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Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2
Author’s Accepted Manuscript Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2 Wei Huang, Beijun Zhao, Shifu Zhu, Zhiyu He, Baojun Chen, Zhen Zhen, Yunxiao Pu, Weijia Liu www.elsevier.com/locate/jcrysgro
PII: DOI: Reference:
S0022-0248(16)30070-7 http://dx.doi.org/10.1016/j.jcrysgro.2016.02.037 CRYS23224
To appear in: Journal of Crystal Growth Received date: 27 October 2015 Revised date: 14 February 2016 Accepted date: 29 February 2016 Cite this article as: Wei Huang, Beijun Zhao, Shifu Zhu, Zhiyu He, Baojun Chen, Zhen Zhen, Yunxiao Pu and Weijia Liu, Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2, Journal of Crystal Growth, http://dx.doi.org/10.1016/j.jcrysgro.2016.02.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2 Wei Huang, Beijun Zhao*, Shifu Zhu, Zhiyu He, Baojun Chen, Zhen Zhen, Yunxiao Pu, Weijia Liu College of Materials Science and Engineering, Sichuan University, Chengdu 610064, China Abstract: Chalcopyrite of CdGeAs2 single crystal was grown by a modified vertical Bridgman method with sufficient size and quality, and its optical, electrical and thermodynamic properties are characterized. The transmission is recorded in the 2.3-18 μm range, and the band-gap at room temperature is at 0.56 eV. Non-ideal transparency near 5.5 μm which limited its application severely exists in the front of the crystal. The crystal is p type at room temperature with hole concentrations varying from 1014 to 1016 cm−3. From the results of X-ray diffraction measurements carried out over the range 25~450 ℃ and thermal dilatometer tests, the thermal expansion coefficients are evaluated. And on this basis the Grüneisen parameters at different temperatures are evaluated and also exhibit anisotropic behavior (γa > γc). It is found that γa, γc, and γV have some difference between these two kinds of test methods. Using these Grüneisen parameters, lattice thermal conductivities have been deduced by two correction formulas. Meanwhile, specific heat capacity and thermal conductivity of [204] have been obtained as a function of temperature by experiment.
Keywords: CdGeAs2 single crystals; Vertical Bridgman technique; thermal expansion coefficients; Grüneisen parameters; thermal conductivities.
1. Introduction
*
Corresponding author. Fax: +86 28 85412745. E-mail address: [email protected] 1
Ternary compounds with chalcopyrite structure, which were first discovered by Hahn et al.[1] in 1953, have received a great of attentions because of their applications in optical devices, detectors and solar cells[2, 3]. Cadmium germanium arsenide (CdGeAs2) is an ideal candidate material for nonlinear optical (NLO) applications, since it has the highest NLO coefficient (d36 = 236 pm/V) [4, 5]. Its wide transparency range (2.3-18 μm), accompanying by significant birefringence, makes this material very promising in frequency conversion applications in the infrared range [6]. Since the 70s, researchers have struggled to grow large and high quality single crystals of this compound, but they were plagued by polycrystallinity and cracking [7-13] due to highly anisotropic thermal expansion coefficients [14]. Therefore, the investigation of anisotropic thermal expansion coefficients has great significance for the improvement of the CdGeAs2 crystal growth. Newmann [15] and Kumar et al. [16] have evaluated the average linear thermal expansion coefficient αL based respectively on empirical and modified empirical relation from first principles. Kildal [17] and Kozhina et al.[18] reported the experimental average thermal expansion coefficients in two main directions (namely, αa and αc) in the range 100~300 ℃ and 20~400 ℃, respectively. However, in many cases, the results seem inadequate in practice, and it is expected to reveal thermal expansion mechanism in specific directions and at specific temperatures which is conducive to the crystal growth. With the development of horizontal gradient freeze method [19, 20] and modified vertical Bridgman method [21, 22], single crystals of this ternary chalcopyrite have been grown without severe cracking. Meanwhile, up to now, structural [10, 23, 24], electronic [2, 25] and optical properties [6, 8, 26] of CdGeAs2 have been studied for the last four decades by a number of experimental and theoretical methods. However, the thermal conductivity of the I-III-VI2 or II-VI-V2 chalcopyrite crystal, which is an important thermodynamic parameters in the aspect of laser application, were usually discussed at ambient temperature or under ambient temperature in the literature, and the derivation formulas of thermal conductivity were different [27-30] each other. To achieve high power laser output, the investigation of the thermal conductivity is indispensable. 2
Herein, we address the interesting about the CdGeAs2 crystal growth by the modified VB method as well as optical, electrical and thermodynamics characterization, and have deeply discussed the properties of anisotropic thermal expansion, heat capacity and thermal conductivity.
2. Experimental Section 2.1. Crystal growth The starting material was synthesized from high purity 6 N Cd spheroids, 6 N Ge and 6 N As lumps about 120 g at the molar ratio of 1:1.005:2.01. After weighing in a glove box, the charges were loaded into a quartz ampoule and sealed under 10-5 Pa at once. Then the ampoule was put into a rotatable tube furnace and the synthesis was mainly carried out with such steps: 50 ℃/h to 650 ℃, 8-h soak and 20 ℃/h to 950 ℃,24-h soak. In order to make the raw materials mix uniformly and drive the residual Cd and As vapor out of melt, the furnace rotating and the temperature oscillation (between 800 to 1040 ℃) have been carried out several times [31, 32]. When the synthesis process finished, the furnace were cooled down to room temperature at the speed of 20 ℃/h. The polycrystalline material was taken out of the synthesis ampoule and ground into powder. Meanwhile, the polycrystalline XRD and EDS analysis demonstrated that there is no impurity phases in the powder. Then the polycrystalline powder was reloaded into a carbon-coated quartz ampoules. After being evacuated to 10-5 Pa, the polycrystalline powder was sealed into the quartz ampoule. Then the quartz ampoule was placed into a three-zone furnace. The temperatures of the upper, middle and lower zones were raised to the special temperatures at the rate of 12 ℃ /min, respectively. After a lot of growth runs, we find that if the temperature of the upper zone is higher than 720 ℃, the obtained products are most polycrystal. But if the temperature of the upper zone is too low, the appropriate temperature gradients is hard to achieve. A suitable temperature gradient not only can provide a sufficient nucleation driving force, but also restrain the formation of polycrystalline boules. Therefore, we have carried out a lot of growth runs with the temperature gradients of 3
10-35 ℃. At last, temperature gradients of 15-25 ℃/cm are adopted, which is different from the small temperature gradients (1-3℃/cm) in the horizontal gradient freeze method [20] and is smaller than that reported in the early vertical Bridgman method (40 ℃/cm) [9]. The growth position of the ampoule was adjusted to the place where the indication of the monitoring thermocouple was about 670 °C. The entire charge was heated above the melting point. Over 100°C of supercooling (down to 547°C) took place before the onset of crystallization [20]. So when the material in the seed well solidified, the ampoule was pulled back to the position where the temperature is about 650 °C. Then the material in the seed well was reheated to partially melt back and the growth ampoule was mechanically descended at the rate of 0.21-0.25 mm/h. After the growth process finished, the temperature was cooled down to room temperature at 20 ℃/h. Thus, CdGeAs2 single crystals could be obtained reproducibly, which is shown in Figure .1.
2.2. Characterization fThe optical transmittance spectra o as-grown CdGeAs2 wafers were recorded using a Shimadzu fourier-transform-infrared (FTIR) spectrometer in the range of 2.3 μm~18 μm. The samples are approximately 2 mm thick which were polished smoothly. Electrical properties were obtained using Ecopia HMS-300 Hall-effect measurement system. Temperature could be controlled at 77 and 300 K, and the measurements were carried out at a magnetic field of 0.554 T. A circular diamond saw was used to cut the as-grown boule, verifying the orientation of the surface by XRD. The CdGeAs2 crystals were carefully examined by the powder XRD (DX-2000, Dandong, China) method at room temperature. The diffraction patterns were recorded over the 2θ range of 10-90°. The high temperature XRD measurements were performed in a DX-2700 diffractometer with Cu Kα radiation, and Graphite monochromator was adopted to select single wavelength of about 1.54 Å. The temperature stages for X-ray diffraction measurements were determined at 25, 100, 200, 300, 400, and 450 ℃, respectively. Prior to thermal conductivities measurements, a series of thermogravimetry and differential scanning calorimetry (TG-DSC) 4
measurements in the shielding nitrogen ambient were carried out in a SDT Q600 apparatus (TA Instruments, USA) for heat capacity, and the sample size is Ф5 mm×1 mm. After careful calibration of the calorimeter and using Al2O3, as standard the heat capacities could be measured with a high-accuracy. Thermal conductivities were determined at 26.6, 100.6, 200.4, 300.4 and 400.3 ℃ by NETZSCH LFA 457 which adopt the InSb detector and Ar protective atmosphere with 50.00 ml/min. The diameter and thickness of test sample were 12.7 mm and 1.8 mm, respectively. Due to the size limitation, thermal conductivity experimental only with [204] direction could be obtained. The [204] direction is just the growth direction in this VB method so that the sample has a large enough diameter.
