Solid State Communications 149 (2009) 1008–1011
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Debye temperature and melting point of ternary chalcopyrite semiconductors
V. Kumar ∗,1 , A.K. Shrivastava, Rajib Banerji, D. Dhirhe
Department of Electronics and Instrumentation, Indian School of Mines University, Dhanbad 826 004, India
article
info
Article history:
Received 20 February 2009
Received in revised form
2 April 2009
Accepted 6 April 2009 by P. Chaddah
Available online 10 April 2009
PACS:
62.20.Dc
63.70.+h
65.40.Ba
74.45 Gm
abstract
II IV V
Debye temperature (θD ) and melting point (Tm ) of AI BIII CVI
2 and A B C2 chalcopyrite semiconductors have
been discussed. Four simple relations have been proposed to calculate the values of θD . Two are based on
plasmon energy data and one each on Tm and molecular weight (W ). We have also proposed two simple
relations to calculate Tm of these semiconductors. One is based on plasmon energy and the other on W.
The calculated values of θD and Tm from all equations are compared with the experimental values and the
values reported by different workers. Reasonably good agreement has been obtained between them.
© 2009 Elsevier Ltd. All rights reserved.
Keywords:
A. Ternary chalcopyrite semiconductors
A. Tetrahedral semiconductors
D. Debye temperature
D. Melting temperature
1. Introduction
The ternary chalcopyrite semiconductors crystallize in the
tetragonal structure (space group (I42d)) with four formula units
in each unit cell, which is a ternary analog of the diamond
structure. The chalcopyrite structure is essentially a superlattice of
the zincblende structure obtained by doubling its unit cube along
the Z -axis that becomes the c-axis of the chalcopyrite structure.
II IV V
Recently the AI BIII CVI
2 and A B C2 ternary chalcopyrites, listed in
Table 1, have received much attention because of their potential
applications in the fields of nonlinear optics, light emitting diodes,
laser diodes and solar cells. In spite of their wide technological
applications, the thermodynamical and optical properties of these
semiconductors have still not been sufficiently investigated. The
Debye temperature (ΘD ) is an important parameter of a solid.
It is frequently found in equations describing properties, which
arise from the vibrations of the atomic lattice (heat) and in
theories involving phonons. There has been a number of methods
to calculate the Debye temperature of these semiconductors. A
relation, which is often used, correlates the Debye temperature
∗
Corresponding author. Tel.: +91 0326 2296622; fax: +91 0326 2296622.
E-mail address: [email protected] (V. Kumar).
1 Visiting Fellow at Institute of Microwave and Photonics, School of Electronics
and Electrical Engineering, University of Leeds, LS2 9JT, UK during January and
February 2009.
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2009.04.003
and bulk modulus by a square-root law. The experimental values
of Debye temperature, usually derived from the specific heat
measurement at low temperature, have been reported in the
literature [1–3] but for a limited number of these compounds.
The values of ΘD have been calculated from microhardness [1],
melting point [4] and using elastic constants in place of the
bulk modulus for hexagonal and cubic crystals [5,6]. Later on
attempts have been made to develop empirical relations between
ΘD and compressibility, and ΘD and microhardness by Rincon
et al. [7–9] for a large number of these compounds. Linear relations
between ΘD and the mean atomic weight, and between the melting
point (Tm ) and mean atomic weight have been obtained by other
workers [10,11]. In almost all methods proposed earlier, a large
variation between the calculated and experimental values has been
obtained. This may be due to a large experimental uncertainty
of the results obtained by different workers for both the Debye
temperature and melting temperature. In this paper, we have given
a number of empirical relations for the calculation of ΘD and Tm of
ternary chalcopyrites, which fits experimental data well.
Recently, Kumar et al. [12–14] have developed various models based on plasma oscillation theory of solids for the calculation of bulk modulus, microhardness, and thermal expansion
coefficient, heat of formation, ionicity and bond length of semiconductors. The plasmon energy (h̄ωp ) is related to effective number
of electrons in a semiconductor. The bond length (d) is also related
to the number of valence electrons and hence h̄ωp . Phillips [15] and
Cohen [16] have shown that bulk modulus (B) is also correlated to
15.09
13.66
16.21
15.16
14.50
16.10
14.76
13.63
15.21
14.23
13.04
16.24
15.89
14.88
17.19
17.02
16.19
16.64
15.52
15.55
14.82
16.05
15.35
15.66
14.90
14.82
14.12
CuInSe2
CuInTe2
AgAlS2
AgAlSe2
AgAlTe2
AgGaS2
AgGaSe2
AgGaTe2
AgInS2
AgInSe2
AgInTe2
AgFeS2
CuTlS2
CuTlSe2
CuFeS2
AII BIV CV2
ZnSiP2
CdSiP2
ZnGeP2
CdGeP2
ZnSnP2
CdSnP2
ZnSiAs2
CdSiAs2
ZnGeAs2
CdGeAs2
ZnSnAs2
CdSnAs2
Ref. [6].
