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Cryst. Res. Technol. 45, No. 9, 920 – 924 (2010) / DOI 10.1002/crat.201000268 © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Debye temperature and melting point of II-VI and III-V semiconductors V. Kumar*, V. Jha, and A. K. Shrivastava Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad 826 004, India Received 10 May 2010, revised 9 June 2010, accepted 12 June 2010 Published online 2 July 2010 Key words Debye temperature, melting point, II-VI and III-V zincblende semiconductors, binary tetrahedral semiconductors. In this paper, simple relations are proposed for the calculation of Debye temperature θ D and melting point T m of II-VI and III-V zincblende semiconductors. Six relations are proposed to calculate the value of θ D . Out of these six relations, two are based on plasmon energy data and the others on molecular weight, melting point, ionicity and energy gap. Three simple relations are proposed to calculate the value of T m . They are based on plasmon energy, molecular weight and ionicity of the semiconductors. The average percentage deviation of all nine equations was calculated. In all cases, except one, it was estimated between 3.34 to 17.42 % for θ D and between 2.37 to 10.45 % for T m . However, in earlier correlations, it was reported between 10.59 to 33.38% for θ D and 6.96 to 14.95% for T m . The lower percentage deviation shows a significant improvement over the empirical relations proposed by earlier workers. The calculated values of θ D and T m from all equations are in good agreement with the available experimental values and the values reported by different workers. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In the recent years, a considerable amount of experimental and theoretical work has been done at the understanding of thermal properties of II-VI and III-V groups of tetrahedral semiconductors. Comprehensive, up-to-date information of band parameters for III-V zincblende compounds have been given by Vurgaftman et al. [1]. Various electronic and optical properties of II-VI semiconductors [2] and their applications in optoelectronic devices [3,4], quantum wells [5], quantum wires [6], quantum dots [7], lasing devices, 980 nm pump lasers [8] and detectors for nuclear power plant [9] have been reported in the literature [2-9]. In spite of their potential applications, the Debye temperature and melting point of these semiconductors have still not been sufficiently studied. The Debye temperature θ D is an important parameter in thermal phenomenon such as specific heat, thermal conductivity and other properties which arise from the vibrations of the atomic lattice. Knowledge of θ D also helps us in selecting the materials for substrate for epitaxial growth and fabrication of integrated circuit (IC). The information about the experimental values of θ D is usually obtained from the low temperature specific heat measurement [10-12] and elastic constant data [13]. There have been number of theories to calculate the Debye temperature of II-VI and III-V semiconductors. The most popular relation between the Debye temperature and bulk modulus is followed by a square-root law [14]. The values of θ D have been calculated from melting point, compressibility and microhardness data [10,15-18]. Linear relations between θ D and number of parameters like mean atomic weight M, ionicity f i , melting temperature T m and energy gap E g have been obtained by Nomura et al. [19] and other researchers [20,21]. Nomura et al. [19] have also obtained a linear relation between T m and M for ternary semiconductors. Van Vechten [22] has predicted melting point of II-VI and III-V crystals using scaling theory. An artificial neural network method based on cellular model has been investigated to predict the melting point of A-B type intermetalic compounds [23]. In almost all methods proposed earlier, a large variation between the calculated and experimental values has been obtained for a limited number of semiconductors. Recently, Kumar et al. [24-26] have developed various correlations based on plasma ____________________ * Corresponding author: e-mail: [email protected]

Debye temperature and melting point of II-VI and III-V semiconductors

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Cryst. Res. Technol. 45, No. 9, 920 – 924 (2010) / DOI 10.1002/crat.201000268

© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Debye temperature and melting point of II-VI and III-V

semiconductors

V. Kumar*, V. Jha, and A. K. Shrivastava

Department of Electronics and Instrumentation, Indian School of Mines, Dhanbad 826 004, India

Received 10 May 2010, revised 9 June 2010, accepted 12 June 2010

Published online 2 July 2010

Key words Debye temperature, melting point, II-VI and III-V zincblende semiconductors, binary

tetrahedral semiconductors.