3. Results and Discussion 3.1 Optical and electrical properties The IR transmittance of a CdGeAs2 wafer is shown in Figure. 2 and the maximum value is up to 51.6% in the range of 2.3-18 μm. Absorption coefficients (α) were estimated from the transmission spectrum of CdGeAs2 single crystals. The inset shows the value of (αhν)2 vs photon energy at room temperature. The useful transmission range is limited by the strong absorption below 2.3 μm due to transitions across the band-gap and by two-phonon absorption at 12.5 and 13.5 μm, which are present in all of samples, are probably due to three-phonon processes. The absorption coefficients at 5.5 μm is close to the lowest optical losses (0.18 cm-1) measured by K.
Nagashio [33]. The broad band peaking near 5.5 μm has been ascribed to transitions between a split-off valence band and the highest valence band at k=0 [2]. Since the EPR hyperfine interactions suggested the acceptor was located on the As site. Lihua Bai [34] believed the shallow acceptor was a Group IV atom on the Group V site which was most likely the Ge-on-As antisite defect. The intensity of this absorption band decreases with decreasing hole concentration [17] as well. Therefore, different part of the crystal with different hole concentration have different degree of absorption. That is why there exists difference between curve 1 and curve 2 in the Figure .2, which describe the optical transmission and absorption located the center 5
part of the crystal and shoulder position. The Hall measurement data indicate that the as-grow crystal samples are consistently p-type with hole concentration varying from 107 to 1010 cm−3 and mobility about 102 cm2/Vs at 77 K, as shown in Table I. At 300 K, the hole concentration decreases and the range is from 1014 to 1016 cm−3. But the resistivity is 1.298~74.44 Ω·cm at room temperature which is geared to a small order of magnitude. At 300 K, the resistivity increases and the range is from 105 to 108 Ω·cm. Compared with the values of early reports, the hole concentration measured in this work is smaller and the resistivity is larger, which is shown in Table .I. The actual hole concentration for samples with low absorption at 5.5 μm can be calculated by an equation proposed by Bai et al [26]. Reported values of the mobility ratio b for CdGeAs2 are about 12[35, 36] and 30[17] and two lines which describe the relation of hole concentration and absorption coefficient are linear least-squares fit to the data points yielding the relation p = (0.93+4.62α)×1015 cm−3 and p = (1.06+4.59α)×1015 cm−3 which is close to the relation obtained by Bai et al[26]. Table .I Electrical properties of the samples This work
Electrical properties
Ref. [37]
300 K
77 K
Bulk carrier
8.819×1014~
4.276×107~5.
concentration（cm-3）
1.977×1016
366×1010
Mobility （cm2/Vs）
13.66~402.2
17.78~895.8
Ref.[17]
300 K
77 k
300 K
77 k
0.4×1016
-
1016~1017
-
181
-
12~30
-
9
1.2×104
1~10
102~105
-
-
102
107
5
2.175×10 ~1.
Resistivity （Ω·cm）
1.298~74.44
Average Hall
3.158×102~
1.163×108~1.
coefficient（cm3/C）
7.078×103
460×1011
663×108
3.2 Determination of thermal expansion coefficients In our previous work, in the range of 25–450 ℃, X-ray powder diffraction measurements of CdGeAs2 crystal have been carried out[38]. XRD patterns collected at different temperatures are presented in Figure. 3 along with the PDF No. 73-0402. Accurate lattice parameters including two lattice constants a and c, cell volume V, axial ratio η, and tetragonal distortion μ are determined. And the average thermal expansion coefficients are evaluated, 13.9×10-6 K-1 for αa and 2.8×10-6 K-1 for αc, 6
respectively. Meanwhile, the Grüneisen parameters behavior on temperatures dependence which also exhibits anisotropic are evaluated (γa>γc). In this work, we employed WinTA 100 dilatometer manufactured by Bähr Company to directly measure the bulk crystals thermal expansion coefficients. The thermal expansion coefficients in linear connection with temperature measured by this two methods are shown in Figure .4. Among the coefficients, αa and αc characterize the linear thermal expansion parallel to a- and c-axis, respectively; αV characterizes the volume thermal expansion of unit cell, (αV =2αa +αc); αη characterizes the anisotropy of thermal expansion. As seen from Figure .4, it should be noted that the results derived from the two method at a given temperature may a little differ from each other. The greens curves which represent thermal expansion coefficients measured directly by thermal dilatometer are always under the lines which represent the results of the XRD. The orange lines are obtained by linear fitting extrapolation. The interrelated relations are listed as follow:
powder :
c 8.10 106 1.04 108 t Bulk : c 2.33 106 3.00 109 t a 1.08 105 8.73 109 t a 5.56 106 6.27 109 t 2.67 106 1.91108 t 3.23 106 9.27 109 t V 29.64 106 7.09 109 t V 13.45 106 9.54 109 t
The difference between these results may be due to the small grain size of the powder. Of course, the thermal expansion of the bulk crystals measured directly dilatometer will be subject to greater restrictions. Therefore, either αa or αc, the strength of the thermal expansion measured from bulk crystal is smaller than that measured from powder. The corresponding results are presented in Table II together with literature values. As seen from it, αa increases while αc decreases with elevating temperature. Moreover, αa is always larger than αc in the whole range. Despite all that, both αa and αc achieve positive values. Hence, it may be supposed the specimen may expand simultaneously in all directions while heated. The absolute values of thermal expansion anisotropy,
, which can reflect the tetragonal distortion, namely αμ, are found increasing with temperature. Therefore, the value αμ is also observed increasing with temperature, 7
which indicates the tetragonal structure is getting more and more distorted on heating. Strong thermal strain is generated from the large longitudinal temperature gradient and thermal expansion coefficient. When the thermal strain exceeds the plastic deformation range of crystal, crystal crack would occur. What’s more, CdGeAs2 crystal also has strong thermal expansion anisotropy. Difference value of αa and αc is very large and the degree of thermal expansion on different direction are different. That would make the thermal strain very uneven, the causing the severe cracking phenomenon. To solve this problem, seed with c axis orientation can also be adopted, which is because the CdGeAs2 crystal c axis with smaller thermal expansion coefficient. On large longitudinal temperature gradient, the thermal strain is minimum. At the same time, due to the radial temperature gradient is small, thermal strain is also small. It is found that the results measured directly by thermal dilatometer in this work are very close to those from other studies, on the contrary, the results obtained by high temperature XRD are slightly larger than that by other authors [15-18]. The causes of such differences are not only the grain size which we have mentioned, but also the measuring conditions and the quality of specimens. Table II. Thermal expansion coefficients of lattice parameters at variable temperatures along with literature values. Thermal expansion coefficients (unit: ×10-6K-1) ` Temperature(K)
αa
300 373 473 573 673 723
13.4 14.0 14.9 15.8 16.6 17.1 a
373~573 293~673b Theorc Theord a
Powder αc αη 5.0 4.2 3.2 2.2 1.1 0.6
-8.4 -9.8 -11.7 -13.6 -15.5 -16.5
Bulk αV
αa
αc
αη
αV
31.8 32.2 33.0 33.8 34.3 34.8
7.44 7.90 8.53 9.15 9.78 10.09
1.43 1.21 0.91 0.61 0.31 0.16
-6.01 -6.69 -7.61 -8.54 -9.47 -9.93
16.31 17.01 17.96 18.92 19.87 20.35
Literature values 8.5 1.0 NG 11.4 1.0 NG 9.5 1.7 NG 11.02 3.19 NG
NG NG NG NG
Data from Ref. [17].
b
Data from Ref. [18]. 8
c
Data from Ref. [15]. Theor. means coefficients are derived from theoretical estimation.
d
Data from Ref. [16].
e
NG means no given.
In some cases, the thermal expansion coefficients in specific direction are defined as follows [39]:
hkl c cos2 a sin 2 aa a cos2
(1)
where φ is the angle between the direction [hkl] and the c-axis. According to the fitting curves of αa and αc, the directional coefficients αhkl are plotted against cos2φ in Figure. 5. On this plot, αhkl corresponds to αa when cos2φ=0 whereas to αc when cos2φ=1. As can be seen, the values of αhkl are all positive, which means it expands in all directions. It is obviously that all the curves seem to intersect at one point nearby cos2φ=0.67, i.e., φ≈35°. That’s to say, there exists specific directions and specific planes where the thermal expansion coefficients are the same and vary in a small range or even tend to be constant in spite of the changing temperatures. It is still exist differences that this point tested by CdGeAs2 powder is at cos2φ=0.5, i.e., φ≈45° [38]. To validate this deduction, we measured the thermal expansion coefficients in [204] direction which is shown in Figure. 6. The green squares are the testing data of the bulk crystal along the [204] direction. This thermal expansion coefficients is still in linear connection with temperature: 204 2.06 106 4.42 109 t . Meanwhile, the pink line is on behalf of the derivative result from the axial thermal expansion coefficients.
From
the
comparative
analysis
of
derivative
line
204 3.85 106 1.37 109 t and the actual measure value, this derivation is an actual and effective method to calculate the thermal expansion coefficients in a given direction.
