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Structural and electronic properties of liquid InSb alloy: An ab initio molecular-dynamics simulation
Chemical Physics Letters 408 (2005) 348–353 www.elsevier.com/locate/cplett
Structural and electronic properties of liquid InSb alloy: An ab initio molecular-dynamics simulation Changqiao Zhang *, Yunhe Wei, Chenfu Zhu College of Chemistry and Chemical Engineering, Shangdong University Jinan, Shandong 250100, PR China Received 31 March 2005 Available online 13 May 2005
Abstract The structural and electronic properties of liquid InSb have been simulated by using ab initio molecular dynamics. The calculated results indicate that the covalent bonds of In–Sb similar to those of c-InSb are preserved in liquid state, and the Sb clusters have open structure while the In clusters have close-packed structure. 2005 Elsevier B.V. All rights reserved.
1. Introduction InSb is one of the popular III–V semiconductors and its structural, electronic properties are similar to those of IV semiconductors. In the crystalline phase, the In–Sb bond has a weak ionic nature, and its Phillips ionicity is 0.32 [1]. Zinc-blende-type InSb crystalline phase is stable at ambient condition with a coordination number of 4. Upon melting, the InSb becomes metallic, similar to Si and Ge, and with the density increasing 12.5% [2], and parts of the covalent bonds are broken by the thermal motion of atoms, compositional defects or ÔwrongÕ bonds exist (bonds between the same type atoms are absent in the crystalline state). Such defects are thought to be considerable influence on electronic properties. Many physical properties of l-InSb have been measured, such as the velocity of sound, [3] adiabatic compressibility [3], and the resistivity [4], etc. For example, the temperature coefficient of the resistivity (TCR) of l-InSb increases with increasing temperature near the melting point and then is independent of temperature above 890 K [4]. But as Sb content less than 40 at.% in l-In1x Sbx, the TCR is constant over whole temperature *
Corresponding author. Fax: +86 531 8564464. E-mail addresses: [email protected], (C. Zhang).
[email protected]
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.04.060
range. This phenomenon has already been observed for many liquid-Sb-based alloys on the Sb-rich side, and it shows more obviously with increasing Sb content [5–8]. For pure l-In, the curve of TCR versus temperature is constant over the whole temperature range [4]. But for l-Sb, this curve shows a turning point interpreted as the weak Perierls distortion surviving in liquid Sb [4]. Therefore, these experimental results suggest that the abnormal changes observed in the resistivity versus temperature for liquid-Sb-based alloys should be related to the peculiar local structure of Sb atoms in these liquid alloys. In order to explain the abnormal change of resistivity versus temperature, Wang et al. [9] investigated the structure of l-InSb with the extended X-ray-absorption fine-structure method and indicated that there were heterogeneous atomic coordinations and some tetrahedral units in l-InSb near the melting point. Because of great difficulties in experiment, the report of the partial structure factors or partial pair correlation functions for the l-InSb alloy has not been seen so far. The experimental results quoted above indicate that the liquid structure of InSb is complex. To gain more insight into the structural and chemical bonding associated with l-InSb, we have performed ab initio molecular-dynamics (AIMD) simulations for l-InSb at 813 K slightly higher than the melting point. The AIMD
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
349
simulations have been proved to be very reliable for the prediction of the structural properties of a variety materials, including liquid metals and semiconductors, and previous AIMD simulations of liquids, such as Si [10], Ge [11], Ga [12], GaSe [13], GaAs [14,15], CdTe [16], GeTe [17] etc., have demonstrated close agreement with experimental results. To our knowledge, there are no AIMD simulations for structure of l-InSb. In this paper, we will focus on the following two problems. First, the anomalous changes of physical properties with temperature in l-InSb should be ascribed to its complex liquid structure. Thus, we will investigate the partial structural functions of l-InSb, and then compare them with the pure element liquids (l-In, l-Sb) and the crystalline phase of InSb. Second, we will study the electronic structures of l-InSb alloy. These results will help us to understand the mechanism of the semiconductor-metal transition in melting and the anomalous changes of physical properties with temperature near melting point. The paper is organized as follows: In Section 2 we discuss the theoretical framework and in Section 3 we present the results of static structure and compare with the experimental results. In Section 4 we report the electronic structure of l-InSb alloy. The final part is the conclusion.
Our simulations for l-InSb, l-In and l-Sb have been performed by using the cubic supercell which contains 80 atoms and the periodic boundary conditions are imposed. The temperatures are set at 813 K for l-InSb, 973 K for l-In, 933 K for l-Sb and the experimental density is used (for l-InSb q = 6.51 g cm3 [24], for l-In q = 6.61 g cm3, [25] and for l-Sb q = 6.38 g cm3 [25]). The simulations are performed in a canonical ensemble with a Nose´ thermostat for temperaturecontrol [26]. The equation of motion is solved via the velocity Verlet algorithm with a time step 3 fs. The C point alone is used to sample the Brillouin Zone of the supercell. To prepare the liquid, the atoms are initially arranged in cubic supercell randomly. Then, the system is heated up to 2500 K by rescaling the ionic velocities. After equilibrating for 3 ps at this temperature, we gradually reduce the temperature to 813 K. After equilibrating for 3 ps again, the quantities of interest are obtained by averaging with configurations from another 12 ps. For l-In and l-Sb, we only repeat this procedure and change the final temperature into 973 K and 933 K, respectively. In all simulations, the energy conservation is excellent; we find that the drift is smaller than 0.5 mev/ atom/ps.