19.0287
21.0183
20.1056
22.1959
16.12
CuInS2
a
20.6229
21.7371
24.8672
17.7778
14.30
CuGaTe2
25.0588
21.4010
23.5469
22.4256
24.7589
25.0466
27.6664
20.9261
23.9747
29.4794
21.2796
24.4025
29.6777
23.7547
27.2044
32.4322
24.1793
29.5060
21.1619
26.9589
18.4881
21.5333
26.3380
18.8358
21.7185
17.25
15.86
14.35
17.10
15.92
AI BIII CVI
2
CuAlS2
CuAlSe2
CuAlTe2
CuGaS2
CuGaSe2
3
V
(10−30 m3 )
2
h̄ωp (eV)
[12,13]
1
Compds.
61.62
97.12
38.66
62.11
86.43
49.35
72.80
38.85
50.61
49.98
61.74
61.50
73.26
60.55
72.58
71.95
82.71
83.48
95.23
56.96
83.01
106.46
45.88
49.74
73.19
97.51
60.43
83.88
108.20
71.70
95.15
119.47
84.07
108.39
4
M
(10−3 kg)
1.2084
1.4208
1.2471
1.2766
1.1391
1.1319
1.0184
1.8630
1.5653
1.6109
1.3616
1.4761
1.2026
1.0058
1.7146
1.5859
1.2323
1.0491
1.4262
1.1190
0.9062
1.2479
1.0252
0.8509
1.1452
0.9283
1.4247
1.0190
1.9385
1.3850
1.0804
1.6590
1.2863
5
M −1/2 V 1/6 (h̄ωp )1.16666
(kg2/3 m5/3 × 10−26 )
Table 1
II IV V
Debye temperature and Melting temperature of AI BIII CVI
2 and A B C2 semiconductors.
Eq. (10)
10
Eq. (11)
11
229.9
251.9
331.3
428
340
434.4
155.9
182.4
255
221.9
191.4, 195.1
273.3
226.2
356
262
463, 540
376, 427
392, 420
304, 324
323, 352
298, 264
339, 386
289
302, 310
260, 253
258, 268
255, 221
245, 242
205, 89, 111, 135, 199
260, 316, 294, 356
311, 286, 313
241, 210, 237
191, 158, 181
282, 261, 215, 270, 259, 276
210, 225, 161, 156, 228
172,129, 125, 122, 172
242, 201, 238
186, 138
159, 113
284, 231, 221, 272, 264,
284 ± 10
225, 170, 172, 207, 219
185, 194, 129, 156, 174
202,190, 146, 177, 200
372, 308, 386, 375
277, 224, 272, 294
213, 303, 207
330, 347, 272, 340, 320, 338
259, 288, 195, 246, 239, 258
463.55
377.57
390.74
318.73
–
274.49
337.65
285.67
294.18
254.47
252.39
219.61
295.30
240.00
200.21
342.52
317.50
246.01
208.96
285.21
223.10
180.07
249.16
204.13
168.86
228.39
184.52
284.91
202.88
388.79
276.88
215.29
332.28
256.93
444.79
369.32
410.24
308.39
31.12
244.73
356.58
292.93
321.12
252.01
244.73
181.07
287.43
272.74
230.36
327.29
288.27
242.11
214.42
281.55
225.33
177.91
244.21
203.09
153.16
239.17
179.17
282.39
206.03
329.81
271.46
208.13
323.52
274.00
448.56
393.82
396.76
342.01
343.11
346.59
347.76
291.49
294.45
239.71
240.79
186.06
298.15
238.54
184.88
323.50
314.65
261.00
205.36
290.22
225.11
180.91
264.38
210.76
155.11
236.11
180.47
289.76
263.46
340.01
286.35
225.56
315.56
261.91
470.1
420.4
384.1
298.2
347.8
210.3
388.7
316.1
326.8
246.6
288.6
221.0
240.6
173.3
268.1
210.2
168.7
216.87
188.8
162.0/
158.2
252.8
192.1/
188.8
346.9
255.9
222.4
341.3
274.5/
268.1
216.8/
216.26
264.14
1523a
1393
1298
1073
1203
843
1310a
1120
1148
938
1048
871
1223
1002
1313
1123
987
1145
1053
965/
953
1263
1064/
1053
1570
1273
1163
1553
1334/
1313
1145/
1143
1300
12
7
Eq. (9)
9
Eq. (8)
8
6
1572.6
1302.8
1449.1
1085.0
1094.8
857.5
1269.9
1116.7
1184.6
1018.1
1000.6
847.4
1299.0
1174.9
1100.4
1280.9
1129.8
1002.3
1180.5
1069.9
935.7
1248.4
1040.9
1397.9
1133.7
1562.0
1358.7
1141.0
1540.2
1368.9
Eq. (12)
13
1637.7
1328.3
1344.9
1035.4
1041.6
1061.1
1293.9
1141.4