3.3 Temperature dependence of Grüneisen parameters Generally, crystal properties that depend on the thermal motion of atoms are much influenced by the anharmonicity of lattice vibration, which means the nonlinearity of interatomic forces with respect to atomic displacements [40]. The anharmonic properties are usually described by Grüneisen parameters which are 9
dimensionless quantity characterizing the volume or strain dependence of the lattice vibrational mode frequencies. As for CdGeAs2 crystal, two independent principal Grüneisen parameters, γa and γc, must be introduced. According to Barron et al.[41], the two Grüneisen parameters are determined as follows:
a
Vm c11 c12 a c13 c Cp
c
Vm 2c13 a c33 c Cp
(2)
(3)
where Vm is the molar volume, Cp is the molar specific heat at constant pressure, cij is the elastic stiffness, and αa and αc are the thermal expansion coefficients along a-axis and c-axis, respectively. The volume Grüneisen parameter γV is a linear combination of γa and γc weighted by the respective linear compressibilities χa and χc as well as the volume compressibility χV. The corresponding relations are given as follows [41, 42]:
V
2 a a c c
(4)
a
c33 c13 c33 c11 c12 2c132
(5)
c
c11 c12 2c13 c33 c11 c12 2c132
(6)
V 2 a c
c11 c12 2c33 4c13 c33 c11 c12 2c132
(7)
Up to date, the data of elastic stiffnesses at high temperatures are inadequate. The only reported data at variable temperatures are measured below 290 K [43]. Using extrapolation method, we calculated the elastic stiffnesses at different temperatures [38] from low temperature values. The heat capacity Cp has been measured in this work which is detailed in the following sections. Here, in order to compare the previous results, we still considered it as a constant equaling to 100 J/(mol K)[44, 45]. The molar volume Vm is now a temperature dependence parameter and can be easily calculated from lattice parameters which can be deduced from the αa and αc. Finally, the Grüneisen parameters γa, γc and γV are obtained according to Eqs. (2)–(7). 10
The Grüneisen parameters measured by high-temperature XRD and thermal dilatometer are plotted against temperature in Figure.7. It can be seen that γa, γc and γV are all positive and that γa is always larger than γc in the whole temperature range. The room temperature values of γa, γc and γV are evaluated to be 0.73, 0.60, and 0.67, respectively, which are very close to the literature values (0.82 for γa, 0.66 for γc and 0.75 for γV)[45] and a little smaller than the values which are measured using CdGeAs2 powder in our previous work and evaluated to be 1.40, 1.20, and 1.32, respectively. Meanwhile, it is found the values of γV measured by us are smaller likewise than that are calculated by Yu et al. Yet for all that, one could see the average Grüneisen parameters γa and γc are anisotropic and, therefore, the anharmonic forces contributing to crystal properties are also anisotropic for CdGeAs2 crystal.
3.4 The deduction and calculation of thermal conductivity Steigmeier calculated the lattice thermal conductivity for III-V compounds at high temperature using the modification equation [46], 3
3 k M 2 T 3
l 41 3 5 h
(8)
where M is the mean atomic mass, δ is the cube root of the atomic volume, and γ is the Grüneisen parameter. The lattice thermal conductive κl is obtained by subtracting the electronic contribution κel from the total thermal conductivity κ. However, judging from the thermal conductivity experimental result, this equation is not appropriate for II-IV-V2 compounds. Meanwhile, the modification equations proposed by Masim [27] and Valeri-Gil [29] are diverse from each other. Therefore, we use two modification equations to revise the results κl of bulk and powder, respectively, and the lattice thermal conductivity with different direction including κl-a, κl-c and κl-V and the thermal conductivity experimental with [204] direction are shown in Figure. 8. 3 3 k M l 41 3 2 5 h 1 T 3
11
(9)
3 M 3 k l 41 3 2 5 h 1 2 T 3
(10)
It can be seen that κ204, κl-a, κl-c and κl-V are all positive quantities and that κl-c is always larger than κl-c in the whole temperature range. With such a relationship, the lattice thermal conductivity of all the II-IV-V2 compounds which have different MδΘ3 can be deduced [27]. In addition, the electronic thermal conductivity of the intrinsic samples could be computed by using the formula [47]
A A k2 npb p p el 2 T n n 2 e nb p
Eg T Bn Bp kT
2
(11)
where n and p refer to electrons and holes, respectively; EgT is the band gap at the temperature T; σ is the electrical conductivity, and b is the mobility ratio. For Boltzmann statistics,
5 5 An, p r , Bn, p r 2 2
(12)
r is given by τ=τ0Eτ. In this case, we can assess the κel at room temperature, i.e. 300 K. Earlier in the article, the energy gap Eg and the relevant electrical parameters have been provided. Reported values of the mobility ratio b for CdGeAs2 are about 12 [35, 36] and 30 [17].
3.5 Temperature dependence of heat capacity Now we return to discuss about the thermal conductivity experimental with [204] direction κ204. To study the influence of the anion on lattice anharmonicity in the CdGeAs2 crystal, we have measured the heat capacity of it. The resulting temperature dependence of the heat capacity and molar heat capacity at constant pressure Cp, of CdGeAs2 is shown in Figure. 9. To analyze the temperature dependence of Cp and to evaluate the contribution to Cp due to lattice anharmonicity we use the general relation [48]: N C p T 12 R F T ckT k k 1
(13)
where R is the molar gas constant and F(Θ/T) the Debye function describing the 12
temperature dependence of the heat capacity in the harmonic approximation. The polynomial of Nth order in the temperature T represents the contribution to Cp, due to lattice anharmonicity. In this work, the Cp dependence on temperature is obtained by a least-squares fit relation,
C p 150.02 1.69 101T 1.15 104 T 2 3.91107 T 3
(14)
which is shown in the Figure. 9. And the specific heat Cp is nearly equal to 100 J/(mol K) from 300 K to 700 K which demonstrates the previous approximate is appropriate. In this case, κ204 can be obtained by such relation DC p , where ρ is density and D is thermal diffusivity which can be measured directly by laser flash method. After Capel model and pulse modification, thermal conductivity can be measured indirectly.
4. Summary Either αa or αc, the strength of the thermal expansion measured from bulk crystal is smaller than that measured from powder. Base on the anisotropic thermal expansion measured by these two different method, the Grüneisen parameters dependence on temperature are evaluated, and also exhibit anisotropic behavior (γa > γc). As well, the thermal expansion, γa, γc, and γV have similar difference between these two kinds of test method. The difference between these results may be due to the grain size of the powder which is in microns, and the constraints of lattice is small. And two correction formula are proposed to deduce the lattice thermal conductivities for II-IV-V2 compounds. At last, the thermal conductivity dependence on temperature with [204] direction has been measured which is demonstrated that the deduction is very close to the experimental results.
Acknowledgments This work is supported by the National Natural Science Foundation Key Programs of China (no. 50732005) and the 863 High-Tech program of China (no. 007AA03Z443).
