as Cijklmn...... ¼ ð@n u=@eij @ekl @emn . . . . . . . . .Þ
ð1Þ
where u is the crystal free-energy density and eij is strain tensor. 0 ) Second and third order elastic constants at 0 K (Cij0 and Cijk have been obtained using electrostatic and Born–Mayer potentials and taking interactions up to next nearest-neighbours. Temperature variation of second and third order elastic constants has been Table 1 Second order elastic constants (1012 dynes/cm2 or 102 GPa) of calcium chalcogenides in B1 and B2 phase at 300 K. Compounds
B1 phase
B2 phase
C11
C12
C44
C11
C12
C44
CaS Present TB theorya GGAa LDAa
2.02 2.02 1.08 1.22
0.60 0.73 0.32 0.39
0.65 0.67 0.36 0.41
1.13 – 1.13 1.24
0.39 – 0.34 0.53
0.33 – 0.39 0.55
CaSe Present TB theorya GGAa LDAa
1.55 1.55 0.95 1.06
0.18 0.55 0.25 0.23
0.32 0.53 0.27 0.20
0.99 – 0.99 1.08
0.32 – 0.28 0.45
0.39 – 0.32 0.47
CaTe Present TB theorya GGAa LDAa
1.13 1.16 0.89 0.97
0.18 0.45 0.14 0.17
0.29 0.43 0.18 0.32
0.93 – 0.93 1.11
0.23 – 0.14 0.17
0.21 – 0.20 0.26
a
CaSe B1 B2 CaTe B1 B2
along [0 01]. along ½1 1 0. Nonlinearity coupling constants DL, DS and DS for calcium chalcogenides in B1 and B2 phases along [1 0 0] and [11 0] directions at 300 K.
s—polarization
s*—polarization
Table 4 ¨ Square average Gruneisen parameter for longitudinal wave /gj2 SL , average square i ¨ Gruneisen parameter for longitudinal wave /gji S2L
Compounds BaS B1 B2 BaSe B1 B2
Ref. [1].
BaTe B1 Table 2 Second order elastic constants (1012 dyne/cm2 or 102 GPa) of barium chalcogenides in B1 and B2 phase at 300 K. Compounds
B1 phase C12
C44
C11
C12
C44
BaS Present TB theorya GGAa LDAa
1.79 1.79 1.01 1.15
0.46 0.58 0.09 0.15
0.41 0.53 0.09 0.14
1.13 – 1.05 1.31
0.14 – 0.11 0.13
0.13 – 0.10 0.11
BaSe Present TB theorya GGAa LDAa
1.27 1.39 0.94 0.99
0.40 0.45 0.08 0.08
0.36 0.42 0.09 0.08
1.12 – 1.13 1.13
0.13 – 0.12 0.12
0.12 – 0.11 0.11
BaTe Present TB theorya GGAa LDAa
1.05 1.15 0.78 0.87
0.15 0.15 0.04 0.05
0.16 0.14 0.04 0.05
0.41 – 0.80 0.95
0.15 – 0.07 0.12
0.26 – 0.06 0.41
Ref. [2].
B2
SL /gj2 i
/gji S2L
/gji S2S
/gji S2S
DL
DS
DS
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
17.30 15.43 0.94 1.06
1.98 2.58 0.17 0.26
0.08 0.67 0.04 0.15
– 39.15 – 1.93
147.30 127.97 7.82 8.63
0.77 6.06 0.37 1.43
– 352.39 – 17.73
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
28.00 24.78 0.90 1.04
2.32 3.14 0.24 0.36
0.09 2.15 0.04 0.22
– 67.66 – 1.80
243.19 211.05 7.27 8.04
0.83 19.36 0.43 1.98
– 609.00 – 16.27
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
17.26 15.32 1.68 2.14
1.91 2.50 1.28 1.78
0.08 0.69 0.34 4.49
– 39.55 – 1.33
148.21 128.56 10.63 12.93
0.77 6.24 2.73 40.46
– 355.95 – 12.00
Nonlinearity coupling constants DL, DS and DS for barium chalcogenides in B1 and B2 phases along [1 0 0] and [11 0] directions at 300 K.
B2 phase
C11
a
and shear wave /gji S2S ,
/gji S2S .
Table 5 Debye temperature, YD and average Debye velocity /VS at 300 K and molecular weight of CaX and BaX in B1 and B2 phase (X ¼ S, Se and Te). Compounds B1 phase
B2 phase 5
YD (K) /VS (10 cm/s) YD (K) /VS (105 cm/s) Mol. wt. (M) CaS CaSe CaTe BaS BaSe BaTe
455 328 248 298 237 196
2.75 2.04 1.59 1.88 1.24 1.23
337 264 210 200 170 143
3.98 3.01 2.29 1.02 0.87 0.77
72.13 119.03 167.67 169.39 216.28 264.92
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obtained by adding vibrational contribution to elastic constants using lattice dynamics developed by Leibried et al. and Ludwig et al. [14,15]. Using the theory discussed in Refs. [16,17] tested by us for FCC and BCC phase, second and third order elastic constants at 0 K are given as 0 C11
0 C12
! pffiffiffi 3e2 ð2Þ 3Q ðr1 Þ 2Q ðr2 Þ 1 1 3 1 ¼ 4 S5 þ þ þ þ qr0 q qr0 2r0 q qr0 8r0
¼
0 C44
0 C112
3e2 Q ðr2 Þ 1 1 ¼ 4 Sð1;1Þ þ þ qr0 2r0 q 8r0 5
B1 phase BScrew
CaS CaSe CaTe BaS BaSe BaTe
¼
0 C116
15e2 Q ðr1 Þ ¼ 4 S7ð2;1Þ 9q 8r0
BScrew
pffiffiffi pffiffiffi! 3 3 3 þ þ 2 qr0 q r02
15e2 ð1 1 1Þ Q ðr1 Þ S 9q 8r04 7
pffiffiffi pffiffiffi! 3 3 3 þ þ qr0 q2 r02
Here Q ðr1 Þ is the short range repulsive interaction energy (r1 ¼ O3r0 and r2 ¼ 2r0, r0 is nearest neighbour distance) and the lattice sums, which give contribution to the potential at the unit cell (being considered at origin) from all other unit cells of the crystal are ¼ 0:354190; Sð1;1Þ ¼ 0:346708 S01 ¼ Z0 ¼ 1:017678; Sð2Þ 5 5
B2 phase BEdge
pffiffiffi pffiffiffi! 3 3 3 þ þ 2 qr0 q r02 ! Q ðr1 Þ 3 6 4 þ þ 2q qr0 q2 r02
15e2 Q ðr1 Þ ¼ 4 Sð3Þ 9q 8r0 7
0 0 0 C123 ¼ C456 ¼ C144 ¼
Table 6 Phonon viscosity due to screw (BScrew) and edge (BEdge) dislocations for CaX and BaX (X ¼ S, Se and Te) at 300 K for longitudinal (in cP) and shear (in mP ) waves. Compounds
0 C111
79
BEdge
¼ 0:540901; S7ð2;1Þ ¼ 0:093356; Sð1;1;1Þ ¼ 0:16000 Sð3Þ 7 7
Long.