2. Computational methods
3. Structural properties
Our calculations have been performed by using Vienna ab initio simulation program VASP [18,19]. The electronic exchange and correlation are described in the generalized gradient approximation (GGA), using the PW91 functional due to Perdew and Wang [20]. The solution of the generalized Kohn–Sham equations is performed with an efficient iterative matrix-diagonalization routine based on a sequential band-by-band residual minimization method (RMM) [18] applied to the one-electron energies. An improved Pulay-mixing [21] is used to achieve self-consistency of charge-density and potential [18]. The ultrasoft pseudopotentials (USPP) of Vanderbilt type are used to describe the electron–ion interaction [22,23]. The electronic wave functions are expanded in the planewave basis set. To obtain an appropriate plane wave cutoff, we have evaluated the total energy for zincblende structure of InSb as a function of plane-wave cutoff energy and the results indicate that taking a cutoff energy of 200 eV gives a total energy convergence of 1.0 meV compared to the total energy at a cutoff of 400 eV. With this energy cutoff and USPP, we obtain the equilibrium lattice parameter of zinc-blende structure of InSb to be 0.6615 nm which is about 2.0% larger than the experimental value (0.6476 nm). All of following calculations have been performed with this energy cutoff 200 eV and the USPP.
In a molecular dynamics simulation of liquid state, the structure factor, S(k), serves as a connection with experimental results. In our paper, the total structure factor is expressed as a linear combination of the partial structural factors Sij(K), normalized by the neutron scattering lengths of In and Sb. SðkÞ ¼
XX i
j
1=2 1=2
ci cj
bi bj S ij ðkÞ; þ cj b2j
ci b2i
ð1Þ
where bi is the neutron scattering length of the ith species (bIn = 4.065 fm, [27] bSb = 5.25 fm [28]), ci is the concentration. The calculated total structure factor is shown in Fig. 1 in which we compare the theoretical S(k) with the experimental result obtained from X-ray diffraction [29] and find a good agreement between them. To investigate the structure of l-InSb in details, we have also studied the structures of l-In and l-Sb of which the pair correlation functions are in good agreement with the experimental results [25] (in Fig. 2). Our calculated g(r) of l-Sb by using GGA method is similar to that of LDA method by Seita [30]. The noticeable character in g(r) of l-Sb is a small shoulder at r = 0.43 nm which is the position of the second-neighbor distance in crystalline state (rhombohedral A7-structure). Thus, there are covalent bonds of Sb–Sb surviving in l-Sb near the melting point.
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C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353 3.0
2.5
2.5
2.0 1.5
S (k)
g (r)
2.0
1.0
1.5
0.5 1.0
0.0 0.0
0.5
0.2
0.4
0.6
0.8
1.0
r (nm) Fig. 3. Total pair correlation function of l-InSb.
0.0 0
10
20
30
40
50
60
70
80
-1
K (nm ) Fig. 1. Structure factor of l-InSb. Full lines – ab initio MD, circles – X-ray diffraction [29].
3
Sb
g (r)
2
1
0
In g (r)
2
The partial pair correlation functions of l-InSb, l-In and l-Sb are shown in Fig. 4. For l-InSb, the nearestneighbor distances of In–In, In–Sb and Sb–Sb are 0.315, 0.31 and 0.305 nm, respectively. The nearestneighbor distance of l-In is 0.305 nm, and for l-Sb this value is 0.31 nm. Furthermore, the gInSb(r) has the highest first-peak among the three partial pair correlation functions indicating the heterocoordination preference in l-InSb. Both the gInIn(r) and gInSb(r) have prominent troughs between the first and second peaks. Obviously the small hump between the first and second peaks in the total g(r) originates from the gSbSb(r). It is instructive to compare the correlation between the same type atoms in l-InSb with that of pure l-In and l-Sb. We find that the first and second peaks of gSbSb(r) in l-InSb are nearly coincidence with that of l-Sb and the only difference is that the shoulder on the right-hand side of the first peak of gSbSb(r) in l-Sb changes into a small hump in l-InSb. Also the overall features of gInIn(r) of l-InSb are similar
1 3 2
0.2
0.4
0.6
0.8
1.0
r (nm)
gInSb (r)
0 0.0
Fig. 2. Pair correlation functions for l-In and l-Sb. Full line – ab inititio MD, circles – X-ray diffraction data of Waseda [25].
1 0
l-Sb l-InSb
1
0
l-In l-InSb
2
gInIn (r)
The total pair correlation function of l-InSb, which is obtained by weighting the partial pair correlation functions with the neutron scattering length as in the calculation of the total structure factor, is shown in Fig. 3. The total pair correlation function of l-InSb shows characters of non-simple liquid structure in three aspects: (1) There is a small hump between the first and second peaks. (2) The trough between the first and second peaks is less prominent than that of simple liquids. (3) The ratio r2/r1, where r1 and r2 are the positions of the first and second peaks, respectively, is 2.01 which is larger than that of simple liquids (1.84–1.90) [25].
gSbSb (r)
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
r (nm) Fig. 4. The partial pair correlation functions of l-InSb alloy and the pair correlation functions of l-In and l-Sb.