13
References: [1] Hahn H, Frank G, Klingler W, Störger AD, Störger G. Untersuchungen über ternäre Chalkogenide. V. Über einige ternäre Chalkogenide mit Chalkopyritstruktur. Zeitschrift für anorganische und allgemeine Chemie 1953;279:241. [2] Kildal H. Band structure of CdGeAs2 near k=0., vol. 10, 1974. p.5082. [3] Gentile AL. Devices using ternary or multinary compounds. Progress in Crystal Growth and Characterization 1984;10:241. [4] Byer RL, Kildal H, Feigelson RS. CdGeAs2 -A New Nonlinear Crystal Phasematchable at 10.6μ m. Appl Phys Lett 1971;19:237. [5] Shay JL, Wernick JH. Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications. Oxford and New York: Pergamon Press, 1975. [6] Iseler GW, Kildal H, Menyuk N. Optical and electrical properties of CdGeAs 2. Journal of Electronic Materials 1978;7:737. [7] Feigelson RS, Route RK, Swarts HW. Solution growth of CdGeAs 2. J Cryst Growth 1975;28:138. [8] Isomura S, Takahashi S. Electrical and Optical Properties of CdGeAs 2. Japanese journal of applied physics 1977;16:1723. [9] Feigelson RS, Route RK. Vertical bridgman growth CdGeAs2 with control of interface shape and orientation. Journal of Crystal Growth 1980;49:261. [10] Baumgartner FP, Lux-Steiner M, Bucher E. Growth of CdGeAs2 single crystals by the chemical vapor transport method. J Electron Mater 1990;19:777. [11] Manimaran M, Ramasamy P. Growth and microhardness studies of CdGeAs 2 single crystals. J Mater Sci Lett 1994;13:1729. [12] Saghir MZ, Labrie D, Ginovker A, Paton BE, George AE, Olson K, Simpson AM. Float-zone crystal growth of CdGeAs2 in microgravitynumerical simulation and experiment. J Cryst Growth 2000;208:370. [13] Feigelson RS. Improving optical transparency in CdGeAs2 crystals by controlling crystalline defects. J Cryst Growth 2006;292:179. [14] Kildal H, Mikkelsen JC. Efficient doubling and CW difference frequency mixing in the infrared using the chalcopyrite CdGeAs2., vol. 10, 1974. p.306. [15] Neumann H. Trends in the thermal expansion coefficients of the A IBIIIC2VI and AIIBIVC2V chalcopyrite compounds. Kristall und Technik 1980;15:849. [16] Kumar V, Sastry BSR. Relationship between the thermal expansion coefficient, plasmon energy, and bond length of ternary chalcopyrite semiconductors. J Phys Chem Solids 2002;63:107. [17] Iseler GW, Kildal H, Menyuk N. Optical and electrical properties of CdGeAs2. J Electron Mater 1978;7:737. [18] David NN. Nonlinear Optical Crystals: A Complete Survey. New York: Springer, 2005. [19] Pandey R, Ohmer MC, Gale JD. A theoretical study of native acceptors in CdGeAs 2. 1998;10:5525. [20] Schunemann PG, Pollak TM. Single crystal growth of large, crack-free CdGeAs2. Journal of Crystal Growth 1997;174:272. [21] He Z, Zhao B, Zhu S, Li J, Zhang Y, Du W, Huang W, Chen B. Preparation and characterization of CdGeAs2 crystal by modified vertical Bridgman method., vol. 314, 2011. p.349. 14
[22] Huang W, Zhao B, Zhu S, He Z, Chen B, Tang J, Liu W. Growth and characterizations of CdGeAs2 single crystal by descending crucible with rotation method. Rare Metals 2014;33:210. [23] Červinka L, Kašpar J. A high-temperature x-ray study of CdGeAs2. Czechoslovak Journal of Physics B 1970;20:101. [24] Abrahams SC, Bernstein JL. Piezoelectric nonlinear optic CuGaSe2 and CdGeAs2 Crystal structure, chalcopyrite microhardness, and sublattice distortion. J Chem Phys 1974;61:1140. [25] Limpijumnong S, Lambrecht WRL. Band structure of CdGeAs 2 near the fundamental gap. Phys Rev B 2002;65:165204. [26] Bai L, Xu C, Gilesa NC, Nagashiob K, Feigelson RS. Correlation of the electrical and optical properties of p-type CdGeAs2. J Appl Phys 2006;99:13512. [27] Wasim SM. Thermal conductivity of ternary Compounds. physica status solidi (a) 1979;51:K35. [28] Bellabarba C, Wasim SM. Thermal conductivity of AgInTe2. physica status solidi (a) 1981;66:K105. [29] Valeri-Gil ML, Rincón C. Thermal conductivity of ternary chalcopyrite compounds. Mater Lett 1993;17:59. [30] Rincón C, Valeri-Gil ML, Wasim SM. Room-Temperature Thermal Conductivity and Grüneisen Parameter of the I-III-VI2 Chalcopyrite Compounds. Physica Status Solidi (a) 1995;147:409. [31] He Z, Zhao B, Zhu S, Li J, Zhang Y, Du W, Huang W, Chen B. Preparation and characterization of CdGeAs2 crystal by modified vertical Bridgman method. J Cryst Growth 2011;314:349. [32] Huang W, Zhao B, Zhu S, He Z, Chen B, Tang J, Liu W. Growth and characterizations of CdGeAs2 single crystal by descending crucible with rotation method. Rare Metals 2014;33:210. [33] Nagashio K, Watcharapasorn A, Zawilskia KT, DeMattei RC, Feigelson RS, Bai L, Giles NC, Halliburton LE, Schunemann PG. Correlation between dislocation etch pits and optical absorption in CdGeAs2. J Cryst Growth 2004;269:195. [34] Bai L, Garces NY, Yang N, Schunemann PG, Setzler SD, Pollak TM, Halliburton LE, Giles NC. Optical and EPR Study of Defects in Cadmium Germanium Arsenide. Materials Research Society 2003;744:M8. [35] Ptak AJ, Jain S, Steven KT, Myer TH, Schunemann PG, Setzler SD, Pollak TM. Temperature dependent Hall measurements made on CdGeAs2. Mat. Res. Soc. Proc 2000;607:427. [36] Bairamov BH, Rud VY, Rud YV. Properties of dopants in ZnGeP 2 CdGeAs2, AgGaS2 and AgGaSe2. Mrs Bull 1998;23:41. [37] Fischer DW, M CO, McCrae JE. Influences of temperature and transport properties on the birefringence of CdGeAs2. J Appl Phys 1997;81:3579. [38] Liu W, Zhao B, Zhu S, He Z, Chen B, Huang W, Tang J, Yu Y. X-ray study of thermal expansion behaviors and Grüneisen parameters of cadmium germanium arsenide crystal over the temperature range 25–450 ℃. J Appl Phys 2013:53513. [39] Bodnar IV, Orlova NS. X-Ray Study of the Thermal Expansion Anisotropy in AgGaS2 and AgInS2 Compounds over the Temperature Range from 80 to 650 K. physica status solidi (a) 1985;91:503. [40] Ashcroft NW, Mermin ND. Solid State Physics. New York: Saunders College, 1976. [41] Barron THK, Collins JG, White GK. Thermal expansion of solids at low temperatures., vol. 29, 1980. p.609. [42] Yu Y, Zhao B, Zhu S, Gao T, Hou H. Ab initio vibrational and dielectric properties of chalcopyrite CdGeAs2. Solid State Sci 2011;13:422. 15
[43] Hailing T, Saunders GA, Lambson WA, Feigelson RS. Elastic behaviour of the chalcopyrite CdGeAs2. Journal of Physics C: Solid State Physics 1982;15:1399. [44] Yu Y, Zhao BJ, Zhu SF, Gao T, Hou HJ, He ZY. Theoretical study of elastic and thermodynamic properties of chalcopyrite CdGeAs2. Physica B: Condensed Matter 2013;417:83. [45] Neumann H. Grüneisen parameters in AgGaS2 and CdGeAs2. Cryst Res Technol 1983;18:K126. [46] Steigmeier EF, Kudman I. Thermal Conductivity of III-V Compounds at High Temperatures. Physical Review 1963;132:508. [47] Madelung O. Handbuch der physik. Berlin: Springer-Verlag, 1957. [48] Möller W, Kühn G, Neumann H. Heat capacity and lattice anharmonicity in CdBIVC2V chalcopyrite compounds. Cryst Res Technol 1987;22:533.
Figure. 1. Photograph of crack-free CdGeAs2 single crystals and an embryo of SHG device. Figure. 2. IR transmission spectra recorded for a 2-mm-thick CdGeAs2 samples, the corresponding absorption coefficient and absorption coefficient vs photon energy spectra of CdGeAs2 single crystals. Figure. 3. Overlay of X-ray diffraction patterns of CdGeAs2 crystal measured at different temperature. The lower part is the PDF card data measured at room temperature. Figure. 4. Variation of lattice parameters αa, αc, αV and αη as a function of temperature for CdGeAs2. Dots represent the high temperature experimental data, while blue solid lines are the polynomial fit curves. The green square represent the data measured by dilatometer, while orange solid lines are the polynomial fit curves. The polynomial expressions, for each parameter, are presented. Figure .5. Variation of directional coefficient αhkl at different temperatures as a function of cos2φ. Figure. 6. Thermal expansion coefficients in specific direction α204 as a function of temperature for CdGeAs2. Figure .7. Variation of the Grüneisen parameters γa, γc, and γV as a function of temperature. Figure .8. Lattice thermal conductivity including κl-a, κl-c and κl-V and the thermal conductivity experimental with [204] direction. Figure .9. Temperature dependence of the heat capacity and molar heat capacity at constant pressure Cp.
16
17
18
19
20
21
Table .I Electrical properties of the samples This work
Electrical properties
300 K
77 K
8.819×1014~
Bulk carrier -3
Ref. [36]
16
concentration（cm ）
1.977×10
Mobility （cm2/Vs）
13.66~402.2
4.276×107~5. 366×1010 17.78~895.8
Ref.[17]
300 K
77 k
300 K
77 k
0.4×1016
-
1016~1017
-
181
-
12~30
-
9
1.2×104
1~10
102~105
-
-
102
107
5
2.175×10 ~1.
Resistivity （Ω·cm）
1.298~74.44
Average Hall
3.158×102~
1.163×108~1.
coefficient（cm3/C）
7.078×103
460×1011
663×108
22
Table II. Thermal expansion coefficients of lattice parameters at variable temperatures along with literature values. Thermal expansion coefficients (unit: ×10-6K-1) ` Temperature(K)
αa
300 373 473 573 673 723
13.4 14.0 14.9 15.8 16.6 17.1 a
373~573 293~673b Theorc Theord a
Powder αc αη 5.0 4.2 3.2 2.2 1.1 0.6
-8.4 -9.8 -11.7 -13.6 -15.5 -16.5
Literature values 8.5 1.0 NG 11.4 1.0 NG 9.5 1.7 NG 11.02 3.19 NG
Bulk αV
αa
αc
αη
αV
31.8 32.2 33.0 33.8 34.3 34.8
7.44 7.90 8.53 9.15 9.78 10.09
1.43 1.21 0.91 0.61 0.31 0.16
-6.01 -6.69 -7.61 -8.54 -9.47 -9.93
16.31 17.01 17.96 18.92 19.87 20.35
NG NG NG NG
Data from Ref. [17].
b
Data from Ref. [18].
c
Data from Ref. [15]. Theor. means coefficients are derived from theoretical estimation.
d
Data from Ref. [16].
e
NG means no given.
