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Thermal conductivity of ternary chalcopyrite compounds
Materials Letters 17 ( 1993 ) 59-62 Nosh-Holland
Thermal conductivity of ternary chalcopyrite compounds M.L. Vale&Gil
and C. Rincon
Centro de Estudios de Sem~cond~ctores, De~~rta~le~t~ de Fisica, F&&ad MPrida. Venezuela
de Ciencias, ~~n~versidadde Los Andes,
Received 28 January 1993; in final form 28 March 1993
The Leibfiied-Schlomann model for the thermal conductivity of dielectric crystals, as modified by Steigmeier and Julian, is used to calculate the thermal conductivity K of ternary chalcopyrite compounds at room temperature in terms of the Griineisen anharmonicity parameter y. A good fit to the experimentat data of K is obtained by using the values y= 1.64 and 1.25 for II-IVV, and I-III-VIZ compounds, respectivety. These are in very good agreement with values of y obtained from studies of Raman modes as a function of pressure in these compounds. The results indicate that in the range between the Debye and room temperature the thermal conductivity is determined only by three-phonon processes.
1. Introduction The temperature dependence of the thermal conductivity K of I-III-VI2 chalcopyrite semiconductor compounds has been reported by various authors [ l61. It is observed that in the low-temperature range the temperature variation of K can be explained by considering the scattering of phonons by the boundary of the sample, point defects and normal and umklapp processes. On the other hand, at ordinary and high temperatures the electronic contribution to the measured thermal conductivity is found to be less than 2%. Hence, K varies inversely with temperature, apparently indicating that in this range the thermal conductivity is determined only by lattice anharmonicity effects. In CuInSez and Cub& this dependence remains valid up to about 400 K and above this temperature scattering by point defects begins to dominate giving rise to a dependence of the form T-“. with O
obtained in these materials from studies of pressure dependence of the Raman modes [ lo- 12 1. This discrepancy can be due to the fact, as noted by Julian [ 131 as well as by Roufosse and Klemens [ 141, that a numerical error was made in determining K in ref. [ 8 1. Hence the values of 1’determined in ref. [ 7 ] are probably overestimated. In the present work, using the expression derived by Leibfried and Schlijmann and including the modifications made by Steigmeier and Julian, the thermal conductivity and the Griineisen constant of ternary chalcopyrite compounds at room temperature are calculated. The values of K and y obtained are in good agreement with experimental data. Also, the variation with temperature of the parameter y for CuInSez and CuInS, has been obtained from the data available at high temperatures.
2. Theory The high-temperature intrinsic thermal conductivity (at T> S,, St, being the Debye temperature) of a crystal due to umklapp processes was calculated by Leibfried and SchlSmann by solving the Boltzmann equation. Their theory is based on Peieri’s perturbation formalism for three-phonon processes and included the development of a variational method to
0167-577x/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
59
Volume 17, number 1,2
MATERIALS LETTERS
account for the normal modified by Steigmeier
process [ 81. Their result as [ 9 ] and Julian [ 13 ] is given
July 1993
200.
by K=
(6/5)4”3(KB/h)3i@&$D/T(y+
l/2)
.
(1)
In this expression KB is the Boltzmann constant, ti is the average atomic mass, d3 is the average volume occupied by one atom of the crystal, and y is an overall Griineisen anharmonicity parameter defined as [I51
ciwi y= Qi ’
(2)
were yi is the Griineisen parameter for the ith mode, defined as the negative logarithmic derivative of the mode frequency Wi with respect to the volume Y, i.e. V &Oi Yi=G*=-
Wn 0,) a(ln V) ’
and ci, the contribution of the ith normal the specific heat, is given by Ci=fiWi( a/aT)ni/63
,
(3) mode to
(4)
Wi being the frequency of the ith mode and ni the number of phonons with frequency Oi present in the crystal.