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
to that of l-In except for the first peak of gInIn(r) of l-InSb lower than that of l-In. The bond-angle distributions of In and Sb clusters in l-In, l-Sb and l-InSb are presented in Fig. 5. To emphasize the deviation from random distributions, bondangle distributions are normalized by sin(h). The bInInIn of l-InSb, with cutoff distance Rcut = 0.40 nm (i.e., including essentially the symmetric part of the first peak of pair correlation function), has two peaks, one is near 60 and the other near 120. Such peaks are expected in closed-packed liquid and also found in pure l-In. The two bond-angle distributions (bInInIn) of l-In and l-InSb are almost coincident with each other. This indicates that the local structure of In atoms in l-InSb is similar to that of pure l-In showing as a close-packed structure which is consistent with what has been found in pair correlation function. In contrast to indium, the bSbSbSb of l-Sb and l-InSb, with cutoff distance of 0.34 nm, have two peaks; one is near 90 and the other near 180 indicating that the clusters of Sb atoms have a more open structure. The covalent character of Sb–Sb bonds defines a strong angle correlation peak near 90 and the peak at 180 corresponds to an open angle of the chained structures of Sb atoms. Seifert et al. [30] had also studied the structure of pure l-Sb and found these two peaks. The bSbSbSb of l-InSb is almost the same as that of l-Sb. Thus, the calculated results of pair correlation functions and bond-angle distributions for l-In, l-Sb and l-InSb indicate that the local structures of likeatoms (In–In and Sb–Sb) of l-InSb resemble those of pure element liquids (l-In and l-Sb). To investigate the characters of the bonds of In–Sb, we calculate the bSbInSb and bInSbIn with the cutoff distance of 0.30 nm, i.e., 10% larger than the bond length of c-InSb. The results are shown in Fig. 6. The peak in bSbInSb is at 109, i.e., the bond angle of c-InSb, but for bInSbIn, this peak shifts to 90 due to the close-packed indium atoms. Thus, the bonds of In–Sb similar to that of c-InSb exist in l-InSb. Given the partial pair correlation functions, it is possible to calculate the partial coordination numbers as
InInIn
l-In l-InSb
SbSbSb
l-Sb l-InSb 0
40
80
(deg)
120
160
0
40
80
120
160
(deg)
Fig. 5. Bond-angle distributions of In and Sb clusters in l-in, l-Sb and l-InSb.
351
SbInSb InSbIn
0
40
80
120
160
(deg) Fig. 6. Bond-angle distributions of SbInSb and InSbIn of l-InSb. Cutoff distance is 0.30 nm.
N ab ¼
Z
Rmin
4pr2 qb gab ðrÞ dr;
ð2Þ
0
where RMin is the first minimum coordinate in g(r). We use the RMin of the total pair correlation function to calculate partial coordination numbers, RMin = 0.40 nm. We obtain NInIn = 3.7, NInSb = 4.7 and NSbSb = 3.4. To estimate the compositional defects, a compositional disorder number (CDN), defined as the ratio of homogeneous and heterogeneous bonds [14], (Naa + Nbb)/2Nab, is calculated. The calculated CDN is 0.75 which indicates that the high concentration heteroatomic bonds exist in l-InSb.
4. Electronic structure The structure behavior of the liquid alloys can be understood in terms of the electronic structure. Here we have investigated the electronic density of states (DOS) and the local density of states (LDOS) [31]. The LDOS are obtained by projecting the wave functions onto spherical harmonic centered on each atom with a radius of 0.144 nm for In atoms and 0.14 nm for Sb atoms (covalent radius). Both the DOS and LDOS of l-InSb are obtained by averaging on ten configurations. The DOS and LDOS for l-InSb and c-InSb (zincblende structure) are shown in Fig. 7. In the zincblende structure, a gap at Fermi level as a signature of semiconductor is observed, the sharpening of both the s and p peaks for In and Sb atoms can be found. The main peaks appear at 9.0 eV, 5.5 eV and 2.0 eV originated from the Sb(s), In(s), and In(s, p)-Sb(p) bonding orbitals. Obviously, the In–Sb bonds mainly due to the interaction of In(p) and Sb(p) orbitals and both the In(p) and Sb(p) orbitals have been split into two peaks, one is the bonding state at 2.0 eV and the other is the antibonding sate at +5 eV. In liquid state, the gap at Fermi level is disappeared demonstrating metallic-like behavior, but a small ÔdipÕ is still preserved. From LDOS, we find that this ÔdipÕ results from the
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C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
(a)
(b) Total
Total
In s In p
In s In p
Sb s Sb p
Sb s Sb p
-15
-10
-5
0
5
10
-15
-10
-5
E-Ef (eV)
0
5
10
E-Ef (eV)
Fig. 7. Total electronic density of states and the local density of states for InSb in the zincblende structure (a) and the liquid state (b).