>CdGeAs2 single crystal was grown by VB method>Optical, electrical and thermodynamic properties are characterized > thermal expansion coefficients are measured by two method> Grüneisen parameters have been calculated by thermal expansion coefficients> thermal conductivity are obtained by experiment and theoretical derivation
23
PII: DOI: Reference:
S0022-0248(16)30070-7 http://dx.doi.org/10.1016/j.jcrysgro.2016.02.037 CRYS23224
To appear in: Journal of Crystal Growth Received date: 27 October 2015 Revised date: 14 February 2016 Accepted date: 29 February 2016 Cite this article as: Wei Huang, Beijun Zhao, Shifu Zhu, Zhiyu He, Baojun Chen, Zhen Zhen, Yunxiao Pu and Weijia Liu, Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2, Journal of Crystal Growth, http://dx.doi.org/10.1016/j.jcrysgro.2016.02.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigation of thermodynamics properties of chalcopyrite compound CdGeAs2 Wei Huang, Beijun Zhao*, Shifu Zhu, Zhiyu He, Baojun Chen, Zhen Zhen, Yunxiao Pu, Weijia Liu College of Materials Science and Engineering, Sichuan University, Chengdu 610064, China Abstract: Chalcopyrite of CdGeAs2 single crystal was grown by a modified vertical Bridgman method with sufficient size and quality, and its optical, electrical and thermodynamic properties are characterized. The transmission is recorded in the 2.3-18 μm range, and the band-gap at room temperature is at 0.56 eV. Non-ideal transparency near 5.5 μm which limited its application severely exists in the front of the crystal. The crystal is p type at room temperature with hole concentrations varying from 1014 to 1016 cm−3. From the results of X-ray diffraction measurements carried out over the range 25~450 ℃ and thermal dilatometer tests, the thermal expansion coefficients are evaluated. And on this basis the Grüneisen parameters at different temperatures are evaluated and also exhibit anisotropic behavior (γa > γc). It is found that γa, γc, and γV have some difference between these two kinds of test methods. Using these Grüneisen parameters, lattice thermal conductivities have been deduced by two correction formulas. Meanwhile, specific heat capacity and thermal conductivity of [204] have been obtained as a function of temperature by experiment.
Keywords: CdGeAs2 single crystals; Vertical Bridgman technique; thermal expansion coefficients; Grüneisen parameters; thermal conductivities.
1. Introduction
*
Corresponding author. Fax: +86 28 85412745. E-mail address: [email protected] 1
Ternary compounds with chalcopyrite structure, which were first discovered by Hahn et al.[1] in 1953, have received a great of attentions because of their applications in optical devices, detectors and solar cells[2, 3]. Cadmium germanium arsenide (CdGeAs2) is an ideal candidate material for nonlinear optical (NLO) applications, since it has the highest NLO coefficient (d36 = 236 pm/V) [4, 5]. Its wide transparency range (2.3-18 μm), accompanying by significant birefringence, makes this material very promising in frequency conversion applications in the infrared range [6]. Since the 70s, researchers have struggled to grow large and high quality single crystals of this compound, but they were plagued by polycrystallinity and cracking [7-13] due to highly anisotropic thermal expansion coefficients [14]. Therefore, the investigation of anisotropic thermal expansion coefficients has great significance for the improvement of the CdGeAs2 crystal growth. Newmann [15] and Kumar et al. [16] have evaluated the average linear thermal expansion coefficient αL based respectively on empirical and modified empirical relation from first principles. Kildal [17] and Kozhina et al.[18] reported the experimental average thermal expansion coefficients in two main directions (namely, αa and αc) in the range 100~300 ℃ and 20~400 ℃, respectively. However, in many cases, the results seem inadequate in practice, and it is expected to reveal thermal expansion mechanism in specific directions and at specific temperatures which is conducive to the crystal growth. With the development of horizontal gradient freeze method [19, 20] and modified vertical Bridgman method [21, 22], single crystals of this ternary chalcopyrite have been grown without severe cracking. Meanwhile, up to now, structural [10, 23, 24], electronic [2, 25] and optical properties [6, 8, 26] of CdGeAs2 have been studied for the last four decades by a number of experimental and theoretical methods. However, the thermal conductivity of the I-III-VI2 or II-VI-V2 chalcopyrite crystal, which is an important thermodynamic parameters in the aspect of laser application, were usually discussed at ambient temperature or under ambient temperature in the literature, and the derivation formulas of thermal conductivity were different [27-30] each other. To achieve high power laser output, the investigation of the thermal conductivity is indispensable. 2
Herein, we address the interesting about the CdGeAs2 crystal growth by the modified VB method as well as optical, electrical and thermodynamics characterization, and have deeply discussed the properties of anisotropic thermal expansion, heat capacity and thermal conductivity.
2. Experimental Section 2.1. Crystal growth The starting material was synthesized from high purity 6 N Cd spheroids, 6 N Ge and 6 N As lumps about 120 g at the molar ratio of 1:1.005:2.01. After weighing in a glove box, the charges were loaded into a quartz ampoule and sealed under 10-5 Pa at once. Then the ampoule was put into a rotatable tube furnace and the synthesis was mainly carried out with such steps: 50 ℃/h to 650 ℃, 8-h soak and 20 ℃/h to 950 ℃,24-h soak. In order to make the raw materials mix uniformly and drive the residual Cd and As vapor out of melt, the furnace rotating and the temperature oscillation (between 800 to 1040 ℃) have been carried out several times [31, 32]. When the synthesis process finished, the furnace were cooled down to room temperature at the speed of 20 ℃/h. The polycrystalline material was taken out of the synthesis ampoule and ground into powder. Meanwhile, the polycrystalline XRD and EDS analysis demonstrated that there is no impurity phases in the powder. Then the polycrystalline powder was reloaded into a carbon-coated quartz ampoules. After being evacuated to 10-5 Pa, the polycrystalline powder was sealed into the quartz ampoule. Then the quartz ampoule was placed into a three-zone furnace. The temperatures of the upper, middle and lower zones were raised to the special temperatures at the rate of 12 ℃ /min, respectively. After a lot of growth runs, we find that if the temperature of the upper zone is higher than 720 ℃, the obtained products are most polycrystal. But if the temperature of the upper zone is too low, the appropriate temperature gradients is hard to achieve. A suitable temperature gradient not only can provide a sufficient nucleation driving force, but also restrain the formation of polycrystalline boules. Therefore, we have carried out a lot of growth runs with the temperature gradients of 3
10-35 ℃. At last, temperature gradients of 15-25 ℃/cm are adopted, which is different from the small temperature gradients (1-3℃/cm) in the horizontal gradient freeze method [20] and is smaller than that reported in the early vertical Bridgman method (40 ℃/cm) [9]. The growth position of the ampoule was adjusted to the place where the indication of the monitoring thermocouple was about 670 °C. The entire charge was heated above the melting point. Over 100°C of supercooling (down to 547°C) took place before the onset of crystallization [20]. So when the material in the seed well solidified, the ampoule was pulled back to the position where the temperature is about 650 °C. Then the material in the seed well was reheated to partially melt back and the growth ampoule was mechanically descended at the rate of 0.21-0.25 mm/h. After the growth process finished, the temperature was cooled down to room temperature at 20 ℃/h. Thus, CdGeAs2 single crystals could be obtained reproducibly, which is shown in Figure .1.
2.2. Characterization fThe optical transmittance spectra o as-grown CdGeAs2 wafers were recorded using a Shimadzu fourier-transform-infrared (FTIR) spectrometer in the range of 2.3 μm~18 μm. The samples are approximately 2 mm thick which were polished smoothly. Electrical properties were obtained using Ecopia HMS-300 Hall-effect measurement system. Temperature could be controlled at 77 and 300 K, and the measurements were carried out at a magnetic field of 0.554 T. A circular diamond saw was used to cut the as-grown boule, verifying the orientation of the surface by XRD. The CdGeAs2 crystals were carefully examined by the powder XRD (DX-2000, Dandong, China) method at room temperature. The diffraction patterns were recorded over the 2θ range of 10-90°. The high temperature XRD measurements were performed in a DX-2700 diffractometer with Cu Kα radiation, and Graphite monochromator was adopted to select single wavelength of about 1.54 Å. The temperature stages for X-ray diffraction measurements were determined at 25, 100, 200, 300, 400, and 450 ℃, respectively. Prior to thermal conductivities measurements, a series of thermogravimetry and differential scanning calorimetry (TG-DSC) 4
measurements in the shielding nitrogen ambient were carried out in a SDT Q600 apparatus (TA Instruments, USA) for heat capacity, and the sample size is Ф5 mm×1 mm. After careful calibration of the calorimeter and using Al2O3, as standard the heat capacities could be measured with a high-accuracy. Thermal conductivities were determined at 26.6, 100.6, 200.4, 300.4 and 400.3 ℃ by NETZSCH LFA 457 which adopt the InSb detector and Ar protective atmosphere with 50.00 ml/min. The diameter and thickness of test sample were 12.7 mm and 1.8 mm, respectively. Due to the size limitation, thermal conductivity experimental only with [204] direction could be obtained. The [204] direction is just the growth direction in this VB method so that the sample has a large enough diameter.