3. Results and discussion Using eq. ( 1) and the value of f$, determined in ref. [ 161, we have calculated the thermal conductivity of II-IV-V, and I-II-VI2 compounds at room temperature. The Grtineisen constant y has been used as the adjustable parameter to obtain the best fit with the available experimental data, as suggested in ref. [ 7 1. The calculated values of K are plotted in fig. 1 versus the scaling parameter A?%)&. Tables 1 and 2 give the parameters A& S, and SD used in the calculation of K of the II-IV-V2 and I-III-VII compounds respectively. The last column of the tables shows the reported value of K for different compounds and the corresponding references. It was found that a good fit can be obtained by using y= 1.64 and 1.25 for II-IV-V2 and I-III-VI2 compounds respectively. These values are lower than y=2.9 and 1.7 obtained for these compounds in ref. [ 7 1, but they are in good agreement with those obtained for 60
oM683,(10-23gcrnK3)
Fig. 1. Experimental values of thermal conductivity at 300 K for ternary II-IV-V, and I-III-VI2 compounds as a function of the scaling parameter &?a@‘.The solid lines are the theoretical fits with y= 1.64 for II-IV-V, and 1.25 for I-III-VI2 compounds.
the Ai vibrational mode in some ternary compounds compiled in ref. [ 111 and also shown in table 3. Also, our value of 1.25 for I-III-VI2 is in good agreement with y= 1.34 for CuGa$ obtained from eqs. (2) and (4) with the values of yi for 19 vibrational modes determined in ref. [ 10 ] and considering that at high temperatures niz KBT/fiWi. The main reason for the deviation of the theoretical value of K from the experimental data for some of these compounds, as shown in fig. 1 and tables 1 and 2, could be the fact that y is not exactly the same for all the ternary compounds (see table 3) as assumed in fitting eq. ( 1). Hence, in order to compare the value of y for two different compounds and its temperature dependence, we fit eq. (1) to the experimental data of K for CuInSe, and CuInS, in the temperature range 293 to 400 K as reported in ref. [ 3 1. The constant y has been used as the adjustable parameter and SD is supposed to be constant in this range of temperature. The calculated value of y for these two compounds as a function of T is given in fig. 2. As observed, these values are in good agreement with those obtained previously for I-III-VI2 compounds. In this range of temperature we have
Volume
17, number
I ,2
MATERIALS
Table 1 Parameters M, 6, and S,, used in the calculation of the thermal and reported values for these compounds, respectively Compound
6.40 4.75 8.30 10.21 17.75 8.40 12.16
ZnSiP, MgSiP* ZnGeP, ZnSnPz ZnSnSb, CdSiP* CdGeP, ZnSiAs, ZnGeAs, ZnSnAs, CdGeAs, CdSnAs* ‘) Ref.
[ 171.
b, Ref.
2.67 2.79 2.72 2.82 3.15 2.78 2.97 2.77 2.84 2.94 2.96 3.04
IO.05 II.95 13.86 13.90 15.81
of II-IV-V,
compounds.
K,.,
and K,~ being the calculated
@D (R)
K-1 (W/cm
463 501 392 323 196 376 304 339 302 258 260 255
0.215
_
0.210 0.172 0.124 0.055
0.180”’ _ _
K)
0.157 0.129 0.134 0.119 0.090 0.093 0.102
&I (w/cm
K)
0.110 a) 0.140 a’ 0.110”’ 0.115 b’ 0.092 a’
[ 181.
Table 2 Parameters &%,6, and en used in the calculation of the thermal and reported values for these compounds, respectively Compound
CuAISe, CuAlTe* CuGaS, CuGaSe, CuGaTeZ Cub& CuInSez CuInTez AgAlSz AgAlSe, AgAITe, AgGaS, AgGaSe,, AgGaTez AgInS, AgInSe* a) Ref. [4].
conductivity
6 ( 10-8cm)
M ( lo-zxg)
July 1993
LETTERS
b, Ref. [3].
conductivity
of I-III-VI2
compounds.