1149.4
1000.3
1003.3
854.2
1333.6
1196.7
1054.6
1271.2
1105.1
992.2
1205.3
1068.4
926.4
1217.9
1044.7
1384.8
1125.0
1541.2
1374.3
1185.1
1465.1
1298.2
Eq. (13)
14
Present work
Obs. [11]
Present work
Mes. from Sp. Ht.
at low temp. [8]
Reported [in Refs. [7,8]]
Melting temp. in (K)
Debye temp. (ΘD ) in (K)
V. Kumar et al. / Solid State Communications 149 (2009) 1008–1011
1009
1010
V. Kumar et al. / Solid State Communications 149 (2009) 1008–1011
the bond length as shown in Eqs. (2) and (3) of this paper. Further
it is well known that ΘD is related to bulk modulus and microhardness [1,8]. This shows that there must be a correlation between ΘD
and plasmon energy. In the present paper we propose four empirical relations to calculate the values of Debye temperature. Two are
based on the plasmon energy and one each on melting point and
molecular weight. We have also proposed two relations for the calculation of melting temperature of these semiconductors. One is
correlated with h̄ωp and other with W . For the melting point, we
have obtained a linear relation between Tm and h̄ωp as ΘD is linearly related to Tm . The calculated values of ΘD and Tm from all
equations are compared with the available experimental values
and the values reported by different workers. In each case, good
agreement has been obtained.
2. Calculation of Debye temperature
The relation between Debye temperature and compressibility
(χ ) was given many years ago by Madelung [17] and Einstein [18],
based on a simple lattice model:
√
θD α M −1/2 V 1/6 χ −1/2 α M −1/2 V 1/6 B
(1)
where M is the mean atomic weight per lattice site, V is the mean
atomic volume and B is the bulk modulus (B ≡ 1/χ ). In the case
of ternary chalcopyrites V = a2 c /16, where a and c are the lattice
parameters.
Based on the Phillips [15] model for binary tetrahedral
compounds sharing eight valence electrons per atom, Cohen [16]
has obtained the following relation between bulk modulus and
bond length:
B = 1761d−3.5
(d in Å, B in GPa).
(2)
The above equation is expected to be appropriate for group
IV, III–V, and II–VI materials in the diamond and zincblende
structures. The effect of ionicity has not been explicitly considered
by Cohen [16]; however in the Phillips [15] model, the average of
the ionicity contribution has been taken into account. Considering
the effect of ionicity, a more appropriate empirical relation has
been proposed by Cohen [16]:
B = (1971 − 220λ)d−3.5
(3)
where λ = 0, 1, and 2, respectively, for group IV, III–V and
II IV V
II–VI semiconductors. The AI BIII CVI
2 and A B C2 semiconductors
II VI
are the ternary analogues of the A B and AIII BV semiconductors,
respectively, and also exhibit tetrahedral coordination [19].
Therefore, it is quite reasonable to suppose that Eq. (3) can also
be used to describe the bulk modulus of ternary chalcopyrite.
This assumption has been taken by Kumar et al. [12] and other
workers [19] to calculate the average values of heat of formation
and thermal expansion coefficient of ternary chalcopyrites.