3. Results and Discussion 3.1 Optical and electrical properties The IR transmittance of a CdGeAs2 wafer is shown in Figure. 2 and the maximum value is up to 51.6% in the range of 2.3-18 μm. Absorption coefficients (α) were estimated from the transmission spectrum of CdGeAs2 single crystals. The inset shows the value of (αhν)2 vs photon energy at room temperature. The useful transmission range is limited by the strong absorption below 2.3 μm due to transitions across the band-gap and by two-phonon absorption at 12.5 and 13.5 μm, which are present in all of samples, are probably due to three-phonon processes. The absorption coefficients at 5.5 μm is close to the lowest optical losses (0.18 cm-1) measured by K.
Nagashio [33]. The broad band peaking near 5.5 μm has been ascribed to transitions between a split-off valence band and the highest valence band at k=0 [2]. Since the EPR hyperfine interactions suggested the acceptor was located on the As site. Lihua Bai [34] believed the shallow acceptor was a Group IV atom on the Group V site which was most likely the Ge-on-As antisite defect. The intensity of this absorption band decreases with decreasing hole concentration [17] as well. Therefore, different part of the crystal with different hole concentration have different degree of absorption. That is why there exists difference between curve 1 and curve 2 in the Figure .2, which describe the optical transmission and absorption located the center 5
part of the crystal and shoulder position. The Hall measurement data indicate that the as-grow crystal samples are consistently p-type with hole concentration varying from 107 to 1010 cm−3 and mobility about 102 cm2/Vs at 77 K, as shown in Table I. At 300 K, the hole concentration decreases and the range is from 1014 to 1016 cm−3. But the resistivity is 1.298~74.44 Ω·cm at room temperature which is geared to a small order of magnitude. At 300 K, the resistivity increases and the range is from 105 to 108 Ω·cm. Compared with the values of early reports, the hole concentration measured in this work is smaller and the resistivity is larger, which is shown in Table .I. The actual hole concentration for samples with low absorption at 5.5 μm can be calculated by an equation proposed by Bai et al [26]. Reported values of the mobility ratio b for CdGeAs2 are about 12[35, 36] and 30[17] and two lines which describe the relation of hole concentration and absorption coefficient are linear least-squares fit to the data points yielding the relation p = (0.93+4.62α)×1015 cm−3 and p = (1.06+4.59α)×1015 cm−3 which is close to the relation obtained by Bai et al[26]. Table .I Electrical properties of the samples This work
Electrical properties
Ref. [37]
300 K
77 K
Bulk carrier
8.819×1014~
4.276×107~5.
concentration（cm-3）
1.977×1016
366×1010
Mobility （cm2/Vs）
13.66~402.2
17.78~895.8
Ref.[17]
300 K
77 k
300 K
77 k
0.4×1016
-
1016~1017
-
181
-
12~30
-
9
1.2×104
1~10
102~105
-
-
102
107
5
2.175×10 ~1.
Resistivity （Ω·cm）
1.298~74.44
Average Hall
3.158×102~
1.163×108~1.
coefficient（cm3/C）
7.078×103
460×1011
663×108
3.2 Determination of thermal expansion coefficients In our previous work, in the range of 25–450 ℃, X-ray powder diffraction measurements of CdGeAs2 crystal have been carried out[38]. XRD patterns collected at different temperatures are presented in Figure. 3 along with the PDF No. 73-0402. Accurate lattice parameters including two lattice constants a and c, cell volume V, axial ratio η, and tetragonal distortion μ are determined. And the average thermal expansion coefficients are evaluated, 13.9×10-6 K-1 for αa and 2.8×10-6 K-1 for αc, 6
respectively. Meanwhile, the Grüneisen parameters behavior on temperatures dependence which also exhibits anisotropic are evaluated (γa>γc). In this work, we employed WinTA 100 dilatometer manufactured by Bähr Company to directly measure the bulk crystals thermal expansion coefficients. The thermal expansion coefficients in linear connection with temperature measured by this two methods are shown in Figure .4. Among the coefficients, αa and αc characterize the linear thermal expansion parallel to a- and c-axis, respectively; αV characterizes the volume thermal expansion of unit cell, (αV =2αa +αc); αη characterizes the anisotropy of thermal expansion. As seen from Figure .4, it should be noted that the results derived from the two method at a given temperature may a little differ from each other. The greens curves which represent thermal expansion coefficients measured directly by thermal dilatometer are always under the lines which represent the results of the XRD. The orange lines are obtained by linear fitting extrapolation. The interrelated relations are listed as follow:
powder :
c 8.10 106 1.04 108 t Bulk : c 2.33 106 3.00 109 t a 1.08 105 8.73 109 t a 5.56 106 6.27 109 t 2.67 106 1.91108 t 3.23 106 9.27 109 t V 29.64 106 7.09 109 t V 13.45 106 9.54 109 t
The difference between these results may be due to the small grain size of the powder. Of course, the thermal expansion of the bulk crystals measured directly dilatometer will be subject to greater restrictions. Therefore, either αa or αc, the strength of the thermal expansion measured from bulk crystal is smaller than that measured from powder. The corresponding results are presented in Table II together with literature values. As seen from it, αa increases while αc decreases with elevating temperature. Moreover, αa is always larger than αc in the whole range. Despite all that, both αa and αc achieve positive values. Hence, it may be supposed the specimen may expand simultaneously in all directions while heated. The absolute values of thermal expansion anisotropy,
, which can reflect the tetragonal distortion, namely αμ, are found increasing with temperature. Therefore, the value αμ is also observed increasing with temperature, 7
which indicates the tetragonal structure is getting more and more distorted on heating. Strong thermal strain is generated from the large longitudinal temperature gradient and thermal expansion coefficient. When the thermal strain exceeds the plastic deformation range of crystal, crystal crack would occur. What’s more, CdGeAs2 crystal also has strong thermal expansion anisotropy. Difference value of αa and αc is very large and the degree of thermal expansion on different direction are different. That would make the thermal strain very uneven, the causing the severe cracking phenomenon. To solve this problem, seed with c axis orientation can also be adopted, which is because the CdGeAs2 crystal c axis with smaller thermal expansion coefficient. On large longitudinal temperature gradient, the thermal strain is minimum. At the same time, due to the radial temperature gradient is small, thermal strain is also small. It is found that the results measured directly by thermal dilatometer in this work are very close to those from other studies, on the contrary, the results obtained by high temperature XRD are slightly larger than that by other authors [15-18]. The causes of such differences are not only the grain size which we have mentioned, but also the measuring conditions and the quality of specimens. Table II. Thermal expansion coefficients of lattice parameters at variable temperatures along with literature values. Thermal expansion coefficients (unit: ×10-6K-1) ` Temperature(K)
αa
300 373 473 573 673 723
13.4 14.0 14.9 15.8 16.6 17.1 a
373~573 293~673b Theorc Theord a
Powder αc αη 5.0 4.2 3.2 2.2 1.1 0.6
-8.4 -9.8 -11.7 -13.6 -15.5 -16.5
Bulk αV
αa
αc
αη
αV
31.8 32.2 33.0 33.8 34.3 34.8
7.44 7.90 8.53 9.15 9.78 10.09
1.43 1.21 0.91 0.61 0.31 0.16
-6.01 -6.69 -7.61 -8.54 -9.47 -9.93
16.31 17.01 17.96 18.92 19.87 20.35
Literature values 8.5 1.0 NG 11.4 1.0 NG 9.5 1.7 NG 11.02 3.19 NG
NG NG NG NG
Data from Ref. [17].
b
Data from Ref. [18]. 8
c
Data from Ref. [15]. Theor. means coefficients are derived from theoretical estimation.
d
Data from Ref. [16].
e
NG means no given.
In some cases, the thermal expansion coefficients in specific direction are defined as follows [39]:
hkl c cos2 a sin 2 aa a cos2
(1)
where φ is the angle between the direction [hkl] and the c-axis. According to the fitting curves of αa and αc, the directional coefficients αhkl are plotted against cos2φ in Figure. 5. On this plot, αhkl corresponds to αa when cos2φ=0 whereas to αc when cos2φ=1. As can be seen, the values of αhkl are all positive, which means it expands in all directions. It is obviously that all the curves seem to intersect at one point nearby cos2φ=0.67, i.e., φ≈35°. That’s to say, there exists specific directions and specific planes where the thermal expansion coefficients are the same and vary in a small range or even tend to be constant in spite of the changing temperatures. It is still exist differences that this point tested by CdGeAs2 powder is at cos2φ=0.5, i.e., φ≈45° [38]. To validate this deduction, we measured the thermal expansion coefficients in [204] direction which is shown in Figure. 6. The green squares are the testing data of the bulk crystal along the [204] direction. This thermal expansion coefficients is still in linear connection with temperature: 204 2.06 106 4.42 109 t . Meanwhile, the pink line is on behalf of the derivative result from the axial thermal expansion coefficients.
From
the
comparative
analysis
of
derivative
line
204 3.85 106 1.37 109 t and the actual measure value, this derivation is an actual and effective method to calculate the thermal expansion coefficients in a given direction.