,Q., and K,~ being the calculated
R (lO-23g)
6 (lO@cm)
@D (R)
&al (W/cm
6.42 10.31 14.47 8.45 12.09 16.12 10.06 13.96 18.02 8.26 12.15 16.19 10.03 13.93 17.96 11.90 15.80
2.65 2.86 2.97 2.10 2.81 3.02 2.77 2.91 3.11 2.76 2.88 2.97 2.80 3.12 3.31 2.85 3.23
372 271 213 330 259 202 284 22s 185 311 241 191 282 210 172 242 186
0.063 0.114
_
0.077 0.153
_
0.110 0.075 0.119 0.086 0.066 0.128 0.09 1 0.063 0.117 0.075 0.057
0.064 =’ 0.125” 0.087 ‘) 0.056 d’ _ _ _ _ _ _
‘) Ref. [5].
K)
&Cr, (W/cm
K)
0.091 0.06 1
d, Ref. [2]
found in both cases a nearly linear variation temperature.
of y with
4. Conclusions The results show that values of y and
K
in good
agreement with experimental data at ordinary and high temperatures can be obtained for ternary chalcopyrite compounds with the Leibfried and SchlSmann formula of the intrinsic thermal conductivity when the corrections of Steigmeier and Julian are included in this formula. This apparently indicates that the thermal conductivity of these materials in this 61
Volume 17, number 1,2
MATERIALS LETTERS
Table 3 Values of the Grilneisen anharmonicity parameter y for the A, mode in some ternary chalcopyrite compounds Compound
YI
CuInSez CuAlSz CuGaS, AgGaS, CdSiPz
1.3-1.46 ‘) 1.6 1.5-1.8 1.02 1.65
a) Ref. [ 121.
T(K) Fig. 2. Calculated values of y for CuInSez and CuInSz as a function of temperature. A linear lit to the calculated points is also shown.
range of temperature is entirely due to lattice anharmonicity effects. Also it is found that the Griineisen parameter, which contains the effect of the
62
July 1993
anharmonic forces, is 1.64 and 1.25 for II-IV-VI I-III-VII compounds, respectively.
and
References
[ 1 ] G.P. Sanchez Porras and S.M. Wasim, Phys. Stat. Sol. 59a (1980) 175. (21 SM. Wasim and A. Noguera, Ternary and multinary compounds, eds. S.K. Deb and A. Zunger (Materials Research Society, Pittsburgh, 1987) p. 121. [3] L.A. Makovetskaya, I.V. Bodnar, B.V. Korzum and G.P. Yaroshevich, Phys. Stat. Sol. 74a (1982) 59. [ 41 S.M. Wasim, G. Marcano and G.P. Sanchez Porras, Japan. J. Appl. Phys. 19, Suppl. 19 ( 1980) 133. [ 51 SM. Wasim and A. Noguera, Phys. Stat. Sol. 82a ( 1980) 553. [ 61 C. Bellabarba and S.M. Wasim, Phys. Stat. Sol. 66a ( 198 1) 105. [ 71 S.M. Wasim, Phys. Stat. Sol. 51a (1979) 35. [8] G. Leibfried and E. Schlomann, Nachr. Ges. Wiss. Goett. Math. Phys. 2 (1954) 71. [9] E. Steigmeier, in: Thermal conductivity, Vol. 2, ed. R.P. Type (Academic Press, New York, 1969) p. 212. [ lo] C. Carleone, D. Olego, A. Jayaraman and A. Cardona, Phys. Rev. B 22 (1980) 3877. [ 111 H. Tanino, T. Maeda, H. Fujikake, H. Nakanishi, S. Endo and T. Irie, Phys. Rev. B 45 ( 1992) 13323. [ 121 J. Gonzalez, M. Quintero and C. Rincon, Phys. Rev. B 45 (1992) 7022. [ 131 C.L. Julian, Phys. Rev. A 137 (1965) 128. [ 141 M. Roufosse and P.G. Klemens, Phys. Rev. B 7 (1973) 5379. [ 151 N.W. Ashcroft and N.D. Mermin, Solid state physics (Saunders College, Philadelphia, 1976) pp. 487-509. [ 161 C. Rincon, Phys. Stat. Sol. 134a (1992) 383. [ 171 J.L. Shay and J.H. Wemick, Ternary chalcopyrite semiconductors: growth, electronic properties and applications (Pergamon Press, New York, 1975) p. 175. [ 181 D.P. Spitzer, J. Phys. Chem. Solids 31 (1970) 19.