Shear
Long.
Shear
Long.
Shear
Long.
Shear
4.08 7.92 11.75 11.15 47.55 21.33
0.2 0.5 0.8 0.6 0.16 0.11
36.30 67.23 95.84 125.65 401.34 279.89
45.7 84.6 120.5 160.8 322.78 362.11
2.2 3.6 5.1 2.30 2.99 4.79
0.15 0.18 1.30 1.09 1.77 12.30
0.40 0.67 1.10 4.57 6.04 10.71
0.54 0.81 3.32 5.52 7.93 32.21
Here r0 is nearest neighbour distance; q is hardness parameter pffiffiffi! r0 3 2r0 andQ ðr2 Þ ¼ A exp Q ðr1 Þ ¼ A exp q q Second and third order elastic constants at required temperature (Cij(T) and Cijk(T)) for BCC and FCC phase are obtained by adding
Fig. 1. Temperature variation of (a/f2)L along different directions for longitudinal wave for (a) barium chalcogenides and (b) calcium chalcogenides in FCC phase.
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vibrational contribution to elastic constants to the corresponding values at absolute zero: Cij ðTÞ ¼ Cij0 þ Cijvib:
ð2Þ
0 vib: þ Cijk Cijk ðTÞ ¼ Cijk
ð3Þ
vib: Here Cijvib: and Cijk are vibrational contribution to elastic 0 vib , Cijk have constants. Explicit expressions for Cij0 , Cijvib and Cijk been given in Ref. [18].
Here ½ðaÞScrew Long ::and Shear and ½ðaÞEdge Long :and Shear are the attenuation resulting due to screw and edge dislocation damping for longitudinal and shear waves, which are measured in terms of viscous drag coefficients ðBScrew ; BEdge Þ. According to Mason and Bateman [9–12], second and third ¨ order elastic constants are related to Gruneisen parameters, gj;s i and nonlinearity parameters, DL and DS (which are the measure of the anharmonicity of the lattice). Ultrasonic attenuation due to phonon–phonon interaction for longitudinal and shear waves in the Akhiezer regime ðot51Þ is given by ½ðaÞp2p L ¼
E0 o2 ðDL =3ÞtL 2rVL3
ð4Þ
½ðaÞp2p S ¼
E0 o2 ðDS =3ÞtS 2rVS3
ð5Þ
2.2. Theory of acoustic attenuation Predominant causes for acoustic attenuation in solids are phonon– phonon interaction, electron–phonon interaction, thermoelastic loss and dislocation damping. At higher temperatures, electron mean free path is not comparable to phonon mean free path, thus attenuation due to electron–phonon interaction is not present at higher temperatures. Therefore, phonon–phonon interaction, thermoelastic loss and dislocation damping are the dominant mechanisms that will give rise appreciable attenuation [19] beyond 50 K. The total attenuation in these materials can be expressed as the sum of attenuation due to phonon–phonon interaction, ðaÞp2p , thermoelastic loss, ðaÞth and dislocation damping due to screw and edge dislocations i.e. ðaÞtotal ¼ ½ðaÞp2p L þ ½ðaÞp2p S þ ½ðaÞth þ ½ðaÞScrew Long ::and Shear þ½ðaÞEdge Long :and Shear
Here o( ¼ 2pf, f is the frequency of ultrasonic wave) is angular frequency, E0 is the energy density, DL and DS are the nonlinearity constants and tL and tS are thermal relaxation times for longitudinal and shear waves, r is the density and VL and VS are the velocities for longitudinal and shear waves, respectively. Propagation of acoustic wave through crystal produces thermoelastic loss owing to the conduction of heat from compressed to rarefied regions which is given by [20] ðaÞth ¼
o2 /gji S2 KT 2rVL5
Here K is thermal conductivity and T is absolute temperature.
Fig. 2. Temperature variation of (a/f2)L along different directions for longitudinal wave for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
ð6Þ
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Nonlinearity coupling constant for longitudinal and shear waves is given by [20] DL ¼ 9/ðgji Þ2 SL
DS ¼ 9/ðgji Þ2 SS
3/gji S2L Cv T E0 j i
3/g
S2S E0
Cv T
ð7Þ
ð8Þ
Here Cv is the specific heat per unit volume and /ðgji Þ2 S and ¨ para/gji S2 are square average and average square Gruneisen meters. The Debye average velocity, /VS; is given by [20] 3 ð/VSÞ3
¼
1 2 þ 3 VL3 VS
ð9Þ
The Debye temperature (which is used for obtaining Cv and E0) is given by [21]
YD ¼
‘ ð6p2 na Þ1=3 /VS kB
ð10Þ
Here ‘ ¼ ðh=2pÞ, h being plank constant, kB is Boltzmann constant, and na ¼ Na r=M, Na and M being the Avogadro number and molecular weight. Thermal relaxation time is given by [20]
t ¼ tS ¼
tL 2
¼
3K Cv /VS2
ð11Þ
Dislocation damping due to screw and edge dislocations (which is measured by viscous drag coefficient) also produces
81
appreciable loss due to phonon–phonon interaction. Viscous drag coefficient due to screw and edge dislocations is given by [20] BScrew ¼ 0:071Z
ð12aÞ
BEdge ¼ 0:053Z=ð1 s2 Þ þ 0:0079=ð1 s2 Þðm=BÞw
ð12bÞ
Here
w ¼ ZL ð4=3ÞZS ; ZL ¼ E0 DL tL =3; ZS ¼ E0 DS tS =3 B ¼ ðC11 þ 2C12 Þ=3; m ¼ ðC11 C12 þ C44 Þ=3and
s ¼ C12 =ðC11 þ C12 Þ=3 Here B; m; Z; s; w are the bulk modulus, shear modulus, phonon viscosity, Poisson’s ratio, compressional viscosity, respectively.