rðxÞ ¼
occ unocc X X 2pe2 X jhw jp jw ij2 dðEn Em hxÞ; 3m2 xX m n a¼x;y;z m a n
ð3Þ where Ei and wi are eigenvalues and eigenfunctions, and X is the volume of the supercell. Dipole transition elements, Æwmjpajwnæ, were sampled by the C point of the Brillouin Zone. The calculated conductivities of l-InSb are results of averaging on forty configurations chosen at random from the representative ensembles and illustrated in Fig. 8 which shows a Drude-like falloff. This behavior has also been found in l-Si [10], l-Ge [11] and l-GaAs [14,15] as a signature of metallic liquid. We can extrapolate the frequency dependent conductivity to zero frequency to estimate the dc conductivity. The estimated value 9660 X1 cm1 is in remarkable agreement with the experimental values of 9200–
0.25 0.20
σ (ω) (a.u.)
In(p)–Sb(p) interaction, i.e., both the splits of In(p) and Sb(p) occur at Fermi level. The positions of bonding and antibonding states of In(p) and Sb(p) in the liquid state are similar to those in c-InSb, respectively. In addition, the peaks of In(s) and Sb(s) are also at the same position as in c-InSb. Thus, our calculated results of electronic structure confirm that the In–Sb bonds still exist in lInSb near melting point. Once the energy states and wave functions for a given atomic configuration are obtained, the optical conductivity can be calculated by using Kubo–Greenwood expression [32]. The real part of the conductivity is determined by the sum of all possible dipole transition at a given frequency:
0.15 0.10 0.05 0.00 0
1
2
3
4
5 6 ω (ev)
7
8
9 10
Fig. 8. Real part of optical conductivity for l-InSb. The conductivity is in atomic unit.
9900X1 cm1 [4]. However, considering the several approximations used in our calculation, such a good agreement is to some extent fortuitous. By fitting a Drude curve rðxÞ ¼
rð0Þ 1 þ x 2 s2
ð4Þ
to our data, we extract a relaxation time s1 = 2.59 eV for l-InSb. Although no experimental values of s are available for l-InSb, a value of the same order of magnitude, s1 = 2.99 eV, has been measured in l-Si [33]. Using these values of s, and the free-electron expression l = hkFs/m for the mean free path, we find that the ratios of l/a, where a is the nearest-neighbor distance, is 1.5 for l-InSb, indicating a strong scattering behavior of l-Insb. In l-Si the ratio, calculated in a similar way, is slightly larger, i.e., l/a = 1.8 [10].
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
5. Summary and conclusion We have performed ab initio molecular-dynamics simulations of l-InSb at temperature of 813 K. Our theoretical results of l-InSb are in good agreement with available experimental data. By analyzing the structural parameters of pair correlation functions and bond-angle distributions, we find that the local structures of In and Sb atoms of l-InSb are similar to those of pure element liquids, especially, the cluster of Sb atoms has more opened structure similar to that of l-Sb. The bond-angle distributions of In–Sb–In and Sb–In–Sb and the electronic structure of l-InSb indicate that the In–Sb bonds similar to those of crystalline state are preserved in liquid states. It can be expected that the remnants of covalent bonds of In–Sb and the peculiar local structure of Sb atoms will be gradually destroyed with increasing temperature which may cause anomalous changes in physical properties. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 20133020).
References [1] G. Phillips, Bands and Bonds in Semiconductors, Academic Press, New York, 1973. [2] H.P Mokroskii, A.R. Regel, J. Technol., Phys. 22 (1952) 1282. [3] Y. Tsuchiya, J. Phys. Soc. Jpn. 66 (1997) 120. [4] Q. Wang, X. Chen, K. Lu, J. Phys.: Condens. Matter 12 (2000) 5201. [5] M.E. Bakkali, J.G. Gasser, P. Terzieff, Z. Metallk 84 (1993) 662. [6] S. Ohno, H. Okazaki, S. Tamaki, J. Phys. Soc. Jpn. 36 (1974) 1133.