Thermal conductivity of ternary chalcopyrite compounds M.L. Vale&Gil
and C. Rincon
Centro de Estudios de Sem~cond~ctores, De~~rta~le~t~ de Fisica, F&&ad MPrida. Venezuela
de Ciencias, ~~n~versidadde Los Andes,
Received 28 January 1993; in final form 28 March 1993
The Leibfiied-Schlomann model for the thermal conductivity of dielectric crystals, as modified by Steigmeier and Julian, is used to calculate the thermal conductivity K of ternary chalcopyrite compounds at room temperature in terms of the Griineisen anharmonicity parameter y. A good fit to the experimentat data of K is obtained by using the values y= 1.64 and 1.25 for II-IVV, and I-III-VIZ compounds, respectivety. These are in very good agreement with values of y obtained from studies of Raman modes as a function of pressure in these compounds. The results indicate that in the range between the Debye and room temperature the thermal conductivity is determined only by three-phonon processes.
1. Introduction The temperature dependence of the thermal conductivity K of I-III-VI2 chalcopyrite semiconductor compounds has been reported by various authors [ l61. It is observed that in the low-temperature range the temperature variation of K can be explained by considering the scattering of phonons by the boundary of the sample, point defects and normal and umklapp processes. On the other hand, at ordinary and high temperatures the electronic contribution to the measured thermal conductivity is found to be less than 2%. Hence, K varies inversely with temperature, apparently indicating that in this range the thermal conductivity is determined only by lattice anharmonicity effects. In CuInSez and Cub& this dependence remains valid up to about 400 K and above this temperature scattering by point defects begins to dominate giving rise to a dependence of the form T-“. with O
obtained in these materials from studies of pressure dependence of the Raman modes [ lo- 12 1. This discrepancy can be due to the fact, as noted by Julian [ 131 as well as by Roufosse and Klemens [ 141, that a numerical error was made in determining K in ref. [ 8 1. Hence the values of 1’determined in ref. [ 7 ] are probably overestimated. In the present work, using the expression derived by Leibfried and Schlijmann and including the modifications made by Steigmeier and Julian, the thermal conductivity and the Griineisen constant of ternary chalcopyrite compounds at room temperature are calculated. The values of K and y obtained are in good agreement with experimental data. Also, the variation with temperature of the parameter y for CuInSez and CuInS, has been obtained from the data available at high temperatures.
2. Theory The high-temperature intrinsic thermal conductivity (at T> S,, St, being the Debye temperature) of a crystal due to umklapp processes was calculated by Leibfried and SchlSmann by solving the Boltzmann equation. Their theory is based on Peieri’s perturbation formalism for three-phonon processes and included the development of a variational method to
0167-577x/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
59
Volume 17, number 1,2
MATERIALS LETTERS
account for the normal modified by Steigmeier
process [ 81. Their result as [ 9 ] and Julian [ 13 ] is given
July 1993
200.
by K=
(6/5)4”3(KB/h)3i@&$D/T(y+
l/2)
.
(1)
In this expression KB is the Boltzmann constant, ti is the average atomic mass, d3 is the average volume occupied by one atom of the crystal, and y is an overall Griineisen anharmonicity parameter defined as [I51
ciwi y= Qi ’
(2)
were yi is the Griineisen parameter for the ith mode, defined as the negative logarithmic derivative of the mode frequency Wi with respect to the volume Y, i.e. V &Oi Yi=G*=-
Wn 0,) a(ln V) ’
and ci, the contribution of the ith normal the specific heat, is given by Ci=fiWi( a/aT)ni/63
,
(3) mode to
(4)
Wi being the frequency of the ith mode and ni the number of phonons with frequency Oi present in the crystal.