3. Results and discussions The elastic constants of materials are directly related to their microstructure and are used to obtain the acoustic attenuation, ¨ Debye temperature, Debye average velocity, Gruneisen parameter; nonlinearity coupling constants, thermal relaxation time, phonon viscosity and other physical properties and therefore, these are of great interest in applications where the mechanical strength and durability are important. Second and third order elastic constants have been evaluated using lattice parameters [1,2] and hardness parameters (hardness parameters of CaS, CaSe, CaTe, BaS, BaSe, BaTe are 0.196, 0.214,
Fig. 3. Temperature variation of (a/f2)S along different directions for shear wave for (a) barium chalcogenides and (b) calcium chalcogenides in FCC phase.
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˚ respectively, in FCC phase and 0.457, 0.234, 0.188, 0.172, 0.215 A, 0.452, 0.374, 0.691, 0.699 and 0.622 A˚ in BCC phase). Values of second order elastic constants obtained for calcium and barium chalcogenides at 300 K are listed in Tables 1 and 2 and have been compared with available data (obtained using Tight Binding Theory (TB Theory), Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) models) [1,2]. Present values of C11 are in good agreement with available data. Disagreement in C12 and C44 values may be attributed to the different approaches adopted. Our calculations of second and third order elastic constants are reasonable since we have made calculations for the temperature variation of elastic constants of these alkaline earth chalcogenides. There are no elastic data as a function of temperature for these compounds. Most simple theories are able to get a reasonable estimate of elastic constants at room temperature only by using experimental parameters. Values of second and third order elastic constants have been ¨ used to obtain square average and average square Gruneisen SL , /gji S2L and parameter for longitudinal and shear waves (/gj2 i /gji S2S , /gji S2S ) and nonlinearity coupling constants DL, DS and DS (which are the measure of anharmonicity of the lattice) along different directions of propagation and are presented in Tables 3 and 4. Values of nonlinearity constants are similar to the case of other cubic crystalline materials [9–12]. Values of Debye average velocity and Debye temperature, obtained using Eqs. (9) and (10) in FCC and BCC phases are given in Table 5. Viscous drag coefficients due to screw and edge dislocations, (BScrew and BEdge) obtained using Eqs. (12a) and (12b) are presented in
¨ Table 6. The values of Gruneisen parameters, nonlinearity constants and viscous drag coefficients for these chalcogenides are greater than other semi-conducting materials due to wide band gap [20]. The ultrasonic attenuation due to phonon–phonon interaction for longitudinal and shear waves, [(a/f2)L and (a/f2)S] have been evaluated using Eqs. (4) and (5) by taking values of E0 and Cv from AIP Handbook (E0 and Cv have been obtained as function of YD/T) [22]. The temperature variation of (a/f2)L and (a/f2)S along [1 0 0], [11 0] and [111] directions of propagation are shown in Figs. 1–4. Ultrasonic attenuation due to thermoelastic loss, (a/f2)th obtained using Eq. (6) is shown in Figs. 5 and 6. The acoustical dissipation due to phonon–phonon interaction for longitudinal and shear waves (a/f2)L and (a/f2)S increases up to Debye temperature and then becomes constant. For temperatures TrYD; (a/f2)L and (a/f2)S increase and at temperatures in the range (YD/T)o1, they become constant, because, [(a/f2)] due to phonon–phonon interaction is mainly affected by the specific heat per unit volume, Cv. For (YD/T)eZ1, Cv increases and becomes nearly constant for (YD/T)o1 (Fig. 7) for these chalcogenides of calcium and barium in B1 and B2 phases. Increase in temperature results in enhanced phonon–phonon interaction, which gives rise increase in attenuation for (YD/T)Z1. For temperatures (YD/T) o1, Cv remains nearly constant, which results steady rate of phonon–phonon interaction and hence (a/f2) becomes constant. It is worth mentioning that in most cases, YD for these materials lies in range 150–350 K (in both the phases, except for CaS in FCC phase for which YD ¼ 455 K). Thus (a/f2) vs. T plots attain nearly constant value in this temperature range, except for CaS, for which
Fig. 4. Temperature variation of (a/f2)S along different directions for shear wave for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
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Fig. 5. Temperature variation of (a/f2)th along different directions for (a)barium chalcogenides and (b) calcium chalcogenides in FCC phase.
