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Large elastic anomalies and strong electron-lattice coupling in iron-based superconductor Ba(Fe1−xCox)2As2
Solid State Communications 152 (2012) 680–687
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Large elastic anomalies and strong electron-lattice coupling in iron-based superconductor Ba(Fe1−x Cox )2 As2 Masahito Yoshizawa a,d,∗ , Shalamujiang Simayi a,d , Kohei Sakano a , Yoshiki Nakanishi a,d , Kunihiro Kihou b,d , Chul-Ho Lee b,d , Akira Iyo b,d , Hiroshi Eisaki b,d , Masamichi Nakajima c,d , Shin-ichi Uchida c,d a
Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan
b
National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba 305-8568, Japan
c
Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
d
Transformative Research-project on Iron Pnictides (TRIP), Japan Science and Technology Agency, Tokyo 102-0075, Japan
article
info
Article history: Accepted 6 December 2011 by H. Hosono Available online 11 December 2011 Keywords: A. Iron-based superconductor C. Elastic constant C. Electron-lattice coupling D. Quantum criticality
abstract We investigated the elastic properties of the iron-based superconductor Ba(Fe1−x Cox )2 As2 with various Co concentrations. The elastic constant shows remarkable anomalies associated with the structural phase transition and the superconducting transition. The elastic constant C66 shows a quantum critical behavior, which behaves just like the magnetic susceptibility of unconventional superconductors. It was suggested that the anomalous part in the elastic compliance S66 (1/C66 ) is closely related the emergence of superconductivity. Large Grüneisen parameter was found at the superconducting transition in the longitudinal elastic constant. These results show the strong electron–lattice coupling in this system, and support the prevailing scenario on the relevant role of structural fluctuation coupling to orbitals to understand a whole picture of iron-based superconductor. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction High superconducting transition temperature (Tc) brings large benefits in the consumption of cryogen for the operation of superconductors. High-Tc superconductors realize electricity transport from power plants to megacities with minimizing energy loss. Superconducting quantum interference device (SQUID) provides us with diagnostic tools for brain disease (MEG: Magnetoenthphalography) and heart disease (MCG: Magnetocardiography) [1]. Since the discovery of La2−x Bax CuO4 (LBCO) by Bednorz and Müller, oxide superconductors have provided much interest in higher Tc superconductivity and opportunities in novel superconductor applications. Together with the investigation of mechanisms of the superconductivity, various applications have been developed around the world. Iron-based superconductor is a new member in high-Tc superconductors. Since the discovery of LaFeAsOF [2], the emergence of superconductivity has been reported in many systems with different structures, such as 122-type BaFe2 As2 [3], 111-type LiFeAs and LiFeP [4], and 11-type FeSe [5]. One of the particular interests concerns their structure. As a common property, they contain a Fe
∗ Corresponding author at: Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan. E-mail address: [email protected] (M. Yoshizawa). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.006
atom, which is surrounded by a tetrahedron of pnictogen atoms. This fact has evoked novel interest in the mechanism of superconductivity. Up to date the highest Tsc in iron-based superconductors have been raised to 55 K [6]. It is the second highest temperature superconductor behind the cuprates. Iron-based superconductors have been studied actively worldwide as new high-temperature superconductors [7]. Among the many iron-based superconductors, Ba(Fe1−x Cox )2 As2 is suitable for basic research, because it provides high-quality large single crystals. The crystal structure of the parent compound, BaFe2 As2 , is tetragonal at room temperature, as shown in Fig. 1(a) [8]. This becomes an orthorhombic structure at the structural transition temperature TS = 140 K accompanying the appearance of the long-range antiferromagnetic order [8] as shown in Fig. 1(b), which is viewed from the c-axis [9]. TS decreases when Co is substituted for Fe, and superconductivity appears at x = 0.03 [10,11]. TS falls to zero at x = 0.07, where the superconducting temperature Tsc becomes highest, reaching 25 K. The magnetic ordering temperature TN coincides with TS for x = 0, and for samples in which x is less than 0.03, TN is always lower than TS . The superconducting phase is neighbor to the magnetically ordered phase in Ba(Fe1−x Cox )2 As2 . Such a phase diagram showing the coexistence of magnetic order and superconductivity has been observed in the rare-earth compound CePd2 Si2 and the uranium compound UCoGe, where superconductivity appears near the quantum critical point (QCP) [12,13]. The mechanism of
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Fig. 1. (Color online) (a) Crystal structure of BaFe2 As2 , which belongs to the base-centered tetragonal crystal class I4/mmm, and (b) crystal structure and magnetic order below TS with the crystal symmetry of Fmmm. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
superconductivity has been also discussed based on the origin of the adjacent magnetic and structural phases for iron-based superconductors, which plays an important role for the emergence of superconductivity, magnetic or orbital fluctuation [14,15]. Mazin et al. and Kuroki et al. predicted that a fully gapped signreversing s-wave state (s± state) would be realized by magnetic instability [16,17]. On the other hand, Kontani and Onari, and Yanagi et al. proposed a conventional s-wave state without sign reversal (s++ state) mediated by orbital fluctuation [18,19]. Investigations on the neighboring order and its fluctuation are important to clarify the superconducting properties of iron-based superconductors. Our experimental tool, ultrasonic measurements, provides us with information on changes in the symmetry of the lattice system. Strains introduced into the crystal in the elastic constant measurements deform the crystal locally and break the crystal symmetry (symmetry breaking field). When the system encounters either a charge, orbital (d-electron systems), or electric quadrupole (f -electron systems) order, the elastic stiffness that possesses the same symmetry as, and thus couples with this order tends to exhibit an anomaly (softening in most cases). Accordingly, by examining the temperature (T )-dependence of the anisotropic elastic stiffness Cij , we can obtain fundamental information on the symmetry of the order. Elastic constants of iron-based superconductors have been measured for polycrystalline LaFeAsO [20] by using a RUS (resonant ultrasonic spectroscopy) method for a platy sample. On the Ba(Fe1−x Cox )2 As2 , Fernandes et al. measured the T -dependence of C66 of BaFe2 As2 and Ba(Fe0.8 Co0.2 )2 As2 by using the same method [21], and reported a large elastic softening towards TS in C66 , whose amount is 95% and 18% for BaFe2 As2 and Ba(Fe0.8 Co0.2 )2 As2 , respectively [21]. Because the RUS method cannot provide all elastic constants, the measurement of single crystals has been desired. The measurements of bulk single crystalline Ba(Fe1−x Cox )2 As2 using pulsed ultrasonic spectroscopy with a phase comparison method were reported for underdoped BaFe2 As2 , Ba(Fe0.963 Co0.037 )2 As2 and Ba(Fe0.940 Co0.060 )2 As2 [22], and overdoped Ba(Fe0.9 Co0.1 )2 As2 [23]. These works showed very large elastic softening toward low temperatures. In addition, Goto et al., reported that C11 , C33 , C44 , and 21 (C11 − C12 ) gradually increased as T decreased, and showed no remarkable anomaly at Tsc in contrast with C66 [23]. We measured the elastic properties of Ba(Fe1−x Cox )2 As2 , associated with the superconducting transition [24], and the structural transition [25], so far. The results show that this system is characterized by a strong electron–lattice coupling. In this article, we will review the elastic properties of Ba(Fe1−x Cox )2 As2 .
Fig. 2. (Color online) Elastic strains and elastic constants categorized into the irreducible representation of the tetragonal point group of D4h .