3. Results and discussion Using eq. ( 1) and the value of f$, determined in ref. [ 161, we have calculated the thermal conductivity of II-IV-V, and I-II-VI2 compounds at room temperature. The Grtineisen constant y has been used as the adjustable parameter to obtain the best fit with the available experimental data, as suggested in ref. [ 7 1. The calculated values of K are plotted in fig. 1 versus the scaling parameter A?%)&. Tables 1 and 2 give the parameters A& S, and SD used in the calculation of K of the II-IV-V2 and I-III-VII compounds respectively. The last column of the tables shows the reported value of K for different compounds and the corresponding references. It was found that a good fit can be obtained by using y= 1.64 and 1.25 for II-IV-V2 and I-III-VI2 compounds respectively. These values are lower than y=2.9 and 1.7 obtained for these compounds in ref. [ 7 1, but they are in good agreement with those obtained for 60
oM683,(10-23gcrnK3)
Fig. 1. Experimental values of thermal conductivity at 300 K for ternary II-IV-V, and I-III-VI2 compounds as a function of the scaling parameter &?a@‘.The solid lines are the theoretical fits with y= 1.64 for II-IV-V, and 1.25 for I-III-VI2 compounds.
the Ai vibrational mode in some ternary compounds compiled in ref. [ 111 and also shown in table 3. Also, our value of 1.25 for I-III-VI2 is in good agreement with y= 1.34 for CuGa$ obtained from eqs. (2) and (4) with the values of yi for 19 vibrational modes determined in ref. [ 10 ] and considering that at high temperatures niz KBT/fiWi. The main reason for the deviation of the theoretical value of K from the experimental data for some of these compounds, as shown in fig. 1 and tables 1 and 2, could be the fact that y is not exactly the same for all the ternary compounds (see table 3) as assumed in fitting eq. ( 1). Hence, in order to compare the value of y for two different compounds and its temperature dependence, we fit eq. (1) to the experimental data of K for CuInSe, and CuInS, in the temperature range 293 to 400 K as reported in ref. [ 3 1. The constant y has been used as the adjustable parameter and SD is supposed to be constant in this range of temperature. The calculated value of y for these two compounds as a function of T is given in fig. 2. As observed, these values are in good agreement with those obtained previously for I-III-VI2 compounds. In this range of temperature we have
Volume
17, number
I ,2
MATERIALS
Table 1 Parameters M, 6, and S,, used in the calculation of the thermal and reported values for these compounds, respectively Compound
6.40 4.75 8.30 10.21 17.75 8.40 12.16
ZnSiP, MgSiP* ZnGeP, ZnSnPz ZnSnSb, CdSiP* CdGeP, ZnSiAs, ZnGeAs, ZnSnAs, CdGeAs, CdSnAs* ‘) Ref.
[ 171.
b, Ref.
2.67 2.79 2.72 2.82 3.15 2.78 2.97 2.77 2.84 2.94 2.96 3.04
IO.05 II.95 13.86 13.90 15.81
of II-IV-V,
compounds.
K,.,
and K,~ being the calculated
@D (R)
K-1 (W/cm
463 501 392 323 196 376 304 339 302 258 260 255
0.215
_
0.210 0.172 0.124 0.055
0.180”’ _ _
K)
0.157 0.129 0.134 0.119 0.090 0.093 0.102
&I (w/cm
K)
0.110 a) 0.140 a’ 0.110”’ 0.115 b’ 0.092 a’
[ 181.
Table 2 Parameters &%,6, and en used in the calculation of the thermal and reported values for these compounds, respectively Compound
CuAISe, CuAlTe* CuGaS, CuGaSe, CuGaTeZ Cub& CuInSez CuInTez AgAlSz AgAlSe, AgAITe, AgGaS, AgGaSe,, AgGaTez AgInS, AgInSe* a) Ref. [4].
conductivity
6 ( 10-8cm)
M ( lo-zxg)
July 1993
LETTERS
b, Ref. [3].
conductivity
of I-III-VI2
compounds.