(a/f2)L and (a/f2)S increase up to YD ¼ 455 K and then become nearly constant. (a/f2)th is directly proportional to the rate of heat transfer from the compressed to the rarefied regions. In the low temperature range, 50–200 K, as temperature increases, (a/f2)th increases at faster rate because, thermal conductivity varies rapidly with temperature in this temperature range. Ultrasonic attenuation, (a/f2) increases from CaS (BaS) to CaTe (BaTe) for FCC as well as BCC structured alkaline earth chalcogenides in a given chalcogenide series. (a/f2) is related to the Debye temperature, YD (since Cv and E0 are functions of YD), which is maximum for CaS (BaS) and minimum for CaTe (BaTe) (Table 5). Thus larger the YD, smaller is attenuation. The value of Debye temperature, YD depends on the Debye average velocity, /VS and molecular weight, M1/3 (Eq. (10)). /VS is maximum for CaS (or BaS) and minimum for CaTe (or BaTe), therefore, for larger /VS, attenuation will be smaller. Thus, attenuation increases in chalcogenides series of Ca and Ba with increasing molecular weight. The Debye temperature, YD is related to the Debye average velocity /VS (Eq. (10)) and /VS is obtained in terms of VL and VS, (Eq. (9)) which is related to second order elastic constants and density as VL ¼ O(C11/r) and VL ¼ O(C44/r). Therefore, increase in (a/f2)L, (a/f2)S and (a/f2)th from CaS (or BaS) to CaTe (or BaTe) is mainly attributed to the second order elastic constant (larger the value of second order elastic constant, smaller will be attenuation in a series) and molecular weight. Dislocation damping in terms of viscous drag coefficient due to screw and edge dislocations is appreciable (Table 6) in both the phases. The attenuation due to thermoelastic loss is negligible in
comparison to the attenuation due to phonon–phonon interaction and dislocation damping for longitudinal and shear waves, thus ultrasonic attenuation is mainly governed by the loss due to phonon–phonon interaction phenomena.
4. Conclusions We have discussed the general aspects of acoustical dissipation induced by phonon–phonon interaction, thermoelastic loss and dislocation damping due to screw and edge dislocations in the alkaline earth chalcogenides in a wide temperature range. It is found that evaluated values of ultrasonic attenuation at different temperatures vary nearly in similar fashion for these FCC and BCC structured chalcogenides and ultrasonic attenuation is mainly governed by the loss due to phonon–phonon interaction phenomena. Due to lack of experimental data on attenuation for these chalcogenides, no comparison has been made, yet, on the basis of agreement in values of elastic constants and nature of ultrasonic attenuation in other similar materials, it can be concluded that present approach is justified and present values will be guide to the experimentalists working in this field.
Acknowledgement One of us (RKS) thankfully acknowledges financial support from UGC.
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Fig. 6. Temperature variation of (a/f2)th along different directions for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
References [1] Z. Charifi, H. Baoziz, F.EL. Haj Hassan, N.J. Bouarissa, Phys. Condens. Matter 17 (2005) 4083. [2] F.E. Haj Hassan, H. Akbarzadeh, Comput. Mater. Sci. 38 (2006) 362. [3] D. Rached, M. Rabah, N. Benkhettou, R. Khenata, B. Soudini, Y.Al. Douri, H. Baltache, Comput. Mater. Sci. 37 (2006) 292. [4] A. Bouhemadou, R. Khenata, F. Zegrar, M. Sahnoun, H. Baltache, A.H. Reshhak, Comput. Mater. Sci. 38 (2006) 263. [5] P. Cervantes, Q. Williams, M. Cote, M. Rohlfing, M.L. Cohen, S.G. Louie, Phys. Rev. B 58 (15) (1998) 9793. [6] G.G. Sahasrabudh, S.D. Lambade, J. Acoust. Soc. Amer. 104 (1) (1998) 81. [7] M. Nandanpawar, S. Rajagopalan, Phys. Rev. B 18 (1978) 10. [8] A. Akhiezer, J. Phys. (USSR) 1 (1939) 227. [9] W.P. Mason, J. Acoust. Soc. Amer. 42 (1) (1967) 253. [10] W.P. Mason, A. Rosenberg, J. Acoust. Soc. Amer. 45 (1969) 470. [11] W.P. Mason, T.B. Bateman, J. Acoust. Soc. Amer. 36 (4) (1964) 645. [12] W.P. Mason, Physical Acoustics, vol. IIIB, Academic Press, New York, 1965. [13] K. Brugger, Phys. Rev. A 133 (1964) 1611. [14] G. Leibried, A. Zur, H.Z. Hahn, Phys. Rev. B 150 (1958) 497. [15] W. Ludwig, G. Leibfried, Solid State Physics, Vol. 12, Academic Press, New York, 1967. [16] R.K. Singh, R.P. Singh, M.P. Singh, P.C. Srivastava, J. Phys. Condens. Matter 20 (2008) 345227. [17] R.K. Singh, R.P. Singh, M.P. Singh, Physica B 404 (2009) 95. [18] P.B. Ghate, Phys. Rev. B 139 (5A) (1965) A1666. [19] D.K. Pandey, D. Singh, R.R. Yadav, Appl. Acoust. 68 (2007) 766. [20] S.K. Kor, U.S. Tondon, G. Rai, Phys. Rev. B 6 (6) (1972) 775. [21] C.Z. Jasiukiewicz, V. Karpus, Solid State Commun. 128 (2003) 167. [22] AIP Handbook, Academic Press, New York, 1981.
Fig. 7. Temperature variation of Cv for barium and calcium chalcogenides in (a) BCC and (b) FCC phase.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Temperature dependent acoustical characterization of alkaline earth monochalcogenides in B1 and B2 phase Rishi Pal Singh, Manish Pratap Singh, Prakash Chandra Srivastava, Rajendra Kumar Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India
a r t i c l e in fo
abstract
Article history: Received 28 January 2009 Received in revised form 2 August 2009 Accepted 5 August 2009
The Born model of ionic solids has been applied to study the temperature dependence of acoustical dissipation due to phonon–phonon interaction, thermo-elastic mechanism and dislocation damping due to screw and edge dislocations in alkaline earth chalcogenides viz. calcium and barium chalcogenides [CaX and BaX; X ¼ S, Se and Te] in FCC as well as BCC phases, in the wide temperature range (50–500 K); for longitudinal and shear modes of propagation along /1 0 0S, /11 0S and /111S directions and for shear modes with different polarizations. Electrostatic and short-range repulsive potentials have been used for evaluating second and third order elastic constants at required temperature, which in turn have been used to obtain acoustical dissipation and related parameters at different temperatures. Hardness parameter, lattice constants and experimental values of second order elastic constants have been used as input parameters. Results have been discussed and temperature variation of acoustical dissipation has been found to exhibit exponential behaviour. & 2009 Elsevier B.V. All rights reserved.
