Electronic and elastic properties of Sr2RuO4 with pressure effects by first principles calculation

Physica B 441 (2014) 62–67

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Electronic and elastic properties of Sr2RuO4 with pressure effects by first principles calculation Xi-Ping Hao a, Hong-Ling Cui a,n, Zhen-Long Lv a,n, Guang-Fu Ji b a

School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Chinese Academy of Engineering Physics, Mianyang 621900, China

b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 July 2013 Received in revised form 17 January 2014 Accepted 7 February 2014 Available online 15 February 2014

In this work, electronic and elastic properties along with the pressure effects of the unconventional superconductor Sr2RuO4 are investigated by first principles calculation. Band structure calculation reveals that Sr2RuO4 is a weak metal. The characteristic of the band structure has also been analyzed. The calculated elastic constants reproduce the experimental data well. Analyses disclose that Sr2RuO4 is mechanically stable but anisotropic. The calculated Debye temperature 475.0 K is in good agreement with the experimental results. Surveys of the pressure effect on the lattice constants reveal that the a axis is less compressible than the c axis, as found in the experiment. Further investigations illustrate that Sr2RuO4 is mechanically stable up to 50 GPa till the elastic constant C44 fails to meet the mechanical stability criterion. The pressure-dependent elastic parameters are also investigated to know their behaviors under pressure. In addition, the bonding properties of Sr2RuO4 at 0 GPa and 50 GPa are analyzed to explain its elastic behavior under pressure. & 2014 Elsevier B.V. All rights reserved.

Keywords: Electronic property Elastic constant First principles calculation

1. Introduction Transition-metal oxides have attracted considerable interest due to their rich physical properties such as metal–insulator transition, exotic superconductivity, spin–charge–orbital ordering, etc. The discovery of superconductivity in Sr2RuO4 at about 1.5 K in 1994 [1] fueled this trend. Till today, most large number of high temperature cuprate superconductors (HTCS) had CuO2 planes as basic structural units. Studies show that the strong hybridization between the Cu 3dx2
y2 and O 2p orbits, the strong electron–electron correlation on the Cu site and the charge-transfer character around the Fermi level play an important role for the occurrence of superconductivity in these materials, but the underlying mechanism is still unclear [2]. Sr2RuO4 is found to be a metallic layered perovskite compound owning the same crystal structure as of the well-known superconductor, La2CuO4, except that Ru is a second-row transition metal with spin S¼ 1, in contrast to the first-row Cu ion (S¼ 1/2) in the cuprates [3]. Sr2RuO4 is also interesting for that some other compounds isostructural to the cuprate superconductors, such as La2NiO4 and Sr2RhO4, are not superconducting [3]. Therefore, its discovery is

n

Corresponding authors. Tel./fax: þ86 379 64160251. E-mail addresses: [email protected] (H.-L. Cui), [email protected] (Z.-L. Lv). http://dx.doi.org/10.1016/j.physb.2014.02.009 0921-4526 & 2014 Elsevier B.V. All rights reserved.

believed to provide a unique opportunity to study the role of CuO2 planes in HTCS. For the above reasons, considerable effort has been put into the investigation of this crystal. Early calculations based on the local density approximation (LDA) indicate that the Fermi surface of Sr2RuO4 contains three sheets: two electron-like sheets located around the Γ-point and a hole-like one centered at the X-point [4,5]. This characteristic was confirmed by the angle-resolved photoemission spectroscopy experiments [6,7] and the de Haas– van Alphen measurements [8,9]. Besides, experiments on its Hall Effect [10], optical conductivity [11], X-ray fluorescence emission [12], magnetoresistance [13], resistivity [14], doping [15], neutron powder diffraction and magnetization [16] and Compton scattering [17] were also performed to explore the corresponding properties. Investigations also reveal that Sr2RuO4 is an unconventional superconductor of the triplet type [18] located near the quantum critical point [19]. To find the origin of the superconductivity, Mackenzie et al. [8] interpreted the observed quasiparticle spectrum of Sr2RuO4 on the basis of a two-dimensional Fermi liquid model. Their results are consistent with Luttinger's theorem and successfully predict the bulk thermodynamic and transport properties at low temperatures. Another model was also proposed [20] taking the advantage of first principles calculation for the electronic structure and magnetic susceptibility. The related results indicate that the superconductivity is rooted in the strong ferromagnetic spin fluctuation, especially