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[7] S. Ohno, H. Okazaki, S. Tamaki, J. Phys. Soc. Jpn. 35 (1973) 1060. [8] A. Bath, J.G. Gasser, J.L. Bretonnet, R. Bianchin, R. Kleim, J. Physique Coll. 41 (1980) C8519. [9] Y.R. Wang, K.Q. Lu, C.X. Li, Phys. Rev. Lett. 79 (1997) 3664. [10] I. Stich, R. Car, M. Parrinello, Phys. Rev. B 44 (1991) 11092. [11] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [12] J.M. Holender, M.J. Gillan, M.C. Payne, A.D. Simpson, Phys. Rev. B 52 (1995) 967. [13] J.M. Holender, M.J. Gillan, Phys. Rev. B 53 (1996) 4399. [14] V. Godlevsky, J.R. Chelikowsky, J. Chem. Phys. 109 (1998) 7312. [15] Q.M. Zhang, G. Chiarotti, A. Selloni, R. Car, M. Parrinello, Phys. Rev. B 42 (1990) 5071. [16] V. Godlevsky, J.J. Derby, J.R. Chelikowsky, Phys. Rev. Lett. 81 (1998) 4959. [17] J.Y. Raty, V. Godlevsky, J.P. Gaspard, C. Bichara, M. Bionducci, R. Bellissent, R. Ce´olin, J.R. Chelikowsky, Ph. Ghosez, Phys. Rev. B 65 (2002) 115205. [18] G. Kresse, J. Furthmu¨ller, Comput. Mater. Sci. 6 (1996) 15. [19] G. Kresse, J. Furthmu¨ller, Phys. Rev. B 54 (1996) 11169. [20] Y. Wang, J.P. Perdew, Phys. Rev. B 44 (1991) 13298. [21] P. Pulay, Chem. Phys. Lett. 73 (1980) 393. [22] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [23] G. Kresse, J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245. [24] Y. Sato, T. Nishizuka, T. Takamizawa, T. Yamamura, Y. Waseda, Int. J. Thermophys. 23 (2002) 235. [25] Y. Waseda, The Structure of Non-crystalline Materials, McGrawHill, New York, 1980. [26] S. Nose, J. Chem. Phys. 81 (1984) 511. [27] L. Koester, K. Knopf, Z. Physik A 297 (1980) 85. [28] L. Koester, K. Knopf, W. Waschkowski, Z. Physik A 323 (1986) 359. [29] K. Tsuji, T. Hattori, T. Kinoshita, SPring-8 User Experiment Report (Japan Synchrotron Radiation Research Institute (JASRI)). [30] K. Seifert, J. Hafner, G. Kresse, J. Non-Cryst. Solids 205–207 (1996) 871. [31] A. Eichler, J. Furthmu¨ller, G. Kresse, Surf. Sci. 346 (1996) 300. [32] (a) R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570; (b) D. Greenwood, Proc. Phys. Soc. 71 (1958) 585. [33] N.F. Mott, E.A. Davis, Electronic Processes in Non-crystalline Materials, Clarendon, Oxford, 1979.
Structural and electronic properties of liquid InSb alloy: An ab initio molecular-dynamics simulation Changqiao Zhang *, Yunhe Wei, Chenfu Zhu College of Chemistry and Chemical Engineering, Shangdong University Jinan, Shandong 250100, PR China Received 31 March 2005 Available online 13 May 2005
Abstract The structural and electronic properties of liquid InSb have been simulated by using ab initio molecular dynamics. The calculated results indicate that the covalent bonds of In–Sb similar to those of c-InSb are preserved in liquid state, and the Sb clusters have open structure while the In clusters have close-packed structure. 2005 Elsevier B.V. All rights reserved.
1. Introduction InSb is one of the popular III–V semiconductors and its structural, electronic properties are similar to those of IV semiconductors. In the crystalline phase, the In–Sb bond has a weak ionic nature, and its Phillips ionicity is 0.32 [1]. Zinc-blende-type InSb crystalline phase is stable at ambient condition with a coordination number of 4. Upon melting, the InSb becomes metallic, similar to Si and Ge, and with the density increasing 12.5% [2], and parts of the covalent bonds are broken by the thermal motion of atoms, compositional defects or ÔwrongÕ bonds exist (bonds between the same type atoms are absent in the crystalline state). Such defects are thought to be considerable influence on electronic properties. Many physical properties of l-InSb have been measured, such as the velocity of sound, [3] adiabatic compressibility [3], and the resistivity [4], etc. For example, the temperature coefficient of the resistivity (TCR) of l-InSb increases with increasing temperature near the melting point and then is independent of temperature above 890 K [4]. But as Sb content less than 40 at.% in l-In1x Sbx, the TCR is constant over whole temperature *
Corresponding author. Fax: +86 531 8564464. E-mail addresses: [email protected], (C. Zhang).
[email protected]
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.04.060
range. This phenomenon has already been observed for many liquid-Sb-based alloys on the Sb-rich side, and it shows more obviously with increasing Sb content [5–8]. For pure l-In, the curve of TCR versus temperature is constant over the whole temperature range [4]. But for l-Sb, this curve shows a turning point interpreted as the weak Perierls distortion surviving in liquid Sb [4]. Therefore, these experimental results suggest that the abnormal changes observed in the resistivity versus temperature for liquid-Sb-based alloys should be related to the peculiar local structure of Sb atoms in these liquid alloys. In order to explain the abnormal change of resistivity versus temperature, Wang et al. [9] investigated the structure of l-InSb with the extended X-ray-absorption fine-structure method and indicated that there were heterogeneous atomic coordinations and some tetrahedral units in l-InSb near the melting point. Because of great difficulties in experiment, the report of the partial structure factors or partial pair correlation functions for the l-InSb alloy has not been seen so far. The experimental results quoted above indicate that the liquid structure of InSb is complex. To gain more insight into the structural and chemical bonding associated with l-InSb, we have performed ab initio molecular-dynamics (AIMD) simulations for l-InSb at 813 K slightly higher than the melting point. The AIMD
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
349
simulations have been proved to be very reliable for the prediction of the structural properties of a variety materials, including liquid metals and semiconductors, and previous AIMD simulations of liquids, such as Si [10], Ge [11], Ga [12], GaSe [13], GaAs [14,15], CdTe [16], GeTe [17] etc., have demonstrated close agreement with experimental results. To our knowledge, there are no AIMD simulations for structure of l-InSb. In this paper, we will focus on the following two problems. First, the anomalous changes of physical properties with temperature in l-InSb should be ascribed to its complex liquid structure. Thus, we will investigate the partial structural functions of l-InSb, and then compare them with the pure element liquids (l-In, l-Sb) and the crystalline phase of InSb. Second, we will study the electronic structures of l-InSb alloy. These results will help us to understand the mechanism of the semiconductor-metal transition in melting and the anomalous changes of physical properties with temperature near melting point. The paper is organized as follows: In Section 2 we discuss the theoretical framework and in Section 3 we present the results of static structure and compare with the experimental results. In Section 4 we report the electronic structure of l-InSb alloy. The final part is the conclusion.