PACS: 62.20Dc 62.65+K 43.35Cg Keywords: Alkaline earth monochalcogenides Elastic constant Phonon–phonon interaction Phonon viscosity Nonlinearity coupling constants
1. Introduction Alkaline earth monochalcogenides form very important closed shell ionic systems crystallizing in NaCl type (B1) structure with six fold coordination and CsCl (B2) type structure with eight fold coordination at ambient conditions [1–3]. There is no d electron in the valence band. Alkaline earth chalcogenides are currently under intense investigations driven by their applications in light emitting diodes (LEDs) and laser diodes (LDs). It is expected that these compounds may provide new II–IV candidates for the fabrication of various electrical and optical devices. Among the wide band gap II–IV semiconductors, calcium chalcogenides CaX and barium chalcogenides BaX (X ¼ S, Se and Te) and their alloys are interesting in connection with optoelectronic applications in blue light wavelength regime. High temperature investigations of alkaline earth chalcogenides have recently attracted interest due to their technological usefulness and interesting physical properties. These compounds are technologically important materials having many applications ranging from catalysis to microelec-
Corresponding author. Tel.: +91 542 2307308; fax: +91 542 2368174.
E-mail address: [email protected] (R.K. Singh). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.08.056
tronics. They also have applications in the area of luminescent devices. Recently, structural, elastic, and electronic properties of these chalcogenides have been investigated extensively [1–5], but, acoustical studies have not been made in these chalcogenides. So, we have chosen this important class of compounds for studying temperature variation of acoustical dissipation. Acoustic ¨ attenuation, phonon viscosity, Gruneisen parameters, nonlinearity parameters and thermal relaxation time, which are of immense importance to explain the microstructure and related physical properties of these compounds, have been obtained over a wide temperature range. Experimental studies of FCC and BCC type compounds reveal that dislocation mechanism yields a negligible contribution to acoustic attenuation in well annealed NaCl (B1) type and CsCl (B2) type crystals and contribution to thermoelastic loss too is negligibly small [6]. Thus the alkaline earth chalcogenides are ideally suited for the study of acoustic attenuation due to phonon–phonon (p–p) interaction mechanism [7]. In the present investigation, using simple potentials (electrostatic and repulsive potentials), second and third order elastic constants (SOEM and TOEM) have been obtained at different temperatures, which in turn have been used to obtain acoustical dissipation and related parameters. The calculation of temperature dependence of acoustical behaviour have been presented for longitudinal and
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shear waves along different directions of propagation viz. [1 0 0], [11 0] and [111] in the temperature range 50–500 K for CaX and BaX (X ¼ S, Se and Te) in FCC and BCC phases in the Akhiezer regime [8] using Mason’s approach [9–12].
Table 3 ¨ Square average Gruneisen parameter for longitudinal wave /gj2 SL , average, square i ¨ Gruneisen parameter for longitudinal wave /gji S2L j i
/g
S2S
and shear wave /gji S2S ,
.
Compounds
SL /gj2 i
/gji S2L
/gji S2S
/gji S2S
DL
DS
DS
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
14.31 12.78 1.45 1.62
1.84 2.37 0.03 0.13
0.08 0.40 0.14 1.04
– 31.73 – 2.74
119.74 103.36 13.10 14.55
0.75 3.60 0.90 9.43
– 285.57 – 24.67
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
13.36 11.90 1.19 1.32
1.77 2.29 0.03 0.07
0.08 0.33 0.05 0.48
– 29.56 – 2.39
112.51 97.19 10.57 11.67
0.75 2.99 0.53 4.32
– 266.04 – 21.53
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
12.60 11.21 1.50 1.89
1.71 2.20 1.13 1.56
0.08 0.29 0.26 3.74
– 27.89 – 1.41
106.85 92.38 9.33 11.16
0.74 2.60 2.38 33.69
– 2.51 – 12.75
2. Theoretical approach CaS B1
2.1. Theory of elastic moduli
B2
According to Brugger’s definition [13], nth order elastic constant is defined as Cijklmn...... ¼ ð@n u=@eij @ekl @emn . . . . . . . . .Þ
ð1Þ
where u is the crystal free-energy density and eij is strain tensor. 0 ) Second and third order elastic constants at 0 K (Cij0 and Cijk have been obtained using electrostatic and Born–Mayer potentials and taking interactions up to next nearest-neighbours. Temperature variation of second and third order elastic constants has been Table 1 Second order elastic constants (1012 dynes/cm2 or 102 GPa) of calcium chalcogenides in B1 and B2 phase at 300 K. Compounds
B1 phase
B2 phase
C11
C12
C44
C11
C12
C44
CaS Present TB theorya GGAa LDAa
2.02 2.02 1.08 1.22
0.60 0.73 0.32 0.39
0.65 0.67 0.36 0.41
1.13 – 1.13 1.24
0.39 – 0.34 0.53
0.33 – 0.39 0.55
CaSe Present TB theorya GGAa LDAa
1.55 1.55 0.95 1.06
0.18 0.55 0.25 0.23
0.32 0.53 0.27 0.20
0.99 – 0.99 1.08
0.32 – 0.28 0.45
0.39 – 0.32 0.47
CaTe Present TB theorya GGAa LDAa
1.13 1.16 0.89 0.97
0.18 0.45 0.14 0.17
0.29 0.43 0.18 0.32
0.93 – 0.93 1.11
0.23 – 0.14 0.17
0.21 – 0.20 0.26
a
CaSe B1 B2 CaTe B1 B2
along [0 01]. along ½1 1 0. Nonlinearity coupling constants DL, DS and DS for calcium chalcogenides in B1 and B2 phases along [1 0 0] and [11 0] directions at 300 K.
s—polarization
s*—polarization
Table 4 ¨ Square average Gruneisen parameter for longitudinal wave /gj2 SL , average square i ¨ Gruneisen parameter for longitudinal wave /gji S2L
Compounds BaS B1 B2 BaSe B1 B2
Ref. [1].