We will discuss the effect of the strong electron–lattice coupling, and the role of the neighboring order and its fluctuation for the emergence of superconductivity in this system. 2. Experimental 2.1. Elastic constant: symmetry and information The strains introduced into the solid were classified into irreducible representations of the point group, as shown in Fig. 2. For tetragonal symmetry, the εZZ and εXX + εYY strains belong to A1g (Γ1 ) representation of D4h point group. The corresponding elastic constants for εZZ and εXX + εYY are C33 and 12 (C11 + C12 ), respectively. 12 (C11 + C12 ) can be observed as a part of longitudinal elastic constant CL propagating along [110] direction, which will appear in Section 7. εXX − εYY for 12 (C11 − C12 ) and εYZ (εZX ) for C44 belong to B1g (Γ3 ) and Eg (Γ5 ), respectively. The deformation patterns of εXX − εYY and εYZ (εZX ) are orthorhombic and monoclinic, respectively. The εXY strain as a perturbation field of C66 belongs to the B2g (Γ4 ), which deforms the crystal symmetry from tetragonal to orthorhombic, as shown in Fig. 2(c). The charge and orbital (electric quadrupole) can couple with the elastic strains possessing the same symmetry. The strains act as a conjugate field for these quantities, and give important information in the corresponding elastic constants, which is just like the magnetic susceptibility for the magnetic systems. 2.2. Ultrasonic measurement We measured the elastic constants for Ba(Fe1−x Cox )2 As2 using an ultrasonic pulse-echo phase comparison method [26] as a
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where the space group is I4/mmm and Fmmm for the high- and low-temperature phases, respectively [8]. The decrease in C66 with decreasing T is prominent for x = 0, 0.037, and 0.060, whose structural transition temperature TS = 141 K (x = 0), 84.7 K (x = 0.037), and 30 K (x = 0.060). Here, we defined TS for the structural transition temperature determined experimentally, where the echo signal disappeared for x = 0 and 0.036, and C66 showed a kink-like anomaly for x = 0.060. The softening of C66 is less prominent as x increases, which is consistent with the disappearance of the structural phase transition. For the superconducting samples, 0.060 6 x 6 0.116, anomalies are observed at Tsc = 24.0 K (x = 0.060), 20.7 K (x = 0.084), 16.5 K (x = 0.098), and 10.5 K (x = 0.116). The underdoped samples show a peak at Tsc and a step-like anomaly below Tsc . On the other hand, C66 increases below Tsc in the overdoped region. The superconducting transition disappears when x > 0.161. C66 for the samples of x = 0.161 and 0.245 shows a normal T -behavior, reflecting phonon anharmonicity. Fig. 3. (Color online) Temperature-dependence of the elastic stiffness C66 of Ba(Fe1−x Cox )2 As2 with various values. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
function of temperature from 5 to 300 K using a cryostat mounted on a Gifford–McMahon (GM) cryocooler. Elastic stiffness was obtained by C = ρv 2 , where ρ is the density and v is either the longitudinal or transverse sound velocity. The corresponding sound velocities can be obtained by choosing the propagation and displacement directions. Tetragonal crystal symmetry has six independent Cij ’s; namely, C11 , C33 , C12 , C13 , C44 , and C66 . The propagation and displacement directions of the sound velocity are respectively [100] and [100] for C11 , [001] and [001] for C 33 , [100] ¯ for and [010] for C66 , [100] and [001] for C44 , [110] and 110 1 (C11 − C12 ), and [110] and [110] for CL = 12 (C11 + C12 + 2C66 ). 2 Here, the XYZ coordinate was defined by the unit cell of the I4/mmm crystal structure [8], where the directions of X , Y and Z coincide with the principal axes of base-centered tetragonal lattice formed by Ba atoms, as shown in Fig. 1 (a). On the other hand, x and y orient toward the nearest Fe–Fe directions in the layered plane for the xyz notation, which rotate 45° from the X and Y directions as shown in Fig. 1(b). The sound velocity was obtained by the time interval of the echo train and the sample length, whose accuracy is within a few percent due to the usage of large crystals. The value of ρ is 6.48 × 103 kg m−3 and 6.55 × 103 kg m−3 for x = 0 and 0.245, respectively. To prevent damage to the sample due to rapid changes in temperature, the rate of change in temperature was carefully controlled so as to be 10 K/h near TS [9]. Highquality large single crystals of Ba(Fe1−x Cox )2 As2 used in this work were grown by the self-flux method. Samples with eight Co concentrations x = 0, 0.037, 0.060, 0.084, 0.098, 0.116, 0.161, and 0.245 were prepared. The Co concentration in the grown crystals was determined by energy-dispersive X-ray spectroscopy (EDS) measurement. The typical size of a sample was 3 mm × 3 mm in the tetragonal ab (XY ) cleavage plane, and 2 mm in thickness on the c (Z )-axis. The experiments detailed are described elsewhere [24,25].
3. Large elastic softening associated with structural phase transition and strong electron–lattice coupling Fig. 3 shows the T -dependence of C66 for samples with x = 0, 0.037, 0.060, 0.084, 0.098, 0.116, 0.161, and 0.245. C66 significantly decreases as T decreases. The softening in C66 corresponds to the symmetry change from tetragonal to orthorhombic. This is consistent with the structural analysis of this material,
4. Origin of C66 softening What is the origin of large C66 softening? Its origin has been discussed based on various mechanisms [27]. Because the elastic softening is a precursor of a structural phase transition, two mechanisms have been proposed, so far, either magnetic origin or orbital origin [14,15,28,29]. Such large elastic anomalies are often observed in the 4f electron system with a bilinear coupling between quadrupole operator O and elastic strain ε with the form of Oε . From time-reversal symmetry, a bilinear coupling between the elastic strain and a single spin is not allowed. First, the effect of spin-nematic order has been considered, where the elastic anomaly above TN is caused by the fluctuation of a pair of magnons at two sublattices [30,21]. Spin-nematic order parameter can couple to the strain with a bilinear form. In this case, the nematic order deforms the crystal through the spin–orbit interaction. Small spin–orbit interaction in the transition metals explains reasonably a small lattice distortion. It would be an enigma whether the large elastic anomalies can be caused by this mechanism. On the other hand, theories based on orbital fluctuation have been proposed. Yanagi et al. considered Ferro-orbital order accompanying large elastic softening [31,32]. Kontani et al. have discussed a two-orbiton process, which is initiated by antiferroquadrupolar fluctuation caused by interband nesting and consisting of a coupling between optical phonon and orbiton at the zone-boundary. They showed that the two-orbiton process brings about very large C66 softening [33,34]. 4.1. Localized picture Besides microscopic theories, we will discuss the origin of the softening in C66 from the viewpoint of experimentalists. The 3d orbitals are split into Eg doublet and T2g triplet by the electric crystalline field (CEF) in a cubic symmetry. The Eg is spilt into two singlets, and the T2g is split into one singlet and one doublet by tetragonal CEF of iron-based materials. Since the remaining doublet can be lifted by the elastic strain, this may cause the elastic anomaly. We would like to consider the CEF effect of the 3d level by referring the work of Hazama et al. [35]. The CEF Hamiltonian for tetragonal symmetry is expressed as HCEF = B2 O02 + B40 O04 + B44 O44
1 O02 = √ 3l2Z − l (l + 1) 3 O04 = 35l4Z − 30l (l + 1) l2Z + 25l2Z − 6l (l + 1) + 3l2 (l + 1)2 O44 =
l4+ + l4− 2
.
(1)
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and the normal contribution (background) S66,0 . For the case of Ba(Fe1−x Cox )2 As2 , we will rewrite S66 =
0 S66
+ S66,cr = S66,0 1 +
EJT T − TC
(5)
with EJT = TC − Θ . EJT stands for the Jahn–Teller energy, an energy scale that corresponds to the strength of the electron–lattice coupling. Eq. (5) has the same form as suscepti the Curie–Weiss bility. In the data analysis, we plot 1/ S66 − S66,0 as a function of temperature, which shows a linear T -dependence. TC can be obtained as the temperature, where the straight line intersects the temperature axis. EJT is the inverse of the temperature gradient of the straight line. Θ can be calculated by TC and EJT . In the case of localized d electrons, for a disordered state, χΓ0 (T ) is expressed as [37]
χΓ (T ) = 0
Fig. 4. (a) Crystalline electric field energy scheme in tetragonal symmetry, and (b) energy splitting of crystalline electric field levels by the application of the elastic strains of εXX − εYY and εXY .