Our simulations for l-InSb, l-In and l-Sb have been performed by using the cubic supercell which contains 80 atoms and the periodic boundary conditions are imposed. The temperatures are set at 813 K for l-InSb, 973 K for l-In, 933 K for l-Sb and the experimental density is used (for l-InSb q = 6.51 g cm3 [24], for l-In q = 6.61 g cm3, [25] and for l-Sb q = 6.38 g cm3 [25]). The simulations are performed in a canonical ensemble with a Nose´ thermostat for temperaturecontrol [26]. The equation of motion is solved via the velocity Verlet algorithm with a time step 3 fs. The C point alone is used to sample the Brillouin Zone of the supercell. To prepare the liquid, the atoms are initially arranged in cubic supercell randomly. Then, the system is heated up to 2500 K by rescaling the ionic velocities. After equilibrating for 3 ps at this temperature, we gradually reduce the temperature to 813 K. After equilibrating for 3 ps again, the quantities of interest are obtained by averaging with configurations from another 12 ps. For l-In and l-Sb, we only repeat this procedure and change the final temperature into 973 K and 933 K, respectively. In all simulations, the energy conservation is excellent; we find that the drift is smaller than 0.5 mev/ atom/ps.
2. Computational methods
3. Structural properties
Our calculations have been performed by using Vienna ab initio simulation program VASP [18,19]. The electronic exchange and correlation are described in the generalized gradient approximation (GGA), using the PW91 functional due to Perdew and Wang [20]. The solution of the generalized Kohn–Sham equations is performed with an efficient iterative matrix-diagonalization routine based on a sequential band-by-band residual minimization method (RMM) [18] applied to the one-electron energies. An improved Pulay-mixing [21] is used to achieve self-consistency of charge-density and potential [18]. The ultrasoft pseudopotentials (USPP) of Vanderbilt type are used to describe the electron–ion interaction [22,23]. The electronic wave functions are expanded in the planewave basis set. To obtain an appropriate plane wave cutoff, we have evaluated the total energy for zincblende structure of InSb as a function of plane-wave cutoff energy and the results indicate that taking a cutoff energy of 200 eV gives a total energy convergence of 1.0 meV compared to the total energy at a cutoff of 400 eV. With this energy cutoff and USPP, we obtain the equilibrium lattice parameter of zinc-blende structure of InSb to be 0.6615 nm which is about 2.0% larger than the experimental value (0.6476 nm). All of following calculations have been performed with this energy cutoff 200 eV and the USPP.
In a molecular dynamics simulation of liquid state, the structure factor, S(k), serves as a connection with experimental results. In our paper, the total structure factor is expressed as a linear combination of the partial structural factors Sij(K), normalized by the neutron scattering lengths of In and Sb. SðkÞ ¼
XX i
j
1=2 1=2
ci cj
bi bj S ij ðkÞ; þ cj b2j
ci b2i
ð1Þ
where bi is the neutron scattering length of the ith species (bIn = 4.065 fm, [27] bSb = 5.25 fm [28]), ci is the concentration. The calculated total structure factor is shown in Fig. 1 in which we compare the theoretical S(k) with the experimental result obtained from X-ray diffraction [29] and find a good agreement between them. To investigate the structure of l-InSb in details, we have also studied the structures of l-In and l-Sb of which the pair correlation functions are in good agreement with the experimental results [25] (in Fig. 2). Our calculated g(r) of l-Sb by using GGA method is similar to that of LDA method by Seita [30]. The noticeable character in g(r) of l-Sb is a small shoulder at r = 0.43 nm which is the position of the second-neighbor distance in crystalline state (rhombohedral A7-structure). Thus, there are covalent bonds of Sb–Sb surviving in l-Sb near the melting point.
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C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353 3.0
2.5
2.5
2.0 1.5
S (k)
g (r)
2.0
1.0
1.5
0.5 1.0
0.0 0.0
0.5
0.2
0.4
0.6
0.8
1.0
r (nm) Fig. 3. Total pair correlation function of l-InSb.
0.0 0
10
20
30
40
50
60
70
80
-1
K (nm ) Fig. 1. Structure factor of l-InSb. Full lines – ab initio MD, circles – X-ray diffraction [29].