BaTe B1 Table 2 Second order elastic constants (1012 dyne/cm2 or 102 GPa) of barium chalcogenides in B1 and B2 phase at 300 K. Compounds
B1 phase C12
C44
C11
C12
C44
BaS Present TB theorya GGAa LDAa
1.79 1.79 1.01 1.15
0.46 0.58 0.09 0.15
0.41 0.53 0.09 0.14
1.13 – 1.05 1.31
0.14 – 0.11 0.13
0.13 – 0.10 0.11
BaSe Present TB theorya GGAa LDAa
1.27 1.39 0.94 0.99
0.40 0.45 0.08 0.08
0.36 0.42 0.09 0.08
1.12 – 1.13 1.13
0.13 – 0.12 0.12
0.12 – 0.11 0.11
BaTe Present TB theorya GGAa LDAa
1.05 1.15 0.78 0.87
0.15 0.15 0.04 0.05
0.16 0.14 0.04 0.05
0.41 – 0.80 0.95
0.15 – 0.07 0.12
0.26 – 0.06 0.41
Ref. [2].
B2
SL /gj2 i
/gji S2L
/gji S2S
/gji S2S
DL
DS
DS
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
17.30 15.43 0.94 1.06
1.98 2.58 0.17 0.26
0.08 0.67 0.04 0.15
– 39.15 – 1.93
147.30 127.97 7.82 8.63
0.77 6.06 0.37 1.43
– 352.39 – 17.73
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
28.00 24.78 0.90 1.04
2.32 3.14 0.24 0.36
0.09 2.15 0.04 0.22
– 67.66 – 1.80
243.19 211.05 7.27 8.04
0.83 19.36 0.43 1.98
– 609.00 – 16.27
[1 0 0] [1 1 0] [1 0 0] [1 1 0]
17.26 15.32 1.68 2.14
1.91 2.50 1.28 1.78
0.08 0.69 0.34 4.49
– 39.55 – 1.33
148.21 128.56 10.63 12.93
0.77 6.24 2.73 40.46
– 355.95 – 12.00
Nonlinearity coupling constants DL, DS and DS for barium chalcogenides in B1 and B2 phases along [1 0 0] and [11 0] directions at 300 K.
B2 phase
C11
a
and shear wave /gji S2S ,
/gji S2S .
Table 5 Debye temperature, YD and average Debye velocity /VS at 300 K and molecular weight of CaX and BaX in B1 and B2 phase (X ¼ S, Se and Te). Compounds B1 phase
B2 phase 5
YD (K) /VS (10 cm/s) YD (K) /VS (105 cm/s) Mol. wt. (M) CaS CaSe CaTe BaS BaSe BaTe
455 328 248 298 237 196
2.75 2.04 1.59 1.88 1.24 1.23
337 264 210 200 170 143
3.98 3.01 2.29 1.02 0.87 0.77
72.13 119.03 167.67 169.39 216.28 264.92
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obtained by adding vibrational contribution to elastic constants using lattice dynamics developed by Leibried et al. and Ludwig et al. [14,15]. Using the theory discussed in Refs. [16,17] tested by us for FCC and BCC phase, second and third order elastic constants at 0 K are given as 0 C11
0 C12
! pffiffiffi 3e2 ð2Þ 3Q ðr1 Þ 2Q ðr2 Þ 1 1 3 1 ¼ 4 S5 þ þ þ þ qr0 q qr0 2r0 q qr0 8r0
¼
0 C44
0 C112
3e2 Q ðr2 Þ 1 1 ¼ 4 Sð1;1Þ þ þ qr0 2r0 q 8r0 5
B1 phase BScrew
CaS CaSe CaTe BaS BaSe BaTe
¼
0 C116
15e2 Q ðr1 Þ ¼ 4 S7ð2;1Þ 9q 8r0
BScrew
pffiffiffi pffiffiffi! 3 3 3 þ þ 2 qr0 q r02
15e2 ð1 1 1Þ Q ðr1 Þ S 9q 8r04 7
pffiffiffi pffiffiffi! 3 3 3 þ þ qr0 q2 r02
Here Q ðr1 Þ is the short range repulsive interaction energy (r1 ¼ O3r0 and r2 ¼ 2r0, r0 is nearest neighbour distance) and the lattice sums, which give contribution to the potential at the unit cell (being considered at origin) from all other unit cells of the crystal are ¼ 0:354190; Sð1;1Þ ¼ 0:346708 S01 ¼ Z0 ¼ 1:017678; Sð2Þ 5 5
B2 phase BEdge
pffiffiffi pffiffiffi! 3 3 3 þ þ 2 qr0 q r02 ! Q ðr1 Þ 3 6 4 þ þ 2q qr0 q2 r02
15e2 Q ðr1 Þ ¼ 4 Sð3Þ 9q 8r0 7
0 0 0 C123 ¼ C456 ¼ C144 ¼
Table 6 Phonon viscosity due to screw (BScrew) and edge (BEdge) dislocations for CaX and BaX (X ¼ S, Se and Te) at 300 K for longitudinal (in cP) and shear (in mP ) waves. Compounds
0 C111
79
BEdge
¼ 0:540901; S7ð2;1Þ ¼ 0:093356; Sð1;1;1Þ ¼ 0:16000 Sð3Þ 7 7
Long.
Shear
Long.
Shear
Long.
Shear
Long.
Shear
4.08 7.92 11.75 11.15 47.55 21.33
0.2 0.5 0.8 0.6 0.16 0.11
36.30 67.23 95.84 125.65 401.34 279.89
45.7 84.6 120.5 160.8 322.78 362.11
2.2 3.6 5.1 2.30 2.99 4.79
0.15 0.18 1.30 1.09 1.77 12.30
0.40 0.67 1.10 4.57 6.04 10.71
0.54 0.81 3.32 5.52 7.93 32.21
Here r0 is nearest neighbour distance; q is hardness parameter pffiffiffi! r0 3 2r0 andQ ðr2 Þ ¼ A exp Q ðr1 Þ ¼ A exp q q Second and third order elastic constants at required temperature (Cij(T) and Cijk(T)) for BCC and FCC phase are obtained by adding
Fig. 1. Temperature variation of (a/f2)L along different directions for longitudinal wave for (a) barium chalcogenides and (b) calcium chalcogenides in FCC phase.