In this paper, simple relations are proposed for the calculation of Debye temperature θD and melting point Tm

of II-VI and III-V zincblende semiconductors. Six relations are proposed to calculate the value of θD. Out of

these six relations, two are based on plasmon energy data and the others on molecular weight, melting point,

ionicity and energy gap. Three simple relations are proposed to calculate the value of Tm. They are based on

plasmon energy, molecular weight and ionicity of the semiconductors. The average percentage deviation of

all nine equations was calculated. In all cases, except one, it was estimated between 3.34 to 17.42 % for θD

and between 2.37 to 10.45 % for Tm. However, in earlier correlations, it was reported between 10.59 to

33.38% for θD and 6.96 to 14.95% for Tm. The lower percentage deviation shows a significant improvement

over the empirical relations proposed by earlier workers. The calculated values of θD and Tm from all

equations are in good agreement with the available experimental values and the values reported by different

workers.

© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In the recent years, a considerable amount of experimental and theoretical work has been done at the

understanding of thermal properties of II-VI and III-V groups of tetrahedral semiconductors. Comprehensive,

up-to-date information of band parameters for III-V zincblende compounds have been given by Vurgaftman et

al. [1]. Various electronic and optical properties of II-VI semiconductors [2] and their applications in

optoelectronic devices [3,4], quantum wells [5], quantum wires [6], quantum dots [7], lasing devices, 980 nm

pump lasers [8] and detectors for nuclear power plant [9] have been reported in the literature [2-9]. In spite of

their potential applications, the Debye temperature and melting point of these semiconductors have still not

been sufficiently studied.

The Debye temperature θD is an important parameter in thermal phenomenon such as specific heat, thermal

conductivity and other properties which arise from the vibrations of the atomic lattice. Knowledge of θD also

helps us in selecting the materials for substrate for epitaxial growth and fabrication of integrated circuit (IC).

The information about the experimental values of θD is usually obtained from the low temperature specific heat

measurement [10-12] and elastic constant data [13]. There have been number of theories to calculate the Debye

temperature of II-VI and III-V semiconductors. The most popular relation between the Debye temperature and

bulk modulus is followed by a square-root law [14]. The values of θD have been calculated from melting point,

compressibility and microhardness data [10,15-18]. Linear relations between θD and number of parameters like

mean atomic weight M, ionicity fi, melting temperature Tm and energy gap Eg have been obtained by Nomura et

al. [19] and other researchers [20,21]. Nomura et al. [19] have also obtained a linear relation between Tm and M

for ternary semiconductors. Van Vechten [22] has predicted melting point of II-VI and III-V crystals using

scaling theory. An artificial neural network method based on cellular model has been investigated to predict the

melting point of A-B type intermetalic compounds [23]. In almost all methods proposed earlier, a large

variation between the calculated and experimental values has been obtained for a limited number of

semiconductors. Recently, Kumar et al. [24-26] have developed various correlations based on plasma ____________________

* Corresponding author: e-mail: [email protected]

Cryst. Res. Technol. 45, No. 9 (2010) 921

www.crt-journal.org © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

oscillations theory of solids for the calculation of refractive index, thermal expansion coefficient and heat of

formation of binary and ternary semiconductors. In the present paper, we propose simple relations for the

calculation of θD and Tm for II-VI and III-V zincblende semiconductors.

2 Calculation of Debye temperature

Based on a simple lattice model, relation between Debye temperature θD and bulk modulus B can be written as

[27,28]: 1/ 61/ 2 1/ 2

DM V B

θ α , (1)

where M is the mean atomic weight per lattice site, V is the mean atomic volume, B is the bulk modulus (B =

1/χ) and χ is the compressibility. In the case of binary zincblende crystals, V = a3/18, where a is the lattice

parameter. Eq. (1) can be written in terms of plasmon energy ћωp for AIBIIIC2VI and AIIBIVC2

V chalcopyrite

semiconductors as [29]:

( )1/ 61.1666

1/ 2

D pM V−⎡ ⎤θ α ω⎣ ⎦� . (2)

The AIBIIIC2VI and AIIBIVC2

V semiconductors are ternary analog of AIIBVI and AIIIBV binary semiconductors,

respectively. It is quite reasonable to suppose that eq. (2) can also be used to describe the θD of binary

semiconductors. To verify eq. (2), linear regression between θD and [M−1/2 V1/6(ћωp)1.1666] has been done using

regression software for binary semiconductors. The following relation between θD and ћωp has been obtained

for AIIBVI and AIIIBV semiconductors.