3
Sb
g (r)
2
1
0
In g (r)
2
The partial pair correlation functions of l-InSb, l-In and l-Sb are shown in Fig. 4. For l-InSb, the nearestneighbor distances of In–In, In–Sb and Sb–Sb are 0.315, 0.31 and 0.305 nm, respectively. The nearestneighbor distance of l-In is 0.305 nm, and for l-Sb this value is 0.31 nm. Furthermore, the gInSb(r) has the highest first-peak among the three partial pair correlation functions indicating the heterocoordination preference in l-InSb. Both the gInIn(r) and gInSb(r) have prominent troughs between the first and second peaks. Obviously the small hump between the first and second peaks in the total g(r) originates from the gSbSb(r). It is instructive to compare the correlation between the same type atoms in l-InSb with that of pure l-In and l-Sb. We find that the first and second peaks of gSbSb(r) in l-InSb are nearly coincidence with that of l-Sb and the only difference is that the shoulder on the right-hand side of the first peak of gSbSb(r) in l-Sb changes into a small hump in l-InSb. Also the overall features of gInIn(r) of l-InSb are similar
1 3 2
0.2
0.4
0.6
0.8
1.0
r (nm)
gInSb (r)
0 0.0
Fig. 2. Pair correlation functions for l-In and l-Sb. Full line – ab inititio MD, circles – X-ray diffraction data of Waseda [25].
1 0
l-Sb l-InSb
1
0
l-In l-InSb
2
gInIn (r)
The total pair correlation function of l-InSb, which is obtained by weighting the partial pair correlation functions with the neutron scattering length as in the calculation of the total structure factor, is shown in Fig. 3. The total pair correlation function of l-InSb shows characters of non-simple liquid structure in three aspects: (1) There is a small hump between the first and second peaks. (2) The trough between the first and second peaks is less prominent than that of simple liquids. (3) The ratio r2/r1, where r1 and r2 are the positions of the first and second peaks, respectively, is 2.01 which is larger than that of simple liquids (1.84–1.90) [25].
gSbSb (r)
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
r (nm) Fig. 4. The partial pair correlation functions of l-InSb alloy and the pair correlation functions of l-In and l-Sb.
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
to that of l-In except for the first peak of gInIn(r) of l-InSb lower than that of l-In. The bond-angle distributions of In and Sb clusters in l-In, l-Sb and l-InSb are presented in Fig. 5. To emphasize the deviation from random distributions, bondangle distributions are normalized by sin(h). The bInInIn of l-InSb, with cutoff distance Rcut = 0.40 nm (i.e., including essentially the symmetric part of the first peak of pair correlation function), has two peaks, one is near 60 and the other near 120. Such peaks are expected in closed-packed liquid and also found in pure l-In. The two bond-angle distributions (bInInIn) of l-In and l-InSb are almost coincident with each other. This indicates that the local structure of In atoms in l-InSb is similar to that of pure l-In showing as a close-packed structure which is consistent with what has been found in pair correlation function. In contrast to indium, the bSbSbSb of l-Sb and l-InSb, with cutoff distance of 0.34 nm, have two peaks; one is near 90 and the other near 180 indicating that the clusters of Sb atoms have a more open structure. The covalent character of Sb–Sb bonds defines a strong angle correlation peak near 90 and the peak at 180 corresponds to an open angle of the chained structures of Sb atoms. Seifert et al. [30] had also studied the structure of pure l-Sb and found these two peaks. The bSbSbSb of l-InSb is almost the same as that of l-Sb. Thus, the calculated results of pair correlation functions and bond-angle distributions for l-In, l-Sb and l-InSb indicate that the local structures of likeatoms (In–In and Sb–Sb) of l-InSb resemble those of pure element liquids (l-In and l-Sb). To investigate the characters of the bonds of In–Sb, we calculate the bSbInSb and bInSbIn with the cutoff distance of 0.30 nm, i.e., 10% larger than the bond length of c-InSb. The results are shown in Fig. 6. The peak in bSbInSb is at 109, i.e., the bond angle of c-InSb, but for bInSbIn, this peak shifts to 90 due to the close-packed indium atoms. Thus, the bonds of In–Sb similar to that of c-InSb exist in l-InSb. Given the partial pair correlation functions, it is possible to calculate the partial coordination numbers as
InInIn
l-In l-InSb
SbSbSb
l-Sb l-InSb 0
40
80
(deg)
120
160
0
40
80
120
160
(deg)
Fig. 5. Bond-angle distributions of In and Sb clusters in l-in, l-Sb and l-InSb.
351
SbInSb InSbIn
0
40
80
120
160
(deg) Fig. 6. Bond-angle distributions of SbInSb and InSbIn of l-InSb. Cutoff distance is 0.30 nm.
N ab ¼
Z
Rmin
4pr2 qb gab ðrÞ dr;
ð2Þ
0
where RMin is the first minimum coordinate in g(r). We use the RMin of the total pair correlation function to calculate partial coordination numbers, RMin = 0.40 nm. We obtain NInIn = 3.7, NInSb = 4.7 and NSbSb = 3.4. To estimate the compositional defects, a compositional disorder number (CDN), defined as the ratio of homogeneous and heterogeneous bonds [14], (Naa + Nbb)/2Nab, is calculated. The calculated CDN is 0.75 which indicates that the high concentration heteroatomic bonds exist in l-InSb.