X.-P. Hao et al. / Physica B 441 (2014) 62–67

at small wave vectors. Besides the LDA method [4,5], its electronic properties were also investigated by the extended Hückel tight-binding approximation [21] and the generalized gradient approximation (GGA) [22], but the GGA calculation predicated a ferromagnetic ground state, opposite to the nonmagnetic result obtained by LDA [4,5]. In the mechanical respect, the compressibility of Sr2RuO4 was investigated by the neutron-powder diffraction experiment [23], which uncovers the nearly linear isotropic compressibility of a axis and c axis in the pressure range of 0–0.62 GPa, although other experiments and calculations show that its magnetoresistance [13], optical [11] and transport properties [14,24] are rather anisotropic. As for the elastic property, only very limited experimental data exist [25–27]; in addition, these values deviate from each other significantly. It is known that the electronic properties play an important role in superconductors, and mechanical properties are also of great importance for the potential application of a material. In view of the disparity existing in the reported electronic properties of Sr2RuO4, and the lack of theoretical investigation on its mechanical properties, we carried out the present study. The rest of this work is organized as follows: computational and theoretical details are described in Section 2. The calculated electronic, elastic properties and the pressure effects are presented and analyzed in Section 3. Finally, conclusions are given in Section 4.

2. Computational and theoretical details 2.1. Computational details The present study was carried out by the plane-wave pseudopotential density functional method, using the Cambridge serial total energy package (CASTEP) [28]. After testing, the generalized gradient approximation (GGA) was adopted with the Wu–Cohen (WC) exchange–correlation functional [29]. The core
electron interaction is represented by the ultrasoft pseudopotential as implemented in CASTEP. In the calculation, the Brillouin zone integration of the primitive cell was performed using 4  4  6 meshes according to the Monkhorst–Pack method [30], and the plane-wave cutoff energy was set to 380 eV. These parameters ensure that the energy, force and stress converge to the values of 5  10
6 eV/atom, 0.01 eV/Å and 0.02 GPa, respectively. In the elastic property calculations, the standard for the convergence of the energy is 1  10
6 eV/atom, and for the force is 0.002 eV/Å. 2.2. Elastic calculations Elastic constants can be defined as the coefficients linking the stress and the strain when an object experiences outer forces within its elastic limit. In theory, the elastic constants Cijkl (i, j, k, l ¼1–6) can be formulated as [31]
 ∂sij ðxÞ C ijkl ¼ ð1Þ ∂ekl X where sij is the applied stress and ekl is the strain and X and x are the coordinates before and after deformation, respectively. In calculating, a set of small strains is imposed on the object to get the corresponding stresses, and then the individual elastic constant can be deduced from these stresses. It is known that the bulk modulus of a material reflects its resistance to the volume change, whereas the shear modulus describes its resistance to the shape change. Both of them are intimately correlated to the mechanical behavior of a material and can be derived from the elastic constants. Generally, for an isotropic polycrystalline material, there are two methods for estimating these

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two moduli: the Voigt's and the Reuss'. In the Voigt's method, the bulk modulus BV and the shear modulus GV for a tetragonal crystal are formulated as [32] BV ¼ ½2ðC 11 þ C 12 Þ þ C 33 þ 4C 13 =9