,Q., and K,~ being the calculated
R (lO-23g)
6 (lO@cm)
@D (R)
&al (W/cm
6.42 10.31 14.47 8.45 12.09 16.12 10.06 13.96 18.02 8.26 12.15 16.19 10.03 13.93 17.96 11.90 15.80
2.65 2.86 2.97 2.10 2.81 3.02 2.77 2.91 3.11 2.76 2.88 2.97 2.80 3.12 3.31 2.85 3.23
372 271 213 330 259 202 284 22s 185 311 241 191 282 210 172 242 186
0.063 0.114
_
0.077 0.153
_
0.110 0.075 0.119 0.086 0.066 0.128 0.09 1 0.063 0.117 0.075 0.057
0.064 =’ 0.125” 0.087 ‘) 0.056 d’ _ _ _ _ _ _
‘) Ref. [5].
K)
&Cr, (W/cm
K)
0.091 0.06 1
d, Ref. [2]
found in both cases a nearly linear variation temperature.
of y with
4. Conclusions The results show that values of y and
K
in good
agreement with experimental data at ordinary and high temperatures can be obtained for ternary chalcopyrite compounds with the Leibfried and SchlSmann formula of the intrinsic thermal conductivity when the corrections of Steigmeier and Julian are included in this formula. This apparently indicates that the thermal conductivity of these materials in this 61
Volume 17, number 1,2
MATERIALS LETTERS
Table 3 Values of the Grilneisen anharmonicity parameter y for the A, mode in some ternary chalcopyrite compounds Compound
YI
CuInSez CuAlSz CuGaS, AgGaS, CdSiPz
1.3-1.46 ‘) 1.6 1.5-1.8 1.02 1.65
a) Ref. [ 121.
T(K) Fig. 2. Calculated values of y for CuInSez and CuInSz as a function of temperature. A linear lit to the calculated points is also shown.
range of temperature is entirely due to lattice anharmonicity effects. Also it is found that the Griineisen parameter, which contains the effect of the
62
July 1993
anharmonic forces, is 1.64 and 1.25 for II-IV-VI I-III-VII compounds, respectively.
and
References
[ 1 ] G.P. Sanchez Porras and S.M. Wasim, Phys. Stat. Sol. 59a (1980) 175. (21 SM. Wasim and A. Noguera, Ternary and multinary compounds, eds. S.K. Deb and A. Zunger (Materials Research Society, Pittsburgh, 1987) p. 121. [3] L.A. Makovetskaya, I.V. Bodnar, B.V. Korzum and G.P. Yaroshevich, Phys. Stat. Sol. 74a (1982) 59. [ 41 S.M. Wasim, G. Marcano and G.P. Sanchez Porras, Japan. J. Appl. Phys. 19, Suppl. 19 ( 1980) 133. [ 51 SM. Wasim and A. Noguera, Phys. Stat. Sol. 82a ( 1980) 553. [ 61 C. Bellabarba and S.M. Wasim, Phys. Stat. Sol. 66a ( 198 1) 105. [ 71 S.M. Wasim, Phys. Stat. Sol. 51a (1979) 35. [8] G. Leibfried and E. Schlomann, Nachr. Ges. Wiss. Goett. Math. Phys. 2 (1954) 71. [9] E. Steigmeier, in: Thermal conductivity, Vol. 2, ed. R.P. Type (Academic Press, New York, 1969) p. 212. [ lo] C. Carleone, D. Olego, A. Jayaraman and A. Cardona, Phys. Rev. B 22 (1980) 3877. [ 111 H. Tanino, T. Maeda, H. Fujikake, H. Nakanishi, S. Endo and T. Irie, Phys. Rev. B 45 ( 1992) 13323. [ 121 J. Gonzalez, M. Quintero and C. Rincon, Phys. Rev. B 45 (1992) 7022. [ 131 C.L. Julian, Phys. Rev. A 137 (1965) 128. [ 141 M. Roufosse and P.G. Klemens, Phys. Rev. B 7 (1973) 5379. [ 151 N.W. Ashcroft and N.D. Mermin, Solid state physics (Saunders College, Philadelphia, 1976) pp. 487-509. [ 161 C. Rincon, Phys. Stat. Sol. 134a (1992) 383. [ 171 J.L. Shay and J.H. Wemick, Ternary chalcopyrite semiconductors: growth, electronic properties and applications (Pergamon Press, New York, 1975) p. 175. [ 181 D.P. Spitzer, J. Phys. Chem. Solids 31 (1970) 19.