Recently, Kumar et al. [12] have given the following equation for
II IV V
the average bond length of AI BIII CVI
2 and A B C2 semiconductors:
d = 15.30(h̄ωp )−2/3
(d in Å, h̄ωp in eV)
(4)
where h̄ωp is the average plasmon energy of ABC2 compounds,
which can be calculated by using the following relation [20]
h̄ωp (eV) = 28.2 Z σ /W
p
(5)
where Z is the effective number of valence electrons taking part in
plasma oscillations, σ is the specific gravity and W is the molecular
II IV V
weight. Using Eq. (5), plasmon energies of AI BIII CVI
2 and A B C2
semiconductors have been calculated and listed in Table 1. The
calculated values of plasmon energies are in excellent agreement
with the values reported by Neumann [19] in the case of AI BIII CVI
2
while in the case of AII BIV CV2 semiconductors the authors have
reported these values in their previous publications [12,13]. The
details of the calculation of h̄ωp are given in our earlier publications
along with the values of W , which are required in Eq. (9) for the
calculation of ΘD [12].
Using Eqs. (3) and (4), we get following relation between B and
h̄ωp for ternary chalcopyrites
B = K1 (h̄ωp )2.3333
(B in GPa, h̄ωp in eV)
(6)
where K1 is the constant and equals to 0.109 and 0.125,
II IV V
respectively, for AI BIII CVI
2 and A B C2 semiconductors. The above
equation (6) is true for the calculation of bulk modulus too. The
calculated values of B from Eq. (6), which are not shown in the
present paper, are in reasonable agreement with the experimental
and reported values of B in the literature [9]. From Eqs. (6) and (1),
we get
θD α [M −1/2 V 1/6 (h̄ωp )1.1666 ].
(7)
−1/2
1/6
1.1666
We have calculated the values of parameter [M
V (h̄ωp )
]
and listed it in Table 1 along with available values of ΘD at 0 K
II IV V
derived from specific heat measurement for AI BIII CVI
2 and A B C2
semiconductors. To verify Eq. (7), linear regression between ΘD
(Column 6 of Table 1) and [M −1/2 V 1/6 (h̄ωp )1.1666 ] (Column 5 of Table 1) has been undertaken. The following straight line expression
for ΘD has been obtained:
θD = κ[M −1/2 V 1/6 (h̄ωp )1.1666 ] − Θ ;
II IV V
(I = 0.914 and 0.909 forAI BIII CVI
2 &A B C2 )
(8)
where κ and Θ are the empirical parameters, which can be
determined from the best-fit data of Debye temperature (ΘD ).
The values of parameters κ and Θ have been found to be,
respectively, 202.19 × 1026 (kg−2/3 M−5/3 K) and 3.15 (K) for
26
AI BIII CVI
(kg−2/3 M−5/3 K) and 74.52 (K) for
2 ; and 288.82 × 10
II IV V
A B C2 semiconductors. The index of determination (I) of Eq. (8) is
II IV V
0.914 and 0.909, respectively, for the AI BIII CVI
2 and A B C2 groups
of semiconductors. This shows that Eq. (6) gives the values of ΘD ,
which are 91.4% and 90.9% close to measured values of ΘD for
II IV V
AI BIII CVI
2 and A B C2 semiconductors, respectively.
The above equation (8) requires experimental data of lattice
constants a and c to calculate the values of V , which are
still not known for some of the ternary chalcopyrites given in
Table 1. Irrespective of this, it requires elaborate computation
for calculating the value of ΘD . For simplicity, we have therefore
simulated the data taking the measured values of ΘD , Tm and
W , and calculated values of h̄ωp using linear regression software.
We obtained the following linear relations between ΘD and other
parameters:
ΘD = −K2 + K3 (h̄ωp );
II IV V
(I = 0.920 and 0.929 for AI BIII CVI
2 and A B C2 )
ΘD = K4 − K5 (W );
II IV V
(I = 0.874 and 0.955 for AI BIII CVI
2 and A B C2 )
ΘD = −K6 + K7 (Tm );
II IV V
(I = 0.941 and 0.887 for AI BIII CVI
2 and A B C2 )
(9)
(10)
(11)
where K2 , K3 , K4 , K5 , K6 and K7 are the constants and their
numerical values are, respectively, 394.0, 41.96, 428.469, 0.572,
132.357 and 0.305 for AI BIII CVI
2 ; and 1103.19, 90.94, 629.45, 1.164,
111.713 and 0.382 for AII BIV CV2 semiconductors. The constants
K2 , K4 , K5 , and K6 are in Kelvin (K), and K3 in K(eV)−1 . The index of
determination is also shown for all the three equations. Similar to
Eq. (10), other workers have also [10,11] obtained linear relations
V. Kumar et al. / Solid State Communications 149 (2009) 1008–1011
between ΘD and mean atomic weight (M), and Tm and M in place of
molecular weight (W ) in the present case. It should be noted that
W is in the denominator of Eq. (5) for the plasmon energy. Using
Eqs. (8)–(11), we have calculated the values of ΘD for all chalcopyrite compounds and listed these in Table 1 along with measured
and reported values of these semiconductors [7,8] for comparison.