ð2Þ

GV ¼ ðM þ 3C 11
3C 12 þ 12C 44 þ 6C 66 Þ=30

ð3Þ

while in the Reuss' method, they are defined as [32] BR ¼ C 2 =M

ð4Þ

GR ¼ 15=½18BV =C 2 þ 6=ðC 11
C 12 Þ þ 6=C 44 þ 3=C 66  2

ð5Þ

ðC 11 þ C 12 ÞC 33
2C 213

where M ¼ C 11 þ C 12 þ 2C 33
4C 13 ; C ¼ Hill [33] has proved that the Voigt's and Reuss' methods represent respectively the upper and lower limits of the true polycrystalline elastic constants. He proposed to use the arithmetic mean of the Voigt's and Reuss' moduli to describe the corresponding polycrystalline moduli, i.e. B ¼ ðBV þBR Þ=2;

G ¼ ðGV þ GR Þ=2

ð6Þ

The velocities of the longitudinal (Vl) and transverse (Vt) acoustic waves, and the average acoustic velocity (Vm) can be calculated by [34] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi V l ¼ ð3B þ 4GÞ=3ρ; V t ¼ G=ρ; V m ¼ ½ð2=V t 3 þ 1=V l 3 Þ=3
1=3 ð7Þ Finally, the Debye temperature (ΘD) can be estimated from the average acoustic velocity via [34] 
 h 3n N 1=3 Vm ð8Þ ΘD ¼ k 4π V where h is the Plank constant, k is the Boltzmann constant, n is the number of atoms in the formula unit, and N is the number of formulas in the cell volume V.

3. Results and discussion 3.1. Crystal structures Sr2RuO4 was refined to be a tetragonal crystal (space group I4/ mmm); the lattice constants at 5 K are a¼3.8628 Å and c¼12.7207 Å; the atom's positions are Sr (0, 0, 0.3533), Ru (0, 0, 0), O1 (0, 0.5, 0) and O2 (0, 0, 0.1615) [3]. It has two formulas in a unit cell. The Ru atoms are centered in the elongated octahedrons formed by six O atoms (four planar ones denoted as O1, and two apical ones denoted as O2, see Fig. 1 for details). Hence there are two types of Ru–O bonds: Ru–O1 and Ru–O2. The relaxed lattice parameters with the available experimental data are listed in Table 1. It can be seen from the table that the relaxed lattice parameters a, c and V have errors not more than 0.80%, 0.40%, and 1.6%, respectively, compared with these experimental data [3,35,36]. The relaxed inner z coordinates of Sr and O1 atoms are also in good agreement with the experimental values. 3.2. Electronic properties The band structure of Sr2RuO4 along the lines of high symmetry points in the Brillouin zone is plotted in Fig. 2. From the figure, one can notice that there are three bands slightly crossing the Fermi level, indicating that Sr2RuO4 has a weak metallicity, which is opposite to Ca2RuO4 because the latter is an insulator [37,38]. The calculation also proves that Sr2RuO4 is nonmagnetic, contrary to the ferromagnetic ground state reported by de Boer and de Groot [22] although the GGA method was used. This difference implies that care must be taken in selecting the exchange–correlation functional on the spin fluctuation superconductors which are

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Fig. 1. Crystal structure and Brillouin zone of Sr2RuO4. The O atoms located in the ab plane are denoted as O1, while others are denoted as O2.

Table 1 Relaxed lattice constants a and c (Å), volume V (Å3), and inner z coordinate of Sr and O1 with the available experimental data.

Present Exp. [3] Exp. [35] Exp. [36]

a0

c0

V0

Sr (z)

O2 (z)

3.8352 3.8628 3.8627 3.8611

12.7088 12.7207 12.7176 12.7217

186.93 189.81 189.75 189.66

0.3526 0.3533 0.3530 0.3533

0.1627 0.1615 0.1618 0.1628

Fig. 3. Partial density of states (PDOS) of Sr2RuO4 at 0 GPa.

very weak hybridization with Ru-4d states. The PDOS of O1 extends widely from
9 eV to 1 eV, having a strong hybridization with Ru-4d states. The differences between the PDOS of O1 and O2 can be interpreted as follows: the bond length of Ru–O1 (1.9176 Å) is shorter than that of Ru–O2 (2.0681 Å), so the interaction in the former is stronger than the latter, resulting in a wider PDOS of O1. Combining with the DOS figure (right panel of Fig. 2), we find a pseudogap existing at
1.5 eV, which means that below
1.5 eV the interaction between Ru and O atoms is bonding, whereas above it, the interaction is anti-bonding. 3.3. Elastic properties