3.3 Temperature dependence of Grüneisen parameters Generally, crystal properties that depend on the thermal motion of atoms are much influenced by the anharmonicity of lattice vibration, which means the nonlinearity of interatomic forces with respect to atomic displacements [40]. The anharmonic properties are usually described by Grüneisen parameters which are 9
dimensionless quantity characterizing the volume or strain dependence of the lattice vibrational mode frequencies. As for CdGeAs2 crystal, two independent principal Grüneisen parameters, γa and γc, must be introduced. According to Barron et al.[41], the two Grüneisen parameters are determined as follows:
a
Vm c11 c12 a c13 c Cp
c
Vm 2c13 a c33 c Cp
(2)
(3)
where Vm is the molar volume, Cp is the molar specific heat at constant pressure, cij is the elastic stiffness, and αa and αc are the thermal expansion coefficients along a-axis and c-axis, respectively. The volume Grüneisen parameter γV is a linear combination of γa and γc weighted by the respective linear compressibilities χa and χc as well as the volume compressibility χV. The corresponding relations are given as follows [41, 42]:
V
2 a a c c
(4)
a
c33 c13 c33 c11 c12 2c132
(5)
c
c11 c12 2c13 c33 c11 c12 2c132
(6)
V 2 a c
c11 c12 2c33 4c13 c33 c11 c12 2c132
(7)
Up to date, the data of elastic stiffnesses at high temperatures are inadequate. The only reported data at variable temperatures are measured below 290 K [43]. Using extrapolation method, we calculated the elastic stiffnesses at different temperatures [38] from low temperature values. The heat capacity Cp has been measured in this work which is detailed in the following sections. Here, in order to compare the previous results, we still considered it as a constant equaling to 100 J/(mol K)[44, 45]. The molar volume Vm is now a temperature dependence parameter and can be easily calculated from lattice parameters which can be deduced from the αa and αc. Finally, the Grüneisen parameters γa, γc and γV are obtained according to Eqs. (2)–(7). 10
The Grüneisen parameters measured by high-temperature XRD and thermal dilatometer are plotted against temperature in Figure.7. It can be seen that γa, γc and γV are all positive and that γa is always larger than γc in the whole temperature range. The room temperature values of γa, γc and γV are evaluated to be 0.73, 0.60, and 0.67, respectively, which are very close to the literature values (0.82 for γa, 0.66 for γc and 0.75 for γV)[45] and a little smaller than the values which are measured using CdGeAs2 powder in our previous work and evaluated to be 1.40, 1.20, and 1.32, respectively. Meanwhile, it is found the values of γV measured by us are smaller likewise than that are calculated by Yu et al. Yet for all that, one could see the average Grüneisen parameters γa and γc are anisotropic and, therefore, the anharmonic forces contributing to crystal properties are also anisotropic for CdGeAs2 crystal.
3.4 The deduction and calculation of thermal conductivity Steigmeier calculated the lattice thermal conductivity for III-V compounds at high temperature using the modification equation [46], 3
3 k M 2 T 3
l 41 3 5 h
(8)
where M is the mean atomic mass, δ is the cube root of the atomic volume, and γ is the Grüneisen parameter. The lattice thermal conductive κl is obtained by subtracting the electronic contribution κel from the total thermal conductivity κ. However, judging from the thermal conductivity experimental result, this equation is not appropriate for II-IV-V2 compounds. Meanwhile, the modification equations proposed by Masim [27] and Valeri-Gil [29] are diverse from each other. Therefore, we use two modification equations to revise the results κl of bulk and powder, respectively, and the lattice thermal conductivity with different direction including κl-a, κl-c and κl-V and the thermal conductivity experimental with [204] direction are shown in Figure. 8. 3 3 k M l 41 3 2 5 h 1 T 3
11
(9)
3 M 3 k l 41 3 2 5 h 1 2 T 3
(10)
It can be seen that κ204, κl-a, κl-c and κl-V are all positive quantities and that κl-c is always larger than κl-c in the whole temperature range. With such a relationship, the lattice thermal conductivity of all the II-IV-V2 compounds which have different MδΘ3 can be deduced [27]. In addition, the electronic thermal conductivity of the intrinsic samples could be computed by using the formula [47]
A A k2 npb p p el 2 T n n 2 e nb p
Eg T Bn Bp kT
2
(11)
where n and p refer to electrons and holes, respectively; EgT is the band gap at the temperature T; σ is the electrical conductivity, and b is the mobility ratio. For Boltzmann statistics,
5 5 An, p r , Bn, p r 2 2
(12)
r is given by τ=τ0Eτ. In this case, we can assess the κel at room temperature, i.e. 300 K. Earlier in the article, the energy gap Eg and the relevant electrical parameters have been provided. Reported values of the mobility ratio b for CdGeAs2 are about 12 [35, 36] and 30 [17].
3.5 Temperature dependence of heat capacity Now we return to discuss about the thermal conductivity experimental with [204] direction κ204. To study the influence of the anion on lattice anharmonicity in the CdGeAs2 crystal, we have measured the heat capacity of it. The resulting temperature dependence of the heat capacity and molar heat capacity at constant pressure Cp, of CdGeAs2 is shown in Figure. 9. To analyze the temperature dependence of Cp and to evaluate the contribution to Cp due to lattice anharmonicity we use the general relation [48]: N C p T 12 R F T ckT k k 1
(13)
where R is the molar gas constant and F(Θ/T) the Debye function describing the 12
temperature dependence of the heat capacity in the harmonic approximation. The polynomial of Nth order in the temperature T represents the contribution to Cp, due to lattice anharmonicity. In this work, the Cp dependence on temperature is obtained by a least-squares fit relation,
C p 150.02 1.69 101T 1.15 104 T 2 3.91107 T 3
(14)
which is shown in the Figure. 9. And the specific heat Cp is nearly equal to 100 J/(mol K) from 300 K to 700 K which demonstrates the previous approximate is appropriate. In this case, κ204 can be obtained by such relation DC p , where ρ is density and D is thermal diffusivity which can be measured directly by laser flash method. After Capel model and pulse modification, thermal conductivity can be measured indirectly.
4. Summary Either αa or αc, the strength of the thermal expansion measured from bulk crystal is smaller than that measured from powder. Base on the anisotropic thermal expansion measured by these two different method, the Grüneisen parameters dependence on temperature are evaluated, and also exhibit anisotropic behavior (γa > γc). As well, the thermal expansion, γa, γc, and γV have similar difference between these two kinds of test method. The difference between these results may be due to the grain size of the powder which is in microns, and the constraints of lattice is small. And two correction formula are proposed to deduce the lattice thermal conductivities for II-IV-V2 compounds. At last, the thermal conductivity dependence on temperature with [204] direction has been measured which is demonstrated that the deduction is very close to the experimental results.
Acknowledgments This work is supported by the National Natural Science Foundation Key Programs of China (no. 50732005) and the 863 High-Tech program of China (no. 007AA03Z443).