4. Electronic structure The structure behavior of the liquid alloys can be understood in terms of the electronic structure. Here we have investigated the electronic density of states (DOS) and the local density of states (LDOS) [31]. The LDOS are obtained by projecting the wave functions onto spherical harmonic centered on each atom with a radius of 0.144 nm for In atoms and 0.14 nm for Sb atoms (covalent radius). Both the DOS and LDOS of l-InSb are obtained by averaging on ten configurations. The DOS and LDOS for l-InSb and c-InSb (zincblende structure) are shown in Fig. 7. In the zincblende structure, a gap at Fermi level as a signature of semiconductor is observed, the sharpening of both the s and p peaks for In and Sb atoms can be found. The main peaks appear at 9.0 eV, 5.5 eV and 2.0 eV originated from the Sb(s), In(s), and In(s, p)-Sb(p) bonding orbitals. Obviously, the In–Sb bonds mainly due to the interaction of In(p) and Sb(p) orbitals and both the In(p) and Sb(p) orbitals have been split into two peaks, one is the bonding state at 2.0 eV and the other is the antibonding sate at +5 eV. In liquid state, the gap at Fermi level is disappeared demonstrating metallic-like behavior, but a small ÔdipÕ is still preserved. From LDOS, we find that this ÔdipÕ results from the
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C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
(a)
(b) Total
Total
In s In p
In s In p
Sb s Sb p
Sb s Sb p
-15
-10
-5
0
5
10
-15
-10
-5
E-Ef (eV)
0
5
10
E-Ef (eV)
Fig. 7. Total electronic density of states and the local density of states for InSb in the zincblende structure (a) and the liquid state (b).
rðxÞ ¼
occ unocc X X 2pe2 X jhw jp jw ij2 dðEn Em hxÞ; 3m2 xX m n a¼x;y;z m a n
ð3Þ where Ei and wi are eigenvalues and eigenfunctions, and X is the volume of the supercell. Dipole transition elements, Æwmjpajwnæ, were sampled by the C point of the Brillouin Zone. The calculated conductivities of l-InSb are results of averaging on forty configurations chosen at random from the representative ensembles and illustrated in Fig. 8 which shows a Drude-like falloff. This behavior has also been found in l-Si [10], l-Ge [11] and l-GaAs [14,15] as a signature of metallic liquid. We can extrapolate the frequency dependent conductivity to zero frequency to estimate the dc conductivity. The estimated value 9660 X1 cm1 is in remarkable agreement with the experimental values of 9200–
0.25 0.20
σ (ω) (a.u.)
In(p)–Sb(p) interaction, i.e., both the splits of In(p) and Sb(p) occur at Fermi level. The positions of bonding and antibonding states of In(p) and Sb(p) in the liquid state are similar to those in c-InSb, respectively. In addition, the peaks of In(s) and Sb(s) are also at the same position as in c-InSb. Thus, our calculated results of electronic structure confirm that the In–Sb bonds still exist in lInSb near melting point. Once the energy states and wave functions for a given atomic configuration are obtained, the optical conductivity can be calculated by using Kubo–Greenwood expression [32]. The real part of the conductivity is determined by the sum of all possible dipole transition at a given frequency:
0.15 0.10 0.05 0.00 0
1
2
3
4
5 6 ω (ev)
7
8
9 10
Fig. 8. Real part of optical conductivity for l-InSb. The conductivity is in atomic unit.
9900X1 cm1 [4]. However, considering the several approximations used in our calculation, such a good agreement is to some extent fortuitous. By fitting a Drude curve rðxÞ ¼
rð0Þ 1 þ x 2 s2
ð4Þ
to our data, we extract a relaxation time s1 = 2.59 eV for l-InSb. Although no experimental values of s are available for l-InSb, a value of the same order of magnitude, s1 = 2.99 eV, has been measured in l-Si [33]. Using these values of s, and the free-electron expression l = hkFs/m for the mean free path, we find that the ratios of l/a, where a is the nearest-neighbor distance, is 1.5 for l-InSb, indicating a strong scattering behavior of l-Insb. In l-Si the ratio, calculated in a similar way, is slightly larger, i.e., l/a = 1.8 [10].
C. Zhang et al. / Chemical Physics Letters 408 (2005) 348–353
5. Summary and conclusion We have performed ab initio molecular-dynamics simulations of l-InSb at temperature of 813 K. Our theoretical results of l-InSb are in good agreement with available experimental data. By analyzing the structural parameters of pair correlation functions and bond-angle distributions, we find that the local structures of In and Sb atoms of l-InSb are similar to those of pure element liquids, especially, the cluster of Sb atoms has more opened structure similar to that of l-Sb. The bond-angle distributions of In–Sb–In and Sb–In–Sb and the electronic structure of l-InSb indicate that the In–Sb bonds similar to those of crystalline state are preserved in liquid states. It can be expected that the remnants of covalent bonds of In–Sb and the peculiar local structure of Sb atoms will be gradually destroyed with increasing temperature which may cause anomalous changes in physical properties. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 20133020).
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