Fig. 2. Band structure (left panel) and total density of states (right panel) of Sr2RuO4.

located near the quantum critical point. Another characteristic is that the bands along the Γ–Z and X–P directions are very flat, while the bands along the Γ–X, Γ–N and P–N directions are rather fluctuant. These behaviors can be understood by inspecting the crystal structure of Sr2RuO4 with its Brillouin zone; Γ–Z and X–P directions are actually parallel to the c axis of the crystal, but Sr2RuO4 has a layered structure with a weak interaction along this direction. These weak interactions ultimately lead to the flat behavior of the corresponding bands. As for the three fluctuant directions, they are mainly parallel to the ab plane, so the strong in-plane interaction in this plane makes the related bands seem quite diverging. To reveal its bonding properties, we now investigate the partial density of states (PDOS). From the PDOS (Fig. 3), we find that Sr-d orbits are mainly located at 2 eV above the Fermi level, which almost have no contribution to the density of states at the Fermi level, indicating the ionic characteristic of Sr2RuO4. The PDOS of O2 has a narrow distribution from
7 eV to the Fermi level, having a

The tetragonal Sr2RuO4 has six independent elastic constants: C11, C33, C44, C66, C12 and C13. The obtained elastic constants, along with the available experimental values, are listed in Table 2. For Sr2RuO4, the standard for mechanical stability is C11 40, C33 4 0, C44 4 0, C66 40, C11
C12 40, C11 þC33
2C13 4 0, 2(C11 þ C12) þC33 þ4C13 40 [32]. It is found that the calculated values meet the above criterion, indicating that Sr2RuO4 is mechanically stable. The values of C44, C66, and C13 agree well with the available experimental data [26,27], and the change tendency of these Cij is also in agreement with those reported. It is also noticed that our results are considerably larger than those reported by Okuda et al. [25], so the results of the latter are doubtful. The value of the elastic constant C11 (264.0 GPa) is greater than that of C33 (256.4 GPa), indicating that the a axis is less compressible than the c axis. The reason should be that C11 is mainly determined by the strong covalent bond (Ru–O1) that is along the a axis, while C33 is mainly determined by the weak covalent bond (Ru–O2) and ionic bond (Sr–O2) which are along the c axis. In the case of the shear modulus, we find that C12 (136.9 GPa) 4 C13 (73.9 GPa)4C44 (64.4 GPa)4C66 (59.2 GPa). The smallest value occurs on the elastic constant C66, implying that the shear deformation (corresponding to C66) is most easy to take place compared with other deformations. These large differences among the elastic constants also indicate that Sr2RuO4 is mechanically anisotropic. The calculated bulk modulus (B) and shear modulus (G) of Sr2RuO4 are 149.5 GPa and 69.5 GPa, respectively. The bulk modulus reproduces the experimental values of 142–149 GPa well [23], which is considerably larger than the shear modulus. This result indicates that the shear deformation is easier to occur, and the parameter

X.-P. Hao et al. / Physica B 441 (2014) 62–67

Table 2 Calculated elastic constants Cij (i, j¼ 1–6), along with the experimental data of Sr2RuO4, unit GPa.

Present Exp. [25] Exp. [26] Exp. [27] a

C11

C33

C44

C66

C12

C13

264.0 107.4 235(7) 232(2)

256.4 78.3 2.4(2)a 208(2)

64.4

59.2

73.9 16.9

68(2) 65.7(4)

65(2) 61.2(4)

136.9 45.2 128(4) 106(2)

71(2)

Measured at 300 K with the value of 2.4(2)  1012 dynes/cm2.