3. Calculation of melting point
The melting points of numerous chalcopyrites have been given
by Shay and Wernick [21] particularly for those chalcopyrites
whose ΘD or ΘD,XR (X-ray diffraction data of Debye temperature)
data are available. In Eq. (11), we have obtained a linear relation
between ΘD and Tm , which shows that there must be a linear
relation between Tm and h̄ωp as ΘD is proportional to h̄ωp (Eq. (9)).
Using regression software, simulation has been done between the
known values of Tm , W and h̄ωp , and the following new relations
have been obtained for Tm :
Tm = −K8 + K9 (h̄ωp )
.
(I
0 93 for Cu–BIII CVI
2
II IV
A B –P2 and AII BIV –As2 )
=
(12)
and
Ag–BIII CVI
2 ;
and 0.88 and 0.90 for
Tm = −K10 + K11 (W)
(13)
(I = 0.88 for
0.90 for
0.87 for A B –P2 and
0.99 for AII BIV –As2 ).
The values of constants K8 , K9 , K10 and K11 are respectively,
942.02, 145.16, 1816.50 and 1.78 for Cu–BIII CVI
2 ; 535.44, 112.82,
1624.12 and 1.46 for Ag–BIII CVI
;
3959.43,
325.03,
2660.27 and 6.58
2
for AII BIV –P2 ; and 2244.49, 218.97, 2061.76 and 3.17 for AII BIV –As2
groups of semiconductors. The constants K8 , K10 and K11 are in K
and K9 is in K(eV)−1 . A similar trend has also been obtained by
Nomura et al. [10] and Matsushita et al. [11] between Tm and M.
The index of determination of both (12) and (13) are also shown
above.
Cu–BIII CVI
2 ,
Ag–BIII CVI
2 ;
II IV
4. Conclusion
We have calculated the values of ΘD from Eqs. (8)–(11), and
II IV V
Tm from (12) and (13) for AI BIII CVI
2 and A B C2 semiconductors.
The calculated values are listed in Table 1 along with the
observed values and the values reported by different workers.
Fairly good agreement has been obtained between the calculated,
observed and reported values. In most of the cases, the index
of determination is more than 0.9 and in the worst case, it is
0.874 for some of the calculated values. The later may be due to
uncertainties in the measurement of specific heat data and hence
Debye temperature. It is important to note that earlier workers
[1,2] have used a single proportionality factor in Eq. (1), which has
not been sufficient to yield acceptable values of Θm and Tm . In the
present calculation, we have taken two proportionality factors in
Eqs. (8) to (13), which give better fit to the measured values.
1011
The main advantage of the present model is the simplicity
of the formula, which does not require any experimental data
except the plasmon energy while the earlier models required
the experimental values of specific heat and lattice parameters
a and c of these semiconductors for calculations. Hence it is
possible to predict the values of Debye temperature and melting
temperature of unknown compounds belonging to these groups
of semiconductors from their plasmon energies and molecular
weight.
Acknowledgements
The authors are thankful to Department of Science and
Technology, Government of India for a supporting fund to study
the various Linear and Non-linear Properties of Opto-electronic
Materials. The authors are also thankful to Prof. T. Kumar, Director,
Indian School of Mines University, Dhanbad, for his continuous
encouragement and inspiration in conducting this work. The first
author is grateful to Prof. E. H. Linfield, Director, Institute of
Microwave and Photonics, School of Electronics and Electrical
Engineering, University of Leeds, LS2 9JT, UK for providing facilities
to complete a part of this work as a Visiting Fellow during January
and February 2009. We would also like to thank Prof. Linfield for
his valuable suggestions and help in correcting the English of this
paper.
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