( )1/ 61.16661/ 2

D pM V−⎡ ⎤θ = κ −Θω⎣ ⎦� , (3)

where κ and Θ are the empirical parameters, which have been determined from the best-fit of the experimental

data of θD. The values of proportionality constants κ and Θ have been found to be, respectively, 547.17 × 1026

Kg−2/3⋅M−5/3

⋅K and 261.24 K for AIIBVI, and 384.56 × 1026 Kg−2/3⋅M−5/3

⋅K and 82.73 K for AIIIBV

semiconductors. The calculated values of θD from eq. (3) are listed in table 1 and plot between θD and [M-1/2

V1/6 (ћωp)1.1666] is shown in figure 1.

Fig. 1 Plot between Debye temperature and

[M-1/2 V1/6 (ħωp)1.1666] of III-V semicon-

ductors.

The values of ћωp in eq. (3) can be calculated using the following relation [30]

28.8 / ( )p

Z W eVω = σ� , (4)

where Z is the effective number of valence electrons taking part in plasma oscillations, σ is the specific gravity

and W is the molecular weight. Using eq. (4), plasmon energies of AIIBVI and AIIIBV semiconductors have been

calculated and listed in table 1 [31].

Eq. (3) requires experimental data of lattice constants to calculate the values of V, which may not be known

for some new compounds. In spite of this, it requires elaborate computation for calculating the value of θD. For

simplicity, we have therefore simulated the data taking the known values of θD, Tm and W, and the calculated

values of ћωp, fi and Eg using linear regression software. We obtained the following relations between θD and

other parameters for II-VI and III-V semiconductors:

θD = − K1 + K2 (ћωp), (5)

θD = K3 – K4W, (6)

θD = −K5 + K6Tm, (7)

922 V. Kumar et al.: Debye temperature and melting point of II-VI and III-V semiconductors

© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.crt-journal.org

where K1, K2, K3, K4, K5 and K6 are the constants and their numerical values are, respectively, 1164.54, 98.70,

711.72, 2.42, 537.21 and 0.50 for AIIBVI, and 800.88, 77.48, 864.82, 3.31, 153.40 and 0.354 for AIIIBV

semiconductors.

Table 1 Debye temperature of II-VI and III-V zincblende semiconductors. (* D. E. Jonckheere and T. Eric, Defense Tech.

Infor. Centre, Rep No-GNE/PH/69-5 (JUNE 1969), # U. Rossler (ED), II-VI and I-VII Compounds; Semimagnetic

Compounds, (Springer-Verlag, 1999) p.1-3.)

Debye Temp ӨD (K)

Reported Present work

Comp

d.

ħωp

(eV)

[31]

V

(10−30 m3)

M

(10−3 Kg)

[M−1/2V1/6

((ћωp)1.1666]

(Kg2/3m5/3

×10−26)

W

[32]

fi

[31]

Eg

(eV)

[31] Expt.

[32] [33] [34] Eq.(3) Eq.(5) Eq.(6) Eq.(7) Eq.(8) Eq.(9)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

II-VI - -

BeS 19.52 6.3969 20.54 2.5064 41.08 0.656 10.62 - - 951 1110.19 762.13 612.11 - 805.75 794.64

BeSe 18.39 7.5398 43.99 1.6420 87.97 0.661 9.69 - - 850 637.21 650.6 498.42 - 686.13 668.13

BeTe 16.12 9.8929 68.31 1.1822 136.61 0.672 7.92 - - 766 385.62 426.54 380.48 - 422.95 427.35

ZnS 16.71 8.7932 48.72 1.4341 97.43 0.669 8.37 530 440 529 523.46 484.78 475.48 512.94 494.73 488.56

ZnSe 15.78 10.1140 72.17 1.1260 144.34 0.673 7.67 400,260* 340 364 354.88 392.98 361.74 357.92 399.03 393.34

ZnTe 14.76 12.6162 96.5 0.9345 192.99 0.678 6.91 223 260 317 250.10 292.31 243.78 246.91 279.41 289.95

CdS 14.88 11.0199 72.23 1.0662 144.46 0.677 7.00 219,286# 280 445 322.15 304.15 361.45 337.92 303.33 302.20

CdSe 14.01 12.3025 95.68 0.8794 191.36 0.681 6.37 181 300 302 219.95 218.28 247.73 218.90 207.63 216.49

CdTe 13.09 15.0955 120.0 0.7506 240.00 0.684 5.72 200 160 253 149.47 127.48 129.80 145.39 135.86 129.44