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3.4.2. On the lattice parameters The pressure dependence of the normalized parameters a/a0 and c/c0, and bond lengths of Ru–O1 and Ru–O2 (a0 and c0 are the parameters at equilibrium) are plotted in Fig. 5 which displays that the a axis is less compressible than the c axis as the result obtained from analyzing the elastic constants. The compressibility of the a axis is equal to that of the bond Ru–O1, which lies in the fact that the length of the former is exactly twice as long as the latter. To compare our results quantitatively with the experimental results, we defined the parameter k¼
(dL/dP)/L as in Ref. [23]. Using the calculated

Fig. 5. Pressure dependence of the normalized lattice constants and bond lengths of Ru–O1 and Ru–O2 (denoted as dRu
O1 and dRu
O2 respectively).

Fig. 4. Partial density of states (PDOS) of Sr2RuO4 at 50 GPa.

limiting the stability of Sr2RuO4 is the shear modulus. Brittleness and ductility have a huge influence on the mechanical behavior of a material. Pugh [39] has put forward an empirical criterion using the ratio of B/G to discriminate them; when B/G of a material is beyond 1.75, it is ductile, otherwise it is brittle. For Sr2RuO4, the ratio of B/G is 2.15, which indicates that it is ductile. The calculated velocity of longitudinal acoustic wave (Vl) of Sr2RuO4 is 6328.1 m/s, while that of the transverse acoustic wave (Vt) is 3389.4 m/s. These values give an average velocity (Vm) of 3785.3 m/s. Based on this value, we obtained a value of 475.0 K for the Debye temperature of Sr2RuO4, which agrees reasonably well with the experimental ones: (46575) K [27], (410750) K [40], and (427750) K [41]. Fig. 6. Pressure dependence of the volume of Sr2RuO4 up to 50 GPa.

3.4. Pressure effects 3.4.1. On the bonding properties In order to investigate the pressure effect on the bonding property, we also calculated the PDOS of Sr2RuO4 at 50 GPa, as shown in Fig. 4. From it, we can see that changes have taken place in contrast to the PDOS at 0 GPa; almost all of the main peaks of the PDOS drop; the peaks below the Fermi level move downward with the extension of the PDOS to the lower energy end, while the peaks above the Fermi level move upward to the higher energy end with the expansion of the PDOS. These phenomena imply that the interaction within the crystal becomes strong with the shortening of the bonds under the pressure. For example, the bond length of Ru–O1 is reduced by about 6.1% from 1.9176 Å to 1.800 Å; the bond length of Ru–O2 is shortened by about 7.3% from 2.0681 Å to 1.9638 Å. The distance between Sr and O2 also decreases by about 9.7% from 2.4132 Å to 2.1773 Å. These different relative reductions are originated from the different bonding properties in the crystal as discussed.

Fig. 7. Pressure dependence of the elastic constants and elastic moduli of Sr2RuO4 up to 50 GPa.

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X.-P. Hao et al. / Physica B 441 (2014) 62–67

Table 3 Fitted coefficients for C(P) ¼ A0 þ A1  P
A2  P2; C (GPa) represents the elastic constants or moduli; P (GPa) represents the applied pressure.