13
References: [1] Hahn H, Frank G, Klingler W, Störger AD, Störger G. Untersuchungen über ternäre Chalkogenide. V. Über einige ternäre Chalkogenide mit Chalkopyritstruktur. Zeitschrift für anorganische und allgemeine Chemie 1953;279:241. [2] Kildal H. Band structure of CdGeAs2 near k=0., vol. 10, 1974. p.5082. [3] Gentile AL. Devices using ternary or multinary compounds. Progress in Crystal Growth and Characterization 1984;10:241. [4] Byer RL, Kildal H, Feigelson RS. CdGeAs2 -A New Nonlinear Crystal Phasematchable at 10.6μ m. Appl Phys Lett 1971;19:237. [5] Shay JL, Wernick JH. Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications. Oxford and New York: Pergamon Press, 1975. [6] Iseler GW, Kildal H, Menyuk N. Optical and electrical properties of CdGeAs 2. Journal of Electronic Materials 1978;7:737. [7] Feigelson RS, Route RK, Swarts HW. Solution growth of CdGeAs 2. J Cryst Growth 1975;28:138. [8] Isomura S, Takahashi S. Electrical and Optical Properties of CdGeAs 2. Japanese journal of applied physics 1977;16:1723. [9] Feigelson RS, Route RK. Vertical bridgman growth CdGeAs2 with control of interface shape and orientation. Journal of Crystal Growth 1980;49:261. [10] Baumgartner FP, Lux-Steiner M, Bucher E. Growth of CdGeAs2 single crystals by the chemical vapor transport method. J Electron Mater 1990;19:777. [11] Manimaran M, Ramasamy P. Growth and microhardness studies of CdGeAs 2 single crystals. J Mater Sci Lett 1994;13:1729. [12] Saghir MZ, Labrie D, Ginovker A, Paton BE, George AE, Olson K, Simpson AM. Float-zone crystal growth of CdGeAs2 in microgravitynumerical simulation and experiment. J Cryst Growth 2000;208:370. [13] Feigelson RS. Improving optical transparency in CdGeAs2 crystals by controlling crystalline defects. J Cryst Growth 2006;292:179. [14] Kildal H, Mikkelsen JC. Efficient doubling and CW difference frequency mixing in the infrared using the chalcopyrite CdGeAs2., vol. 10, 1974. p.306. [15] Neumann H. Trends in the thermal expansion coefficients of the A IBIIIC2VI and AIIBIVC2V chalcopyrite compounds. Kristall und Technik 1980;15:849. [16] Kumar V, Sastry BSR. Relationship between the thermal expansion coefficient, plasmon energy, and bond length of ternary chalcopyrite semiconductors. J Phys Chem Solids 2002;63:107. [17] Iseler GW, Kildal H, Menyuk N. Optical and electrical properties of CdGeAs2. J Electron Mater 1978;7:737. [18] David NN. Nonlinear Optical Crystals: A Complete Survey. New York: Springer, 2005. [19] Pandey R, Ohmer MC, Gale JD. A theoretical study of native acceptors in CdGeAs 2. 1998;10:5525. [20] Schunemann PG, Pollak TM. Single crystal growth of large, crack-free CdGeAs2. Journal of Crystal Growth 1997;174:272. [21] He Z, Zhao B, Zhu S, Li J, Zhang Y, Du W, Huang W, Chen B. Preparation and characterization of CdGeAs2 crystal by modified vertical Bridgman method., vol. 314, 2011. p.349. 14
[22] Huang W, Zhao B, Zhu S, He Z, Chen B, Tang J, Liu W. Growth and characterizations of CdGeAs2 single crystal by descending crucible with rotation method. Rare Metals 2014;33:210. [23] Červinka L, Kašpar J. A high-temperature x-ray study of CdGeAs2. Czechoslovak Journal of Physics B 1970;20:101. [24] Abrahams SC, Bernstein JL. Piezoelectric nonlinear optic CuGaSe2 and CdGeAs2 Crystal structure, chalcopyrite microhardness, and sublattice distortion. J Chem Phys 1974;61:1140. [25] Limpijumnong S, Lambrecht WRL. Band structure of CdGeAs 2 near the fundamental gap. Phys Rev B 2002;65:165204. [26] Bai L, Xu C, Gilesa NC, Nagashiob K, Feigelson RS. Correlation of the electrical and optical properties of p-type CdGeAs2. J Appl Phys 2006;99:13512. [27] Wasim SM. Thermal conductivity of ternary Compounds. physica status solidi (a) 1979;51:K35. [28] Bellabarba C, Wasim SM. Thermal conductivity of AgInTe2. physica status solidi (a) 1981;66:K105. [29] Valeri-Gil ML, Rincón C. Thermal conductivity of ternary chalcopyrite compounds. Mater Lett 1993;17:59. [30] Rincón C, Valeri-Gil ML, Wasim SM. Room-Temperature Thermal Conductivity and Grüneisen Parameter of the I-III-VI2 Chalcopyrite Compounds. Physica Status Solidi (a) 1995;147:409. [31] He Z, Zhao B, Zhu S, Li J, Zhang Y, Du W, Huang W, Chen B. Preparation and characterization of CdGeAs2 crystal by modified vertical Bridgman method. J Cryst Growth 2011;314:349. [32] Huang W, Zhao B, Zhu S, He Z, Chen B, Tang J, Liu W. Growth and characterizations of CdGeAs2 single crystal by descending crucible with rotation method. Rare Metals 2014;33:210. [33] Nagashio K, Watcharapasorn A, Zawilskia KT, DeMattei RC, Feigelson RS, Bai L, Giles NC, Halliburton LE, Schunemann PG. Correlation between dislocation etch pits and optical absorption in CdGeAs2. J Cryst Growth 2004;269:195. [34] Bai L, Garces NY, Yang N, Schunemann PG, Setzler SD, Pollak TM, Halliburton LE, Giles NC. Optical and EPR Study of Defects in Cadmium Germanium Arsenide. Materials Research Society 2003;744:M8. [35] Ptak AJ, Jain S, Steven KT, Myer TH, Schunemann PG, Setzler SD, Pollak TM. Temperature dependent Hall measurements made on CdGeAs2. Mat. Res. Soc. Proc 2000;607:427. [36] Bairamov BH, Rud VY, Rud YV. Properties of dopants in ZnGeP 2 CdGeAs2, AgGaS2 and AgGaSe2. Mrs Bull 1998;23:41. [37] Fischer DW, M CO, McCrae JE. Influences of temperature and transport properties on the birefringence of CdGeAs2. J Appl Phys 1997;81:3579. [38] Liu W, Zhao B, Zhu S, He Z, Chen B, Huang W, Tang J, Yu Y. X-ray study of thermal expansion behaviors and Grüneisen parameters of cadmium germanium arsenide crystal over the temperature range 25–450 ℃. J Appl Phys 2013:53513. [39] Bodnar IV, Orlova NS. X-Ray Study of the Thermal Expansion Anisotropy in AgGaS2 and AgInS2 Compounds over the Temperature Range from 80 to 650 K. physica status solidi (a) 1985;91:503. [40] Ashcroft NW, Mermin ND. Solid State Physics. New York: Saunders College, 1976. [41] Barron THK, Collins JG, White GK. Thermal expansion of solids at low temperatures., vol. 29, 1980. p.609. [42] Yu Y, Zhao B, Zhu S, Gao T, Hou H. Ab initio vibrational and dielectric properties of chalcopyrite CdGeAs2. Solid State Sci 2011;13:422. 15
[43] Hailing T, Saunders GA, Lambson WA, Feigelson RS. Elastic behaviour of the chalcopyrite CdGeAs2. Journal of Physics C: Solid State Physics 1982;15:1399. [44] Yu Y, Zhao BJ, Zhu SF, Gao T, Hou HJ, He ZY. Theoretical study of elastic and thermodynamic properties of chalcopyrite CdGeAs2. Physica B: Condensed Matter 2013;417:83. [45] Neumann H. Grüneisen parameters in AgGaS2 and CdGeAs2. Cryst Res Technol 1983;18:K126. [46] Steigmeier EF, Kudman I. Thermal Conductivity of III-V Compounds at High Temperatures. Physical Review 1963;132:508. [47] Madelung O. Handbuch der physik. Berlin: Springer-Verlag, 1957. [48] Möller W, Kühn G, Neumann H. Heat capacity and lattice anharmonicity in CdBIVC2V chalcopyrite compounds. Cryst Res Technol 1987;22:533.
Figure. 1. Photograph of crack-free CdGeAs2 single crystals and an embryo of SHG device. Figure. 2. IR transmission spectra recorded for a 2-mm-thick CdGeAs2 samples, the corresponding absorption coefficient and absorption coefficient vs photon energy spectra of CdGeAs2 single crystals. Figure. 3. Overlay of X-ray diffraction patterns of CdGeAs2 crystal measured at different temperature. The lower part is the PDF card data measured at room temperature. Figure. 4. Variation of lattice parameters αa, αc, αV and αη as a function of temperature for CdGeAs2. Dots represent the high temperature experimental data, while blue solid lines are the polynomial fit curves. The green square represent the data measured by dilatometer, while orange solid lines are the polynomial fit curves. The polynomial expressions, for each parameter, are presented. Figure .5. Variation of directional coefficient αhkl at different temperatures as a function of cos2φ. Figure. 6. Thermal expansion coefficients in specific direction α204 as a function of temperature for CdGeAs2. Figure .7. Variation of the Grüneisen parameters γa, γc, and γV as a function of temperature. Figure .8. Lattice thermal conductivity including κl-a, κl-c and κl-V and the thermal conductivity experimental with [204] direction. Figure .9. Temperature dependence of the heat capacity and molar heat capacity at constant pressure Cp.
16
17
18
19
20
21
Table .I Electrical properties of the samples This work
Electrical properties
300 K
77 K
8.819×1014~
Bulk carrier -3
Ref. [36]
16
concentration（cm ）
1.977×10
Mobility （cm2/Vs）
13.66~402.2
4.276×107~5. 366×1010 17.78~895.8
Ref.[17]
300 K
77 k
300 K
77 k
0.4×1016
-
1016~1017
-
181
-
12~30
-
9
1.2×104
1~10
102~105
-
-
102
107
5
2.175×10 ~1.
Resistivity （Ω·cm）
1.298~74.44
Average Hall
3.158×102~
1.163×108~1.
coefficient（cm3/C）
7.078×103
460×1011
663×108
22
Table II. Thermal expansion coefficients of lattice parameters at variable temperatures along with literature values. Thermal expansion coefficients (unit: ×10-6K-1) ` Temperature(K)
αa
300 373 473 573 673 723
13.4 14.0 14.9 15.8 16.6 17.1 a
373~573 293~673b Theorc Theord a
Powder αc αη 5.0 4.2 3.2 2.2 1.1 0.6
-8.4 -9.8 -11.7 -13.6 -15.5 -16.5
Literature values 8.5 1.0 NG 11.4 1.0 NG 9.5 1.7 NG 11.02 3.19 NG
Bulk αV
αa
αc
αη
αV
31.8 32.2 33.0 33.8 34.3 34.8
7.44 7.90 8.53 9.15 9.78 10.09
1.43 1.21 0.91 0.61 0.31 0.16
-6.01 -6.69 -7.61 -8.54 -9.47 -9.93
16.31 17.01 17.96 18.92 19.87 20.35
NG NG NG NG
Data from Ref. [17].
b
Data from Ref. [18].
c
Data from Ref. [15]. Theor. means coefficients are derived from theoretical estimation.
d
Data from Ref. [16].
e
NG means no given.
>CdGeAs2 single crystal was grown by VB method>Optical, electrical and thermodynamic properties are characterized > thermal expansion coefficients are measured by two method> Grüneisen parameters have been calculated by thermal expansion coefficients> thermal conductivity are obtained by experiment and theoretical derivation
23