Average % Dev 22.73 33.38 14.01 17.42 21.25 15.13 14.16 16.67

III-V

BP 21.71 5.1918 20.87 2.7187 41.78 0.312 8.95 980 - 1096 962.78 881.19 726.50 837.77 893.95 895.62

BAs 20.12 6.0561 42.87 1.7809 85.73 0.320 7.94 625 - 956 602.14 757.99 580.99 660.77 757.38 758.52

AlP 16.65 8.9982 28.98 1.8554 57.95 0.336 5.88 588 - 668 630.79 489.15 672.96 589.97 484.23 478.89

AlAs 15.75 10.0851 50.95 1.3365 101.90 0.340 5.38 417 417 549 431.24 419.42 527.45 559.18 415.94 411.02

AlSb 13.72 12.8314 74.37 0.9804 148.73 0.349 4.31 292 380 482 294.29 262.13 372.41 317.40 262.29 265.78

GaP 16.50 8.9957 50.35 1.3927 100.69 0.337 5.79 446 500 563 452.85 477.52 531.46 466.08 467.15 466.68

GaAs 15.35 10.0368 72.32 1.0878 144.64 0.342 5.16 344 360 404 335.56 388.42 385.95 381.12 381.79 381.16

GaSb 13.38 12.5815 95.74 0.8363 191.47 0.350 4.14 265 266 331 238.88 235.78 230.90 193.51 245.22 242.71

InP 14.76 11.2296 72.90 1.0546 145.79 0.345 4.85 321 420 495 322.83 342.70 382.14 317.40 330.58 339.08

InAs 14.07 12.3536 94.87 0.8883 189.74 0.347 4.49 249 262 328 258.88 289.25 236.63 276.69 296.44 290.22

InSb 12.73 15.1079 118.29 0.7319 236.57 0.353 3.82 202 205 268 198.77 185.42 81.58 129.08 194.01 199.27

Average % Dev 10.59 32.93 3.34 11.00 20.83 13.99 9.74 9.66

The constants K1 and K3 to K6 are in Kelvin, and K2 is in K⋅(eV)−1. Other workers [19,20] have also obtained

similar linear relations between θD and mean atomic weight M in place of molecular weight W in the present

case, i.e. eq. (6). It should be noted that W is in the denominator of eq. (4) for the plasmon energy. Further, we

analysed the data between the known values of θD and fi [31] and obtained the following relation:

θD = K7 – K8 fi , (8)

where K7 and K8 are the constants. The numerical values of K7 and K8 are, respectively, 16500.33 and 23924.66

for AIIBVI, and 6220.35 and 17071.8 for AIIIBV semiconductors. Eq. (8) is similar to the equation proposed by

Rincon [17] for ternary semiconductors.

Nomura et al. [19] have obtained a linear relation between Eg and M. Recently, Kumar et al. [29] proposed

a linear relation between θD and W, which shows that there must be a linear relation between θD and Eg as M

and W are similar parameters. Based on this, we analyzed θD and obtained the following relation between θD

and Eg:

θD = − K9 + K10 Eg . (9)

The values of the constants K9 and K10 are, respectively, 650.03 and 136.03 for AIIBVI, and 319.25 and 135.74

for AIIIBV semiconductors. Using eqs. (3), (5-9), the values of θD have been calculated for all binary

compounds and listed in table 1 along with the experimental values and values reported by other workers [32-

34].

3 Calculation of melting point

A linear relation between θD and Tm (Eq. 7) shows that there must also be a linear relationship between Tm and

ћωp as θD is proportional to plasmon energy. We have analyzed the experimental data of Tm with ћωp, and Tm

with W and fi , and obtained the following relations for Tm:

Tm = −K11 + K12(ћωp), (10)

Tm = K13 – K14W, (11)

Tm = K15 – K16fi . (12)

The values of the constants K11, K12, K13, K14, K15, and K16 are, respectively, 1228.25, 195.61, 2537.32, 5.08,

33161.33 and 46500 for AIIBVI, and 1604.02, 204.70, 2910.48, 9.61, 16820.88 and 44734.3 for AIIIBV

Cryst. Res. Technol. 45, No. 9 (2010) 923

www.crt-journal.org © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

semiconductors. The constants K9, K11, K13, K14, K15 and K16 are in K, whereas K10 and K12 are in K⋅(eV)−1.