The CEF energy scheme is calculated and schematically illustrated in Fig. 4(a), together with Fe 3d electron with intermediate spin state S = 1 [15]. In the case of low spin state, two electrons occupy the B2g level of dXY , and are thermally activated to Eg of dYZ /dZX . Next, we will consider the coupling between the strain εΓ and an order parameter OΓ as H = −λOΓ εΓ . Here, Γ is the irreducible representation, which OΓ and εΓ belong to. The equivalence of the X and Y axes in tetragonal symmetry leads to the degeneration of the dZX and dYZ orbitals. This degeneracy is lifted by εXY or εXX − εYY , and brings about anomalies in the corresponding C66 and 21 (C11 − C12 ). The energy scheme with the strain is shown in Fig. 4(b). We can calculate the elastic constant CΓ for the localized 3delectron system, which is the Jahn–Teller effect based on the localized picture of d electron. CΓ = CΓ ,0 − N λ2 χΓ .
(2)
Here, χ is the strain susceptibility [36]. If we adopt the form of χ 0 (T ) = TA for a single-ion strain susceptibility χ 0 including no
intersite interaction [37], we then get CΓ = CΓ ,0 − N λ2
T − TC χΓ0 = C Γ ,0 0 T −Θ 1 − I χΓ
(3)
where λ, I, and N are the coupling constant, the intersite interaction, and the number of atoms per unit volume. TC and Θ are the transition temperatures in the presence of strain interaction (zero stress state) and for zero strain (clamped state), respectively. It would be noted that, in Eq. (3), the transition undergoes at TC , where the lattice shows instability. Although the transition temperature was already defined as TS in Section 3, we defined TC in Eq. (3), which will be determined from the data analysis. TC differs from TS for general cases, while they coincide with each other for a ferro-order case that is realized in the present case, as mentioned later. Next, we introduce the elastic compliance Sij , which is a component of the inverse Cij matrix. Elastic compliance represents the ‘‘structural’’ susceptibility of elastic systems, and corresponds to the magnetic susceptibility χ in magnetic systems. SΓ = SΓ ,0
T −Θ T − TC
.
(4)
The S66 (= 1/C66 ) can be decomposed into the sum of the anomalous contributions that exhibit critical behavior S66,cr
(OΓ )2 T
≈
⟨φ |OΓ | φ⟩2 T
.
(6)
Here, |φ⟩ is the eigen function of 3d-orbitals, ⟨· · ·⟩ the thermal average, and OΓ the quadrupole operator with irreducible representation Γ . OΓ is described as OXY = lX lY + lY lX , OYZ = lY lZ + lZ lY and O22 = l2X − l2Y for C66 , C44 and CE , respectively, by using angular momentum operator (lX , lY , lZ ) with l = 2. The energy scheme is shown in Fig. 4(b). For C66 , ⟨φ± | OXY |φ± ⟩ = ∓3, where |φ± ⟩ = √1 (|YZ ⟩ ± |ZX ⟩). For CE , ⟨YZ | O22 |YZ ⟩ = 3, 2
and ⟨ZX | O22 |ZX ⟩ = −3. For C44 , ⟨YZ | OYZ |YZ ⟩ is vanished. This may lead to a large elastic softening for either C66 or CE , while no softening for C44 . 4.2. Itinerant (band) picture We know superconducting systems, which show large elastic softening, the A15 compound V3 Si and Laves phase compound CeRu2 being famous examples [38,39]. Both compounds are categorized into conventional superconductors. CeRu2 was precisely investigated, because it was a candidate for the Fulde–Ferrel–Larkin–Ovchinnikov (FFLO) state. In the case of CeRu2 , the related compounds LaRh2 and CeCo2 show also elastic softening toward low temperatures, but CeRh2 shows no elastic anomaly [40,41]. Therefore, the softening in these materials have been ascribed to a large density of states at Fermi energy, which is contributed mainly from d electron component [40,42]. The 3d-orbitals in an iron-based superconductor form bands. According to band calculations, the bands located above the Fermi energy at the Γ -point form hole Fermi surfaces and electron pockets at M-points ofthe zone boundary [19]. In addition, the band nesting along the πa , πa , 0 direction is a key feature in ironbased superconductors. Here, we will consider the effects of bands. The bandwidth is affected by the crystal deformation, because the electron transfer (namely, transfer integral) between iron atoms would be modified by the lattice distortion. Therefore, equivalent four M-points in the tetragonal lattice do not become equivalent under the application of the strain εXY , as shown in Fig. 5 [43]. For example, the width of the bands at the Brillouin zone boundary M1 and M3 in Fig. 5 becomes large, and those of the bands at M2 and M4 become smaller under the application of εXY . This process gains the electronic energy, and looses the elastic energy. The amount of deformation is determined by the energy valance of the electronic and lattice energies. The formula for the elastic constant based on this consideration is as follows [44,45]: C = C0 − (dM1 − dM2 )2
χS0 1 − I χS0
(7)
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Fig. 5. (Color online) Effect of band on the crystal deformation. Four M points in the square lattice do not become equivalent by the strain εXY , which makes the band consisting of dyz /dzx and dx2 −y2 degeneracy lifted [43]. Source: This figure was taken from Ref. [27]. © 2011, Journal of the Physical Society of Japan.
Fig. 6. (Color online) Temperature dependence of the inverse of S66,cr . Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
χ = 0 S
1 kB T
N (E ) =
N0 2
fk (1 − fk ) =
dE N (E ) f (E )
k
1+
E − EF
2 −1
W
where f is the Fermi–Dirac function, d(= dM1 = −dM2 ) is the electron–lattice coupling constant, N0 is the density of states at Fermi energy EF , and I is the intersite interaction. If a large density of states exists at the M-point, an elastic anomaly may be caused. This provides a natural solution to the question of why the anomaly only emerged in C66 . In this situation, no anomaly is expected in 12 (C11 − C12 ), because the bands at four M-points stay equivalent under the presence of εXX − εYY . How about the case of band nesting? We have no analysis tool for the elastic constant based on the band nesting. We suppose that the formula is more complicated than the band Jahn–Teller effect, because the effects of the whole band in the Brillouin zone have to be accounted for. Nevertheless, we infer that it brings about a similar effect to the elastic constant anomaly discussed above. 5. Analysis of C66 5.1. Analysis based on the Jahn–Teller formula Fig. 6 shows the inverse of S66,cr as a function of temperature. Here, we employed the data on Ba(Fe0.755 Co0.245 )2 As2 for S66,0 ,
Fig. 7. (Color online) Theoretical fitting of C66 by the band Jahn–Teller effect under the assumption of Lorentzian density of states. Adjustable parameters are the bandwidth W and the intersite interaction I. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
due to its normal T -dependence, and subtract it from the other data. It can clearly be seen that 1/S66,cr in the underdoped region (x < 0.070) exhibits linear T -dependence. This indicates that S66,cr obeys Eq. (5). In the underdoped samples, TC in Eq. (5) and TS obtained from C66 are 134.3 and 141 K for x = 0, 81.4 and 84.7 K for x = 0.037, 28.2 and 30.0 K for x = 0.060, respectively. The closeness of the values in each case suggests the occurrence of ferro-order. This strongly suggests the crystal deformation of εXY below TS , and the ordering of the parameter possessing the same symmetry as εXY . The amount of EJT is about 50 K, which is almost independent of x. 5.2. Analysis based on the band picture The T -dependence of C66 in the overdoped region cannot be explained by the Jahn–Teller formula, but by a band picture. Fig. 7 shows theoretical curves of Eq. (7) under the assumption of the Lorentzian density of states. In this analysis, the main adjustable parameter is the bandwidth W . We fixed EF /W = 0.4 to obtain the best fitting. The obtained parameters are displayed in Fig. 8. The value of W is proportional to x − 0.07. From the adjustable parameter N0 d2 , the value of d was evaluated under the assumption that N0 is approximately equal to 1/W . The value of d is located in the range of 0.22–0.28 eV/Fe, for overdoped samples. The values of N0 are also evaluated to be 110, 64, 46, and 26 states/eV for the four samples, respectively, when we assume. N0 = 1/W .