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vibrational contribution to elastic constants to the corresponding values at absolute zero: Cij ðTÞ ¼ Cij0 þ Cijvib:
ð2Þ
0 vib: þ Cijk Cijk ðTÞ ¼ Cijk
ð3Þ
vib: Here Cijvib: and Cijk are vibrational contribution to elastic 0 vib , Cijk have constants. Explicit expressions for Cij0 , Cijvib and Cijk been given in Ref. [18].
Here ½ðaÞScrew Long ::and Shear and ½ðaÞEdge Long :and Shear are the attenuation resulting due to screw and edge dislocation damping for longitudinal and shear waves, which are measured in terms of viscous drag coefficients ðBScrew ; BEdge Þ. According to Mason and Bateman [9–12], second and third ¨ order elastic constants are related to Gruneisen parameters, gj;s i and nonlinearity parameters, DL and DS (which are the measure of the anharmonicity of the lattice). Ultrasonic attenuation due to phonon–phonon interaction for longitudinal and shear waves in the Akhiezer regime ðot51Þ is given by ½ðaÞp2p L ¼
E0 o2 ðDL =3ÞtL 2rVL3
ð4Þ
½ðaÞp2p S ¼
E0 o2 ðDS =3ÞtS 2rVS3
ð5Þ
2.2. Theory of acoustic attenuation Predominant causes for acoustic attenuation in solids are phonon– phonon interaction, electron–phonon interaction, thermoelastic loss and dislocation damping. At higher temperatures, electron mean free path is not comparable to phonon mean free path, thus attenuation due to electron–phonon interaction is not present at higher temperatures. Therefore, phonon–phonon interaction, thermoelastic loss and dislocation damping are the dominant mechanisms that will give rise appreciable attenuation [19] beyond 50 K. The total attenuation in these materials can be expressed as the sum of attenuation due to phonon–phonon interaction, ðaÞp2p , thermoelastic loss, ðaÞth and dislocation damping due to screw and edge dislocations i.e. ðaÞtotal ¼ ½ðaÞp2p L þ ½ðaÞp2p S þ ½ðaÞth þ ½ðaÞScrew Long ::and Shear þ½ðaÞEdge Long :and Shear
Here o( ¼ 2pf, f is the frequency of ultrasonic wave) is angular frequency, E0 is the energy density, DL and DS are the nonlinearity constants and tL and tS are thermal relaxation times for longitudinal and shear waves, r is the density and VL and VS are the velocities for longitudinal and shear waves, respectively. Propagation of acoustic wave through crystal produces thermoelastic loss owing to the conduction of heat from compressed to rarefied regions which is given by [20] ðaÞth ¼
o2 /gji S2 KT 2rVL5
Here K is thermal conductivity and T is absolute temperature.
Fig. 2. Temperature variation of (a/f2)L along different directions for longitudinal wave for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
ð6Þ
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Nonlinearity coupling constant for longitudinal and shear waves is given by [20] DL ¼ 9/ðgji Þ2 SL
DS ¼ 9/ðgji Þ2 SS
3/gji S2L Cv T E0 j i
3/g
S2S E0
Cv T
ð7Þ
ð8Þ
Here Cv is the specific heat per unit volume and /ðgji Þ2 S and ¨ para/gji S2 are square average and average square Gruneisen meters. The Debye average velocity, /VS; is given by [20] 3 ð/VSÞ3
¼
1 2 þ 3 VL3 VS
ð9Þ
The Debye temperature (which is used for obtaining Cv and E0) is given by [21]
YD ¼
‘ ð6p2 na Þ1=3 /VS kB
ð10Þ
Here ‘ ¼ ðh=2pÞ, h being plank constant, kB is Boltzmann constant, and na ¼ Na r=M, Na and M being the Avogadro number and molecular weight. Thermal relaxation time is given by [20]
t ¼ tS ¼
tL 2
¼
3K Cv /VS2
ð11Þ
Dislocation damping due to screw and edge dislocations (which is measured by viscous drag coefficient) also produces
81
appreciable loss due to phonon–phonon interaction. Viscous drag coefficient due to screw and edge dislocations is given by [20] BScrew ¼ 0:071Z
ð12aÞ
BEdge ¼ 0:053Z=ð1 s2 Þ þ 0:0079=ð1 s2 Þðm=BÞw
ð12bÞ
Here
w ¼ ZL ð4=3ÞZS ; ZL ¼ E0 DL tL =3; ZS ¼ E0 DS tS =3 B ¼ ðC11 þ 2C12 Þ=3; m ¼ ðC11 C12 þ C44 Þ=3and
s ¼ C12 =ðC11 þ C12 Þ=3 Here B; m; Z; s; w are the bulk modulus, shear modulus, phonon viscosity, Poisson’s ratio, compressional viscosity, respectively.
3. Results and discussions The elastic constants of materials are directly related to their microstructure and are used to obtain the acoustic attenuation, ¨ Debye temperature, Debye average velocity, Gruneisen parameter; nonlinearity coupling constants, thermal relaxation time, phonon viscosity and other physical properties and therefore, these are of great interest in applications where the mechanical strength and durability are important. Second and third order elastic constants have been evaluated using lattice parameters [1,2] and hardness parameters (hardness parameters of CaS, CaSe, CaTe, BaS, BaSe, BaTe are 0.196, 0.214,
Fig. 3. Temperature variation of (a/f2)S along different directions for shear wave for (a) barium chalcogenides and (b) calcium chalcogenides in FCC phase.