A0 A1 A2  10
2 A2/A0  10
3

C11

C33

C44

C66

C12

C13

B

G

264.68 7.93 1.37 5.18

258.09 8.60 2.40 9.30

65.30 0.22 1.04 15.9

59.42 0.96 0.281 4.73

137.03 3.50 0.959 7.00

75.10 2.38 0.254 3.38

151.70 4.46 0.738 4.86

71.69 1.13 1.01 14.1

data, we obtained the results that ka ¼ 1.82, kc ¼2.50, kRu
O1 ¼ 1:82 and kRu
O2 ¼ 1:5 at a low pressure range of 0–5 GPa. These results agree reasonably with the experimental findings [23] (at temperature of 60 K and pressure range of 0–0.62 GPa) that ka ¼ 2.12, kc ¼ 2.49 and kRu
O1 ¼ 2:12, but the value of kRu
O2 ð2:9Þ is not encouraging. We suggest making further measurements on this value in view of the complicated bonding property along the c axis. The ratio of V/V0 (V0 is the volume at equilibrium) as a function of the pressure is drawn in Fig. 6. After fitting these data into the Murnaghan equation of state (EOS) [42] PðVÞ ¼ ½ðV 0 =VÞB0
1B0 =B0 , we got the bulk modulus B0 ¼154.1 GPa together with its first order derivative B0 ¼4.1. Obviously, the modulus obtained by this method agrees well with the value of 149.5 GPa deduced from the elastic constants and the experimental values of 142–149 GPa [23]. 3.4.3. On the elastic properties The pressure dependence of the elastic constants of Sr2RuO4 is displayed in Fig. 7. For tetragonal crystals, the mechanical stability criterion under hydrostatic pressure is: c11 40, c33 40, c44 40, c66 40, c11
c12 4 0, c11 þc33
2c13 40, 2(c11 þc12)þc33 þ 4c13 40, where cii ¼Cii
P (i¼ 1, 2, 3), c12 ¼C12 þP and c13 ¼ C13 þ P [43]. Through calculation, we find under hydrostatic pressure, the obtained elastic constants of Sr2RuO4 satisfy the above conditions up to 50 GPa, which indicates that Sr2RuO4 is mechanically stable in this pressure range. We also find that the criterion c44 40 fails first when the applied pressure surpasses 50 GPa, suggesting that the (001) planes may begin to slip even under a small shear strain eyz (corresponding to the elastic constant c44) for the weakness of the ionic Sr–O2 bonds. This mechanical instability is evidently different from that at 0 GPa (corresponding to C66), which should be caused by the pressure effect. Fig. 7 also shows us intuitively that the elastic constants C11 and C33 increase quickly with the pressure; C12, C13, and B increase moderately; C66 and B increase slowly; while C44 first increases but then decreases beyond 5 GPa. To quantify their pressure-dependences, we performed the second order polynomial fitting using the formula: CðPÞ ¼ A0 þ A1  P
A2  P 2 , where C represents the elastic constants or moduli, while P is the pressure. The obtained coefficients A are listed in Table 3. Evidently, the first order pressure derivatives (A1) give a relation that 8.60 (C33)47.93 (C11)44.46 (B)43.50 (C12)42.38 (C13), suggesting that C33 and C11 have bigger slopes than B, C12, C13 and C66. The relative second pressure derivatives (A2/A0  10
3) give a sequence of 15.9(C44)414.1(G)49.30(C33)47.00(C12)45.18(C11) 44.86(B)44.73(C66)43.38(C13), indicating that C44 and G have a relatively larger nonlinear pressure dependence than other elastic parameters, as shown in Fig. 7.

4. Conclusion In this work, we have studied the electronic and elastic properties, along with the pressure effects, of the unconventional superconductor Sr2RuO4, using the first principle method within the generalized gradient approximation. Studies reveal that Sr2RuO4 is metallic and nonmagnetic and is a covalent–ionic compound with different bonding properties in

different crystal directions. Elastic constants of Sr2RuO4 are calculated. The obtained values give the result that C11(264.4 GPa) 4C33(256.4 GPa) 4 C66(59.2 GPa), which on one hand indicates that Sr2RuO4 is mechanically anisotropic and on the other hand implies that shear deformation (corresponding to C66) is easier to take place than in other types. The bulk modulus calculated from the elastic constants is 149.5 GPa; by fitting P–V curve into the Murnaghan equation of state is 154.1 GPa, both of them agree well with the experimental values of 142–149 GPa. The calculated Debye temperature of 475.0 K is also in good agreement with those derived from experiment. Studies on the pressure effect on the lattice constants demonstrate that the a axis is less compressible than the c axis, conforming to the experimental findings. Further investigations indicate that Sr2RuO4 is mechanically stable up to 50 GPa till the shear deformation corresponding to C44 appears, which is different from the shear instability at 0 GPa, originating possibly from the pressure effect. By fitting the elasticity-relevant parameters with the pressure into the second order polynomial, we find that C11 and C33 have a larger linear dependence on the pressure, while C44 and G have a larger nonlinear dependence.

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