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coupling is not caused by weak spin–orbital coupling in transition metals, but by the orbital nature of this system. These values are considered to support the orbital-based theory. By the way, it is worthwhile to speculate the shrinkage of the effective orbital (quadrupole) moment. The application of the JahnTeller formula in the underdoped samples implies the existence of an orbital (quadrupole) moment. However, it is considered that it is somewhat different from the fully localized picture. Once the system enters the overdoped region, the temperature dependence crosses over to the itinerant picture across the QCP discussed in the next section. It means the system is located on the boundary between the itinerant and localized. If we assume that the value of λ is approximately equal to d = 0.22 eV/Fe for x = 0.084,
(OXY )2 is evaluated to be 1.3. Another evaluation was made by the condition that the Θ in Eq. (3) must be equal to I (OXY )2 for x = 0.084 sample. This gives (OXY )2 = 1.6. Both evaluations give similar values. According to the localized picture, the value of ⟨OXY ⟩ is 3. This suggests
the orbital (quadrupole) moment of
Fig. 8. (Color online) The obtained parameters from the analyses based on the localized and band pictures of Eqs. (5) and (5), respectively. Electron–lattice coupling constants of λ
(OXY )2 was evaluated from EJT . The evaluation method
of d is described in the text. The curves are visual guides.
the quadrupole moment of this system is remarkably reduced. This is consistent with the itinerant electron picture in iron-based material. 6. Structural quantum criticality and superconductivity 6.1. Phase diagram
Fig. 9. (Color online) Phase diagram of Ba(Fe1−x Cox )2 As2 . The magnetic transition temperature TN for x = 0.036 was taken from Ref. [50]. The curves are the visual guides. TS and Tsc obtained in this work are consistent with the values reported elsewhere [51]. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
5.3. Strong electron–lattice coupling and shrinkage of orbital (quadrupole) moment We would like to summarize the values of the electron–lattice coupling λ constants in this system. Fernandes et al. and Kontani et al. estimated the value of λ to be 17 meV/Fe and 0.2 eV/Fe, respectively [21,33]. The parameters obtained in Section 5 are summarized in Fig. 8. For the underdoped samples, λ is obtained from EJT = 50 K. The calculated value of λ
(OXY )2 from EJT is 0.25
eV/Fe for x = 0.037. For overdoped samples, the value of d is located in the range of 0.22–0.28 eV/Fe. These values of the coupling constant are consistent with the orbital-based theory both from the localized picture and band picture. Generally speaking, a coupling between the magnetic moment and the stain is mediated by a spin–orbit interaction. It is considered that such strong electron–lattice
Fig. 9 summarizes the phase diagram of the Ba(Fe1−x Cox )2 As2 system. We found two characteristic temperatures of T ∗ and Tmax . As shown in Fig. 6, 1/S66,cr deviates from T -linear behavior below a certain temperature T ∗ ; i.e., 40 K for x = 0.084, 130 K for x = 0.098, and 190 K for x = 0.116. We found Tmax , which corresponds to the temperature at which S66,cr takes the maximum value (1/S66,cr takes the minimum value), as shown in Figs. 6 and 10(a). It would be notable that T ∗ and Tmax go to zero at the QCP concentration of xC = 0.07. A possible explanation of T ∗ is the crossover from the non-Fermi liquid to the Fermi liquid region. A similar behavior including the x-dependence of the crossover region is observed in BaFe2 (As1−x Px )2 [46]. On the Tmax , there are two possible explanations. For highly correlated electron systems such as CeCu2 Si2 , UPd2 Al3 , and UPt3 , a similar maximum was reported in the magnetic susceptibility χ . The Tmax for these cases has been considered to be a Kondo temperature [47]. In the case of magnetic QCP, the Kondo temperature means the entrance to the novel electronic state at low temperatures; namely the coherent motion of f -electrons. Can it be applied for the iron-based materials, where the role of orbitals is considered to be relevant? The low temperature coherent state may be an orbital liquid from the analogy of Fermi liquid. Another explanation will be made by a pseudo-gap formation. The minimum behavior in the temperature dependence of C66 can be fit by the density of states containing a pseudo-gap. In the case of Ba(Fe1−x Cox )2 As2 , an existence of the pseudo-gap was suggested [48]. However, the temperature, where the pseudogap opens, is located at higher T than TS , and its Co concentration dependence is quite different from that of Tmax . Tmax is considered to be proportional to the bandwidth W , which suggests that Tmax should correlate with W . One of the possible candidates is the Dirac node, which was predicted in iron-based superconductors [49]. This should be clarified by future investigations. In addition, the analysis based on the band picture suggests possible mass enhancement toward QCP in this system. A similar phenomenon was reported for BaFe2 (As1−x Px )2 [52]. The obtained phase diagram and various phenomena near the QCP resemble those of the well-known rare-earth compounds and uranium
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Fig. 11. Temperature dependence of C66 for the overdoped Ba(Fe0.940 Co0.060 )2 As2 near Tsc in expanded scales. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
Fig. 10. (Color online) (a) Temperature dependence of S66,cr for the overdoped samples. (b) Tsc from this work and Ref. [51] and the inverse of the peak value of S66,cr as a function of the distance from the QCP concentration x − xC . The curve of Tsc is a visual guide. Source: These figures were taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
compounds. This coincidence strongly suggests the intimate relationship between superconductivity and QCP in this system as discussed in Section 6.2. It should be noted that the quantum criticality appears in structural fluctuations. On this point, we would like to name the structural quantum criticality. 6.2. Correlation between elastic anomaly and superconductivity We now discuss the relationship between the elastic anomaly and superconductivity. As seen in Fig. 10(b), the amount of 1/S66,cr is proportional to x − xC , where xC = 0.07 for this system. Such behavior on the χ as a function of the distance from QCP has been predicted theoretically for the itinerant electron magnetic QCP system [53]. It is surprising that such well-known behavior holds in this system with the respect of S66 instead of χ for the magnetic system. As shown in the same figure, Tsc decreases with increasing x − xC . So we can recognize an apparent correlation between Tsc and 1/S66,cr that Tsc is a function of 1/S66,cr . The explanation for this interesting fact is speculated to be as follows. As shown in Fig. 11, the underdoped sample exhibits a small anomaly at Tsc , while a large upturn at Tsc is seen in the overdoped samples. Once the system enters the orthorhombic phase from the tetragonal phase, structural fluctuations are suppressed in the ordered phase. In the overdoped samples, however, structural fluctuations still
survive even at Tsc . The amount of the anomaly at Tsc correlates with the peak height of S66,cr , which is the measure of structural fluctuation. The large anomaly at Tsc for the overdoped samples suggests strong coupling between the structural fluctuations and superconductivity. These facts suggest that the origin of S66,cr is deeply related to the emergence of superconductivity. According to the NMR measurement of Ba(Fe1−x Cox )2 As2 , remarkable enhancements of the spin susceptibility in 75 As NMR Knight sift and (1/T1 T )0.5 were observed [54]. These enhancements also show similar quantum critical behavior to the S66 in the elastic measurement. There is no doubt that the QCP behavior observed in NMR and elastic measurements would be expected to be ascribed to the same origin, because the elastic constant is no sensitive probe for magnetism, but is sensitive to orbital (quadrupole), contrary to NMR. In that sense, our measurement preferably observes the orbital (quadrupole) side of the quantum criticality. 7. Elastic anomaly associated with superconducting transition and large Grüneisen parameter The elastic investigation showed strong electron–lattice coupling with respect to the structural transition. The strong electron–lattice coupling in iron-based compound was suggested from another point of view [24]. Fig. 12 shows the temperature dependence of CL = 12 (C11 + C12 + 2C66 ) in the magnetic field, which was applied parallel to the propagation direction of the sound [110]. CL shows a softening with decreasing temperature in the region above 26 K, which is caused by the elastic anomaly of C66 . A remarkable step-like anomaly was observed at Tsc = 24.5 K at 0T . The Tsc tends to decrease as applying the field, which confirms that the step-wise anomaly is originated from the superconductivity. The shape of the elastic anomaly in CL suggests that the coupling between the superconducting order parameter η and elastic strain ε is −ξ η2 ε , which is the same as so called magneto–elastic coupling. This type of coupling is originated from the elastic strain dependence of Tsc . We consider that the CL anomaly at Tsc is not originated from C66 , but from 12 (C11 + C12 ), which comes from the coupling between εXX +εYY (see Fig. 2) and the superconducting order parameter, because the anomaly in C66 for this sample is small as shown in Fig. 11. The magneto–elastic coupling constant ξ is related to Grüneisen Tsc parameter Ωsc for Tsc , which is defined as Ω ≡ − T1 ∂∂ε . By sc
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However, irrespective of whether the spin or orbital mediate superconductivity, the results of the present study suggest that the structural fluctuation must be actively incorporated into their role, and that the orbital degrees of Fe–3d must also be taken into consideration in order to understand the full picture of superconductivity in iron-based compounds.