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˚ respectively, in FCC phase and 0.457, 0.234, 0.188, 0.172, 0.215 A, 0.452, 0.374, 0.691, 0.699 and 0.622 A˚ in BCC phase). Values of second order elastic constants obtained for calcium and barium chalcogenides at 300 K are listed in Tables 1 and 2 and have been compared with available data (obtained using Tight Binding Theory (TB Theory), Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) models) [1,2]. Present values of C11 are in good agreement with available data. Disagreement in C12 and C44 values may be attributed to the different approaches adopted. Our calculations of second and third order elastic constants are reasonable since we have made calculations for the temperature variation of elastic constants of these alkaline earth chalcogenides. There are no elastic data as a function of temperature for these compounds. Most simple theories are able to get a reasonable estimate of elastic constants at room temperature only by using experimental parameters. Values of second and third order elastic constants have been ¨ used to obtain square average and average square Gruneisen SL , /gji S2L and parameter for longitudinal and shear waves (/gj2 i /gji S2S , /gji S2S ) and nonlinearity coupling constants DL, DS and DS (which are the measure of anharmonicity of the lattice) along different directions of propagation and are presented in Tables 3 and 4. Values of nonlinearity constants are similar to the case of other cubic crystalline materials [9–12]. Values of Debye average velocity and Debye temperature, obtained using Eqs. (9) and (10) in FCC and BCC phases are given in Table 5. Viscous drag coefficients due to screw and edge dislocations, (BScrew and BEdge) obtained using Eqs. (12a) and (12b) are presented in
¨ Table 6. The values of Gruneisen parameters, nonlinearity constants and viscous drag coefficients for these chalcogenides are greater than other semi-conducting materials due to wide band gap [20]. The ultrasonic attenuation due to phonon–phonon interaction for longitudinal and shear waves, [(a/f2)L and (a/f2)S] have been evaluated using Eqs. (4) and (5) by taking values of E0 and Cv from AIP Handbook (E0 and Cv have been obtained as function of YD/T) [22]. The temperature variation of (a/f2)L and (a/f2)S along [1 0 0], [11 0] and [111] directions of propagation are shown in Figs. 1–4. Ultrasonic attenuation due to thermoelastic loss, (a/f2)th obtained using Eq. (6) is shown in Figs. 5 and 6. The acoustical dissipation due to phonon–phonon interaction for longitudinal and shear waves (a/f2)L and (a/f2)S increases up to Debye temperature and then becomes constant. For temperatures TrYD; (a/f2)L and (a/f2)S increase and at temperatures in the range (YD/T)o1, they become constant, because, [(a/f2)] due to phonon–phonon interaction is mainly affected by the specific heat per unit volume, Cv. For (YD/T)eZ1, Cv increases and becomes nearly constant for (YD/T)o1 (Fig. 7) for these chalcogenides of calcium and barium in B1 and B2 phases. Increase in temperature results in enhanced phonon–phonon interaction, which gives rise increase in attenuation for (YD/T)Z1. For temperatures (YD/T) o1, Cv remains nearly constant, which results steady rate of phonon–phonon interaction and hence (a/f2) becomes constant. It is worth mentioning that in most cases, YD for these materials lies in range 150–350 K (in both the phases, except for CaS in FCC phase for which YD ¼ 455 K). Thus (a/f2) vs. T plots attain nearly constant value in this temperature range, except for CaS, for which
Fig. 4. Temperature variation of (a/f2)S along different directions for shear wave for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
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83
Fig. 5. Temperature variation of (a/f2)th along different directions for (a)barium chalcogenides and (b) calcium chalcogenides in FCC phase.
(a/f2)L and (a/f2)S increase up to YD ¼ 455 K and then become nearly constant. (a/f2)th is directly proportional to the rate of heat transfer from the compressed to the rarefied regions. In the low temperature range, 50–200 K, as temperature increases, (a/f2)th increases at faster rate because, thermal conductivity varies rapidly with temperature in this temperature range. Ultrasonic attenuation, (a/f2) increases from CaS (BaS) to CaTe (BaTe) for FCC as well as BCC structured alkaline earth chalcogenides in a given chalcogenide series. (a/f2) is related to the Debye temperature, YD (since Cv and E0 are functions of YD), which is maximum for CaS (BaS) and minimum for CaTe (BaTe) (Table 5). Thus larger the YD, smaller is attenuation. The value of Debye temperature, YD depends on the Debye average velocity, /VS and molecular weight, M1/3 (Eq. (10)). /VS is maximum for CaS (or BaS) and minimum for CaTe (or BaTe), therefore, for larger /VS, attenuation will be smaller. Thus, attenuation increases in chalcogenides series of Ca and Ba with increasing molecular weight. The Debye temperature, YD is related to the Debye average velocity /VS (Eq. (10)) and /VS is obtained in terms of VL and VS, (Eq. (9)) which is related to second order elastic constants and density as VL ¼ O(C11/r) and VL ¼ O(C44/r). Therefore, increase in (a/f2)L, (a/f2)S and (a/f2)th from CaS (or BaS) to CaTe (or BaTe) is mainly attributed to the second order elastic constant (larger the value of second order elastic constant, smaller will be attenuation in a series) and molecular weight. Dislocation damping in terms of viscous drag coefficient due to screw and edge dislocations is appreciable (Table 6) in both the phases. The attenuation due to thermoelastic loss is negligible in
comparison to the attenuation due to phonon–phonon interaction and dislocation damping for longitudinal and shear waves, thus ultrasonic attenuation is mainly governed by the loss due to phonon–phonon interaction phenomena.
4. Conclusions We have discussed the general aspects of acoustical dissipation induced by phonon–phonon interaction, thermoelastic loss and dislocation damping due to screw and edge dislocations in the alkaline earth chalcogenides in a wide temperature range. It is found that evaluated values of ultrasonic attenuation at different temperatures vary nearly in similar fashion for these FCC and BCC structured chalcogenides and ultrasonic attenuation is mainly governed by the loss due to phonon–phonon interaction phenomena. Due to lack of experimental data on attenuation for these chalcogenides, no comparison has been made, yet, on the basis of agreement in values of elastic constants and nature of ultrasonic attenuation in other similar materials, it can be concluded that present approach is justified and present values will be guide to the experimentalists working in this field.
Acknowledgement One of us (RKS) thankfully acknowledges financial support from UGC.
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Fig. 6. Temperature variation of (a/f2)th along different directions for (a) barium chalcogenides and (b) calcium chalcogenides in BCC phase.
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Fig. 7. Temperature variation of Cv for barium and calcium chalcogenides in (a) BCC and (b) FCC phase.