Similar to eq. (11), other workers [20,21] have also obtained linear relations between Tm and M in place of W

in the present case. The calculated values of Tm from eqs. (10-12) are listed in table 2 along with the

experimental and reported values. The curves, Tm vs W, and Tm vs fi are also shown in figure 2 and 3,

respectively.

Fig. 2 Plot of melting point and molecular weight of II-VI

semiconductors.

Fig. 3 Plot of melting point and ionicity of II-VI semi-

conductors.

Table 2 Melting point of II-VI and III-V zincblende semiconductors.

Melting Point Tm (K)

Present work

Compd. ħωp

(eV)

[31]

V

(10−30 m3)

M

(10−3

Kg)

[M−1/2V1/6

(ћωp)1.1666 ]

(Kg2/3m5/3 ×10−26)

Expt.

[32]

Reported

[22] Eq.(10) Eq.(11) Eq.(12)

1 2 3 4 5 6 7 8 9 10

II-VI

BeS 19.52 6.3969 20.54 2.5064 - 2590.10 2328.42 2657.33

BeSe 18.39 7.5398 43.99 1.6420 - 2369.06 2089.98 2424.83

BeTe 16.12 9.8929 68.31 1.1822 - 1452 1925.02 1842.64 1913.33

ZnS 16.71 8.7932 48.72 1.4341 2100 2294 2040.43 2041.87 2052.83

ZnSe 15.78 10.1140 72.17 1.1260 1790 2059 1858.51 1803.33 1866.83

ZnTe 14.76 12.6162 96.5 0.9345 1568 1657 1658.99 1555.94 1634.33

CdS 14.88 11.0199 72.23 1.0662 1750 2077 1682.46 1802.72 1680.83

CdSe 14.01 12.3025 95.68 0.8794 1512 1835 1512.28 1564.23 1494.83

CdTe 13.09 15.0955 120.0 0.7506 1365 1631 1332.31 1316.89 1355.33

Average % Dev 14.9 3.11 2.37 2.75

III-V

BP 21.71 5.1918 20.87 2.7187 2800 2250 2840.13 2508.89 2863.77

BAs 20.12 6.0561 42.87 1.7809 2300 2514.65 2086.44 2505.90

AlP 16.65 8.9982 28.98 1.8554 2100 1853 1804.32 2353.46 1790.15

AlAs 15.75 10.0851 50.95 1.3365 2013 1620.09 1931.01 1611.22

AlSb 13.72 12.8314 74.37 0.9804 1330 1344 1204.54 1480.88 1208.61

GaP 16.50 8.9957 50.35 1.3927 1750 1623 1773.62 1942.64 1745.42

GaAs 15.35 10.0368 72.32 1.0878 1510 1538.21 1520.19 1521.75

GaSb 13.38 12.5815 95.74 0.8363 980 983 1134.93 1070.06 1163.88

InP 14.76 11.2296 72.90 1.0546 1330 1357 1417.43 1509.14 1387.55

InAs 14.07 12.3536 94.87 0.8883 1215 1276.18 1086.69 1298.08

InSb 12.73 15.1079 118.29 0.7319 798 745 1001.88 636.55 1029.67

Average % Dev 6.96 9.99 10.25 10.45

4 Conclusions

The values of Debye temperature have been calculated from eqs. (3), (5−9) for AIIBVI and AIIIBV

semiconductors and listed in table 1 along with the available experimental and reported values. The average

percentage deviation were estimated for all of the six equations using the relation: Percentage deviation =

100 × |(θDexp - θDcal)|/ θDexp, and are presented in table 1. The maximum average deviation of 21.25 % was

obtained in the case of eq. (6), but in other cases it was estimated between 3.34 to 17.42% in comparison with

10.59 to 33.38% estimated from the empirical relations proposed by earlier workers [33,34]. The values of

melting point were calculated using eq. (10−12) and are listed in table 2. The average percentage deviation for

Tm was also estimated and is listed in table 2, which is between 2.37 to 10.45% in contrast to 6.96 to 14.9%

estimated by the earlier correlations [22]. The graphical representation of eqs. (3), (11) and (12) are also shown

924 V. Kumar et al.: Debye temperature and melting point of II-VI and III-V semiconductors

© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.crt-journal.org

in figures 1-3, respectively along with the experimental and reported data. The lower percentage deviation

indicates the soundness of the proposed correlations. Thus, the present study show that all the six correlations

proposed for θD and the three for Tm give more accurate results than those correlations proposed by earlier

workers.

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 Optical 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.

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