CL (GPa)
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The authors wish to thank H. Fukuyama, H. Hosono, Y. Õno, H. Kontani, Y. Yanagi, J. Schmalian, T. Goto, P. Thalmeier and B. Lüthi for their valuable discussions. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas, ‘‘Heavy Electrons’’ (No. 20102007), of The Ministry of Education, Culture, Sports, Science and Technology, Japan, and the Transformative Research-project on Iron Pnictides of the Japan Science and Technology Agency.
Temperature (K)
References Fig. 12. (Color online) Temperature dependence of (C11 + C12 + 2C66 )/2 for 6.0% sample in the magnetic field of 0, 3 and 5T is displayed with an expanded scale near Tsc . Source: This figure was taken from Ref. [24]. © 2011, Institute of Physics.
employing the scaling Ansatz, we can get the formula of ∆C = −Ωsc2 ∆CV Tsc [55]. Here, ∆C and ∆CV are jumps of the elastic constant and the specific heat at Tsc , respectively. By using ∆CL = 1 GPa and ∆CV = 0.67 J/mol · K [56], we get |Ωsc | = 62. This value is comparable to Ωsc = 15 and −40 for a and c axes, respectively, which were evaluated from thermal expansion data, [56] and Ehrenfest’s relation. Large Ωsc has been reported in heavy fermion systems [57]. The large value of the iron-based superconductor leads us to infer that this system should be categorized into strongly correlated electron system [58]. It would be interesting that the oxide superconductor La2−x Srx CuO4 also shows a large Ωsc [59]. We suppose that the large Grüneisen parameter may be related to the fact that this system is located closely to the quantum critical point. 8. Summaries and conclusion Iron-based superconductor Ba(Fe1−x Cox )2 As2 shows a remarkable elastic softening in C66 . This softening is a precursor for the tetragonal-orthorhombic transition. Similar elastic softening has been observed in A15 compounds, Laves phase compounds and La1−x Srx CuO4 . The correlation between the superconductivity and the elastic anomaly has attracted much attention, but has not been clarified so far. In most cases showing the elastic anomalies, the lattice instability comes from the phonon anomaly. Such a case is not so interesting from the view of superconductivity, because the energy scale of phonon is not large and cannot act a relevant role in the emergence of high-Tc superconductivity. This consideration leads us to misunderstand the relevance of the signature of the elastic anomaly. We have confidence that our study on iron-based superconductor is spotlighted on the relevance of structural fluctuation through the systematic investigation. If the elastic anomaly is caused by the origin possessing high-energy scale; for example, band, orbital (quadrupole) fluctuations or magnetic fluctuations, it may lead to a high-Tc superconductivity. However, we infer that a key point is the strong electron–lattice coupling. In the case of Ba(Fe1−x Cox )2 As2 , the strong electron–lattice coupling is suggested both for the structural instability in C66 and the anomaly at Tsc in CL . We suppose that it would be a characteristic feature of multi-band systems. Further discussions will be necessary on the role of the spin and orbital in the mechanism of superconductivity.
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Large elastic anomalies and strong electron-lattice coupling in iron-based superconductor Ba(Fe1−x Cox )2 As2 Masahito Yoshizawa a,d,∗ , Shalamujiang Simayi a,d , Kohei Sakano a , Yoshiki Nakanishi a,d , Kunihiro Kihou b,d , Chul-Ho Lee b,d , Akira Iyo b,d , Hiroshi Eisaki b,d , Masamichi Nakajima c,d , Shin-ichi Uchida c,d a
Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan
b
National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba 305-8568, Japan
c
Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
d
Transformative Research-project on Iron Pnictides (TRIP), Japan Science and Technology Agency, Tokyo 102-0075, Japan
article
info
Article history: Accepted 6 December 2011 by H. Hosono Available online 11 December 2011 Keywords: A. Iron-based superconductor C. Elastic constant C. Electron-lattice coupling D. Quantum criticality
abstract We investigated the elastic properties of the iron-based superconductor Ba(Fe1−x Cox )2 As2 with various Co concentrations. The elastic constant shows remarkable anomalies associated with the structural phase transition and the superconducting transition. The elastic constant C66 shows a quantum critical behavior, which behaves just like the magnetic susceptibility of unconventional superconductors. It was suggested that the anomalous part in the elastic compliance S66 (1/C66 ) is closely related the emergence of superconductivity. Large Grüneisen parameter was found at the superconducting transition in the longitudinal elastic constant. These results show the strong electron–lattice coupling in this system, and support the prevailing scenario on the relevant role of structural fluctuation coupling to orbitals to understand a whole picture of iron-based superconductor. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction High superconducting transition temperature (Tc) brings large benefits in the consumption of cryogen for the operation of superconductors. High-Tc superconductors realize electricity transport from power plants to megacities with minimizing energy loss. Superconducting quantum interference device (SQUID) provides us with diagnostic tools for brain disease (MEG: Magnetoenthphalography) and heart disease (MCG: Magnetocardiography) [1]. Since the discovery of La2−x Bax CuO4 (LBCO) by Bednorz and Müller, oxide superconductors have provided much interest in higher Tc superconductivity and opportunities in novel superconductor applications. Together with the investigation of mechanisms of the superconductivity, various applications have been developed around the world. Iron-based superconductor is a new member in high-Tc superconductors. Since the discovery of LaFeAsOF [2], the emergence of superconductivity has been reported in many systems with different structures, such as 122-type BaFe2 As2 [3], 111-type LiFeAs and LiFeP [4], and 11-type FeSe [5]. One of the particular interests concerns their structure. As a common property, they contain a Fe
∗ Corresponding author at: Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan. E-mail address: [email protected] (M. Yoshizawa). 0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.006
atom, which is surrounded by a tetrahedron of pnictogen atoms. This fact has evoked novel interest in the mechanism of superconductivity. Up to date the highest Tsc in iron-based superconductors have been raised to 55 K [6]. It is the second highest temperature superconductor behind the cuprates. Iron-based superconductors have been studied actively worldwide as new high-temperature superconductors [7]. Among the many iron-based superconductors, Ba(Fe1−x Cox )2 As2 is suitable for basic research, because it provides high-quality large single crystals. The crystal structure of the parent compound, BaFe2 As2 , is tetragonal at room temperature, as shown in Fig. 1(a) [8]. This becomes an orthorhombic structure at the structural transition temperature TS = 140 K accompanying the appearance of the long-range antiferromagnetic order [8] as shown in Fig. 1(b), which is viewed from the c-axis [9]. TS decreases when Co is substituted for Fe, and superconductivity appears at x = 0.03 [10,11]. TS falls to zero at x = 0.07, where the superconducting temperature Tsc becomes highest, reaching 25 K. The magnetic ordering temperature TN coincides with TS for x = 0, and for samples in which x is less than 0.03, TN is always lower than TS . The superconducting phase is neighbor to the magnetically ordered phase in Ba(Fe1−x Cox )2 As2 . Such a phase diagram showing the coexistence of magnetic order and superconductivity has been observed in the rare-earth compound CePd2 Si2 and the uranium compound UCoGe, where superconductivity appears near the quantum critical point (QCP) [12,13]. The mechanism of
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Fig. 1. (Color online) (a) Crystal structure of BaFe2 As2 , which belongs to the base-centered tetragonal crystal class I4/mmm, and (b) crystal structure and magnetic order below TS with the crystal symmetry of Fmmm. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
superconductivity has been also discussed based on the origin of the adjacent magnetic and structural phases for iron-based superconductors, which plays an important role for the emergence of superconductivity, magnetic or orbital fluctuation [14,15]. Mazin et al. and Kuroki et al. predicted that a fully gapped signreversing s-wave state (s± state) would be realized by magnetic instability [16,17]. On the other hand, Kontani and Onari, and Yanagi et al. proposed a conventional s-wave state without sign reversal (s++ state) mediated by orbital fluctuation [18,19]. Investigations on the neighboring order and its fluctuation are important to clarify the superconducting properties of iron-based superconductors. Our experimental tool, ultrasonic measurements, provides us with information on changes in the symmetry of the lattice system. Strains introduced into the crystal in the elastic constant measurements deform the crystal locally and break the crystal symmetry (symmetry breaking field). When the system encounters either a charge, orbital (d-electron systems), or electric quadrupole (f -electron systems) order, the elastic stiffness that possesses the same symmetry as, and thus couples with this order tends to exhibit an anomaly (softening in most cases). Accordingly, by examining the temperature (T )-dependence of the anisotropic elastic stiffness Cij , we can obtain fundamental information on the symmetry of the order. Elastic constants of iron-based superconductors have been measured for polycrystalline LaFeAsO [20] by using a RUS (resonant ultrasonic spectroscopy) method for a platy sample. On the Ba(Fe1−x Cox )2 As2 , Fernandes et al. measured the T -dependence of C66 of BaFe2 As2 and Ba(Fe0.8 Co0.2 )2 As2 by using the same method [21], and reported a large elastic softening towards TS in C66 , whose amount is 95% and 18% for BaFe2 As2 and Ba(Fe0.8 Co0.2 )2 As2 , respectively [21]. Because the RUS method cannot provide all elastic constants, the measurement of single crystals has been desired. The measurements of bulk single crystalline Ba(Fe1−x Cox )2 As2 using pulsed ultrasonic spectroscopy with a phase comparison method were reported for underdoped BaFe2 As2 , Ba(Fe0.963 Co0.037 )2 As2 and Ba(Fe0.940 Co0.060 )2 As2 [22], and overdoped Ba(Fe0.9 Co0.1 )2 As2 [23]. These works showed very large elastic softening toward low temperatures. In addition, Goto et al., reported that C11 , C33 , C44 , and 21 (C11 − C12 ) gradually increased as T decreased, and showed no remarkable anomaly at Tsc in contrast with C66 [23]. We measured the elastic properties of Ba(Fe1−x Cox )2 As2 , associated with the superconducting transition [24], and the structural transition [25], so far. The results show that this system is characterized by a strong electron–lattice coupling. In this article, we will review the elastic properties of Ba(Fe1−x Cox )2 As2 .
Fig. 2. (Color online) Elastic strains and elastic constants categorized into the irreducible representation of the tetragonal point group of D4h .
We will discuss the effect of the strong electron–lattice coupling, and the role of the neighboring order and its fluctuation for the emergence of superconductivity in this system. 2. Experimental 2.1. Elastic constant: symmetry and information The strains introduced into the solid were classified into irreducible representations of the point group, as shown in Fig. 2. For tetragonal symmetry, the εZZ and εXX + εYY strains belong to A1g (Γ1 ) representation of D4h point group. The corresponding elastic constants for εZZ and εXX + εYY are C33 and 12 (C11 + C12 ), respectively. 12 (C11 + C12 ) can be observed as a part of longitudinal elastic constant CL propagating along [110] direction, which will appear in Section 7. εXX − εYY for 12 (C11 − C12 ) and εYZ (εZX ) for C44 belong to B1g (Γ3 ) and Eg (Γ5 ), respectively. The deformation patterns of εXX − εYY and εYZ (εZX ) are orthorhombic and monoclinic, respectively. The εXY strain as a perturbation field of C66 belongs to the B2g (Γ4 ), which deforms the crystal symmetry from tetragonal to orthorhombic, as shown in Fig. 2(c). The charge and orbital (electric quadrupole) can couple with the elastic strains possessing the same symmetry. The strains act as a conjugate field for these quantities, and give important information in the corresponding elastic constants, which is just like the magnetic susceptibility for the magnetic systems. 2.2. Ultrasonic measurement We measured the elastic constants for Ba(Fe1−x Cox )2 As2 using an ultrasonic pulse-echo phase comparison method [26] as a
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where the space group is I4/mmm and Fmmm for the high- and low-temperature phases, respectively [8]. The decrease in C66 with decreasing T is prominent for x = 0, 0.037, and 0.060, whose structural transition temperature TS = 141 K (x = 0), 84.7 K (x = 0.037), and 30 K (x = 0.060). Here, we defined TS for the structural transition temperature determined experimentally, where the echo signal disappeared for x = 0 and 0.036, and C66 showed a kink-like anomaly for x = 0.060. The softening of C66 is less prominent as x increases, which is consistent with the disappearance of the structural phase transition. For the superconducting samples, 0.060 6 x 6 0.116, anomalies are observed at Tsc = 24.0 K (x = 0.060), 20.7 K (x = 0.084), 16.5 K (x = 0.098), and 10.5 K (x = 0.116). The underdoped samples show a peak at Tsc and a step-like anomaly below Tsc . On the other hand, C66 increases below Tsc in the overdoped region. The superconducting transition disappears when x > 0.161. C66 for the samples of x = 0.161 and 0.245 shows a normal T -behavior, reflecting phonon anharmonicity. Fig. 3. (Color online) Temperature-dependence of the elastic stiffness C66 of Ba(Fe1−x Cox )2 As2 with various values. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
function of temperature from 5 to 300 K using a cryostat mounted on a Gifford–McMahon (GM) cryocooler. Elastic stiffness was obtained by C = ρv 2 , where ρ is the density and v is either the longitudinal or transverse sound velocity. The corresponding sound velocities can be obtained by choosing the propagation and displacement directions. Tetragonal crystal symmetry has six independent Cij ’s; namely, C11 , C33 , C12 , C13 , C44 , and C66 . The propagation and displacement directions of the sound velocity are respectively [100] and [100] for C11 , [001] and [001] for C 33 , [100] ¯ for and [010] for C66 , [100] and [001] for C44 , [110] and 110 1 (C11 − C12 ), and [110] and [110] for CL = 12 (C11 + C12 + 2C66 ). 2 Here, the XYZ coordinate was defined by the unit cell of the I4/mmm crystal structure [8], where the directions of X , Y and Z coincide with the principal axes of base-centered tetragonal lattice formed by Ba atoms, as shown in Fig. 1 (a). On the other hand, x and y orient toward the nearest Fe–Fe directions in the layered plane for the xyz notation, which rotate 45° from the X and Y directions as shown in Fig. 1(b). The sound velocity was obtained by the time interval of the echo train and the sample length, whose accuracy is within a few percent due to the usage of large crystals. The value of ρ is 6.48 × 103 kg m−3 and 6.55 × 103 kg m−3 for x = 0 and 0.245, respectively. To prevent damage to the sample due to rapid changes in temperature, the rate of change in temperature was carefully controlled so as to be 10 K/h near TS [9]. Highquality large single crystals of Ba(Fe1−x Cox )2 As2 used in this work were grown by the self-flux method. Samples with eight Co concentrations x = 0, 0.037, 0.060, 0.084, 0.098, 0.116, 0.161, and 0.245 were prepared. The Co concentration in the grown crystals was determined by energy-dispersive X-ray spectroscopy (EDS) measurement. The typical size of a sample was 3 mm × 3 mm in the tetragonal ab (XY ) cleavage plane, and 2 mm in thickness on the c (Z )-axis. The experiments detailed are described elsewhere [24,25].
3. Large elastic softening associated with structural phase transition and strong electron–lattice coupling Fig. 3 shows the T -dependence of C66 for samples with x = 0, 0.037, 0.060, 0.084, 0.098, 0.116, 0.161, and 0.245. C66 significantly decreases as T decreases. The softening in C66 corresponds to the symmetry change from tetragonal to orthorhombic. This is consistent with the structural analysis of this material,
4. Origin of C66 softening What is the origin of large C66 softening? Its origin has been discussed based on various mechanisms [27]. Because the elastic softening is a precursor of a structural phase transition, two mechanisms have been proposed, so far, either magnetic origin or orbital origin [14,15,28,29]. Such large elastic anomalies are often observed in the 4f electron system with a bilinear coupling between quadrupole operator O and elastic strain ε with the form of Oε . From time-reversal symmetry, a bilinear coupling between the elastic strain and a single spin is not allowed. First, the effect of spin-nematic order has been considered, where the elastic anomaly above TN is caused by the fluctuation of a pair of magnons at two sublattices [30,21]. Spin-nematic order parameter can couple to the strain with a bilinear form. In this case, the nematic order deforms the crystal through the spin–orbit interaction. Small spin–orbit interaction in the transition metals explains reasonably a small lattice distortion. It would be an enigma whether the large elastic anomalies can be caused by this mechanism. On the other hand, theories based on orbital fluctuation have been proposed. Yanagi et al. considered Ferro-orbital order accompanying large elastic softening [31,32]. Kontani et al. have discussed a two-orbiton process, which is initiated by antiferroquadrupolar fluctuation caused by interband nesting and consisting of a coupling between optical phonon and orbiton at the zone-boundary. They showed that the two-orbiton process brings about very large C66 softening [33,34]. 4.1. Localized picture Besides microscopic theories, we will discuss the origin of the softening in C66 from the viewpoint of experimentalists. The 3d orbitals are split into Eg doublet and T2g triplet by the electric crystalline field (CEF) in a cubic symmetry. The Eg is spilt into two singlets, and the T2g is split into one singlet and one doublet by tetragonal CEF of iron-based materials. Since the remaining doublet can be lifted by the elastic strain, this may cause the elastic anomaly. We would like to consider the CEF effect of the 3d level by referring the work of Hazama et al. [35]. The CEF Hamiltonian for tetragonal symmetry is expressed as HCEF = B2 O02 + B40 O04 + B44 O44
1 O02 = √ 3l2Z − l (l + 1) 3 O04 = 35l4Z − 30l (l + 1) l2Z + 25l2Z − 6l (l + 1) + 3l2 (l + 1)2 O44 =
l4+ + l4− 2
.
(1)
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and the normal contribution (background) S66,0 . For the case of Ba(Fe1−x Cox )2 As2 , we will rewrite S66 =
0 S66
+ S66,cr = S66,0 1 +
EJT T − TC
(5)
with EJT = TC − Θ . EJT stands for the Jahn–Teller energy, an energy scale that corresponds to the strength of the electron–lattice coupling. Eq. (5) has the same form as suscepti the Curie–Weiss bility. In the data analysis, we plot 1/ S66 − S66,0 as a function of temperature, which shows a linear T -dependence. TC can be obtained as the temperature, where the straight line intersects the temperature axis. EJT is the inverse of the temperature gradient of the straight line. Θ can be calculated by TC and EJT . In the case of localized d electrons, for a disordered state, χΓ0 (T ) is expressed as [37]
χΓ (T ) = 0
Fig. 4. (a) Crystalline electric field energy scheme in tetragonal symmetry, and (b) energy splitting of crystalline electric field levels by the application of the elastic strains of εXX − εYY and εXY .
The CEF energy scheme is calculated and schematically illustrated in Fig. 4(a), together with Fe 3d electron with intermediate spin state S = 1 [15]. In the case of low spin state, two electrons occupy the B2g level of dXY , and are thermally activated to Eg of dYZ /dZX . Next, we will consider the coupling between the strain εΓ and an order parameter OΓ as H = −λOΓ εΓ . Here, Γ is the irreducible representation, which OΓ and εΓ belong to. The equivalence of the X and Y axes in tetragonal symmetry leads to the degeneration of the dZX and dYZ orbitals. This degeneracy is lifted by εXY or εXX − εYY , and brings about anomalies in the corresponding C66 and 21 (C11 − C12 ). The energy scheme with the strain is shown in Fig. 4(b). We can calculate the elastic constant CΓ for the localized 3delectron system, which is the Jahn–Teller effect based on the localized picture of d electron. CΓ = CΓ ,0 − N λ2 χΓ .
(2)
Here, χ is the strain susceptibility [36]. If we adopt the form of χ 0 (T ) = TA for a single-ion strain susceptibility χ 0 including no
intersite interaction [37], we then get CΓ = CΓ ,0 − N λ2
T − TC χΓ0 = C Γ ,0 0 T −Θ 1 − I χΓ
(3)
where λ, I, and N are the coupling constant, the intersite interaction, and the number of atoms per unit volume. TC and Θ are the transition temperatures in the presence of strain interaction (zero stress state) and for zero strain (clamped state), respectively. It would be noted that, in Eq. (3), the transition undergoes at TC , where the lattice shows instability. Although the transition temperature was already defined as TS in Section 3, we defined TC in Eq. (3), which will be determined from the data analysis. TC differs from TS for general cases, while they coincide with each other for a ferro-order case that is realized in the present case, as mentioned later. Next, we introduce the elastic compliance Sij , which is a component of the inverse Cij matrix. Elastic compliance represents the ‘‘structural’’ susceptibility of elastic systems, and corresponds to the magnetic susceptibility χ in magnetic systems. SΓ = SΓ ,0
T −Θ T − TC
.
(4)
The S66 (= 1/C66 ) can be decomposed into the sum of the anomalous contributions that exhibit critical behavior S66,cr
(OΓ )2 T
≈
⟨φ |OΓ | φ⟩2 T
.
(6)
Here, |φ⟩ is the eigen function of 3d-orbitals, ⟨· · ·⟩ the thermal average, and OΓ the quadrupole operator with irreducible representation Γ . OΓ is described as OXY = lX lY + lY lX , OYZ = lY lZ + lZ lY and O22 = l2X − l2Y for C66 , C44 and CE , respectively, by using angular momentum operator (lX , lY , lZ ) with l = 2. The energy scheme is shown in Fig. 4(b). For C66 , ⟨φ± | OXY |φ± ⟩ = ∓3, where |φ± ⟩ = √1 (|YZ ⟩ ± |ZX ⟩). For CE , ⟨YZ | O22 |YZ ⟩ = 3, 2
and ⟨ZX | O22 |ZX ⟩ = −3. For C44 , ⟨YZ | OYZ |YZ ⟩ is vanished. This may lead to a large elastic softening for either C66 or CE , while no softening for C44 . 4.2. Itinerant (band) picture We know superconducting systems, which show large elastic softening, the A15 compound V3 Si and Laves phase compound CeRu2 being famous examples [38,39]. Both compounds are categorized into conventional superconductors. CeRu2 was precisely investigated, because it was a candidate for the Fulde–Ferrel–Larkin–Ovchinnikov (FFLO) state. In the case of CeRu2 , the related compounds LaRh2 and CeCo2 show also elastic softening toward low temperatures, but CeRh2 shows no elastic anomaly [40,41]. Therefore, the softening in these materials have been ascribed to a large density of states at Fermi energy, which is contributed mainly from d electron component [40,42]. The 3d-orbitals in an iron-based superconductor form bands. According to band calculations, the bands located above the Fermi energy at the Γ -point form hole Fermi surfaces and electron pockets at M-points ofthe zone boundary [19]. In addition, the band nesting along the πa , πa , 0 direction is a key feature in ironbased superconductors. Here, we will consider the effects of bands. The bandwidth is affected by the crystal deformation, because the electron transfer (namely, transfer integral) between iron atoms would be modified by the lattice distortion. Therefore, equivalent four M-points in the tetragonal lattice do not become equivalent under the application of the strain εXY , as shown in Fig. 5 [43]. For example, the width of the bands at the Brillouin zone boundary M1 and M3 in Fig. 5 becomes large, and those of the bands at M2 and M4 become smaller under the application of εXY . This process gains the electronic energy, and looses the elastic energy. The amount of deformation is determined by the energy valance of the electronic and lattice energies. The formula for the elastic constant based on this consideration is as follows [44,45]: C = C0 − (dM1 − dM2 )2
χS0 1 − I χS0
(7)
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Fig. 5. (Color online) Effect of band on the crystal deformation. Four M points in the square lattice do not become equivalent by the strain εXY , which makes the band consisting of dyz /dzx and dx2 −y2 degeneracy lifted [43]. Source: This figure was taken from Ref. [27]. © 2011, Journal of the Physical Society of Japan.
Fig. 6. (Color online) Temperature dependence of the inverse of S66,cr . Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
χ = 0 S
1 kB T
N (E ) =
N0 2
fk (1 − fk ) =
dE N (E ) f (E )
k
1+
E − EF
2 −1
W
where f is the Fermi–Dirac function, d(= dM1 = −dM2 ) is the electron–lattice coupling constant, N0 is the density of states at Fermi energy EF , and I is the intersite interaction. If a large density of states exists at the M-point, an elastic anomaly may be caused. This provides a natural solution to the question of why the anomaly only emerged in C66 . In this situation, no anomaly is expected in 12 (C11 − C12 ), because the bands at four M-points stay equivalent under the presence of εXX − εYY . How about the case of band nesting? We have no analysis tool for the elastic constant based on the band nesting. We suppose that the formula is more complicated than the band Jahn–Teller effect, because the effects of the whole band in the Brillouin zone have to be accounted for. Nevertheless, we infer that it brings about a similar effect to the elastic constant anomaly discussed above. 5. Analysis of C66 5.1. Analysis based on the Jahn–Teller formula Fig. 6 shows the inverse of S66,cr as a function of temperature. Here, we employed the data on Ba(Fe0.755 Co0.245 )2 As2 for S66,0 ,
Fig. 7. (Color online) Theoretical fitting of C66 by the band Jahn–Teller effect under the assumption of Lorentzian density of states. Adjustable parameters are the bandwidth W and the intersite interaction I. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
due to its normal T -dependence, and subtract it from the other data. It can clearly be seen that 1/S66,cr in the underdoped region (x < 0.070) exhibits linear T -dependence. This indicates that S66,cr obeys Eq. (5). In the underdoped samples, TC in Eq. (5) and TS obtained from C66 are 134.3 and 141 K for x = 0, 81.4 and 84.7 K for x = 0.037, 28.2 and 30.0 K for x = 0.060, respectively. The closeness of the values in each case suggests the occurrence of ferro-order. This strongly suggests the crystal deformation of εXY below TS , and the ordering of the parameter possessing the same symmetry as εXY . The amount of EJT is about 50 K, which is almost independent of x. 5.2. Analysis based on the band picture The T -dependence of C66 in the overdoped region cannot be explained by the Jahn–Teller formula, but by a band picture. Fig. 7 shows theoretical curves of Eq. (7) under the assumption of the Lorentzian density of states. In this analysis, the main adjustable parameter is the bandwidth W . We fixed EF /W = 0.4 to obtain the best fitting. The obtained parameters are displayed in Fig. 8. The value of W is proportional to x − 0.07. From the adjustable parameter N0 d2 , the value of d was evaluated under the assumption that N0 is approximately equal to 1/W . The value of d is located in the range of 0.22–0.28 eV/Fe, for overdoped samples. The values of N0 are also evaluated to be 110, 64, 46, and 26 states/eV for the four samples, respectively, when we assume. N0 = 1/W .
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coupling is not caused by weak spin–orbital coupling in transition metals, but by the orbital nature of this system. These values are considered to support the orbital-based theory. By the way, it is worthwhile to speculate the shrinkage of the effective orbital (quadrupole) moment. The application of the JahnTeller formula in the underdoped samples implies the existence of an orbital (quadrupole) moment. However, it is considered that it is somewhat different from the fully localized picture. Once the system enters the overdoped region, the temperature dependence crosses over to the itinerant picture across the QCP discussed in the next section. It means the system is located on the boundary between the itinerant and localized. If we assume that the value of λ is approximately equal to d = 0.22 eV/Fe for x = 0.084,
(OXY )2 is evaluated to be 1.3. Another evaluation was made by the condition that the Θ in Eq. (3) must be equal to I (OXY )2 for x = 0.084 sample. This gives (OXY )2 = 1.6. Both evaluations give similar values. According to the localized picture, the value of ⟨OXY ⟩ is 3. This suggests
the orbital (quadrupole) moment of
Fig. 8. (Color online) The obtained parameters from the analyses based on the localized and band pictures of Eqs. (5) and (5), respectively. Electron–lattice coupling constants of λ
(OXY )2 was evaluated from EJT . The evaluation method
of d is described in the text. The curves are visual guides.
the quadrupole moment of this system is remarkably reduced. This is consistent with the itinerant electron picture in iron-based material. 6. Structural quantum criticality and superconductivity 6.1. Phase diagram
Fig. 9. (Color online) Phase diagram of Ba(Fe1−x Cox )2 As2 . The magnetic transition temperature TN for x = 0.036 was taken from Ref. [50]. The curves are the visual guides. TS and Tsc obtained in this work are consistent with the values reported elsewhere [51]. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
5.3. Strong electron–lattice coupling and shrinkage of orbital (quadrupole) moment We would like to summarize the values of the electron–lattice coupling λ constants in this system. Fernandes et al. and Kontani et al. estimated the value of λ to be 17 meV/Fe and 0.2 eV/Fe, respectively [21,33]. The parameters obtained in Section 5 are summarized in Fig. 8. For the underdoped samples, λ is obtained from EJT = 50 K. The calculated value of λ
(OXY )2 from EJT is 0.25
eV/Fe for x = 0.037. For overdoped samples, the value of d is located in the range of 0.22–0.28 eV/Fe. These values of the coupling constant are consistent with the orbital-based theory both from the localized picture and band picture. Generally speaking, a coupling between the magnetic moment and the stain is mediated by a spin–orbit interaction. It is considered that such strong electron–lattice
Fig. 9 summarizes the phase diagram of the Ba(Fe1−x Cox )2 As2 system. We found two characteristic temperatures of T ∗ and Tmax . As shown in Fig. 6, 1/S66,cr deviates from T -linear behavior below a certain temperature T ∗ ; i.e., 40 K for x = 0.084, 130 K for x = 0.098, and 190 K for x = 0.116. We found Tmax , which corresponds to the temperature at which S66,cr takes the maximum value (1/S66,cr takes the minimum value), as shown in Figs. 6 and 10(a). It would be notable that T ∗ and Tmax go to zero at the QCP concentration of xC = 0.07. A possible explanation of T ∗ is the crossover from the non-Fermi liquid to the Fermi liquid region. A similar behavior including the x-dependence of the crossover region is observed in BaFe2 (As1−x Px )2 [46]. On the Tmax , there are two possible explanations. For highly correlated electron systems such as CeCu2 Si2 , UPd2 Al3 , and UPt3 , a similar maximum was reported in the magnetic susceptibility χ . The Tmax for these cases has been considered to be a Kondo temperature [47]. In the case of magnetic QCP, the Kondo temperature means the entrance to the novel electronic state at low temperatures; namely the coherent motion of f -electrons. Can it be applied for the iron-based materials, where the role of orbitals is considered to be relevant? The low temperature coherent state may be an orbital liquid from the analogy of Fermi liquid. Another explanation will be made by a pseudo-gap formation. The minimum behavior in the temperature dependence of C66 can be fit by the density of states containing a pseudo-gap. In the case of Ba(Fe1−x Cox )2 As2 , an existence of the pseudo-gap was suggested [48]. However, the temperature, where the pseudogap opens, is located at higher T than TS , and its Co concentration dependence is quite different from that of Tmax . Tmax is considered to be proportional to the bandwidth W , which suggests that Tmax should correlate with W . One of the possible candidates is the Dirac node, which was predicted in iron-based superconductors [49]. This should be clarified by future investigations. In addition, the analysis based on the band picture suggests possible mass enhancement toward QCP in this system. A similar phenomenon was reported for BaFe2 (As1−x Px )2 [52]. The obtained phase diagram and various phenomena near the QCP resemble those of the well-known rare-earth compounds and uranium
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Fig. 11. Temperature dependence of C66 for the overdoped Ba(Fe0.940 Co0.060 )2 As2 near Tsc in expanded scales. Source: This figure was taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
Fig. 10. (Color online) (a) Temperature dependence of S66,cr for the overdoped samples. (b) Tsc from this work and Ref. [51] and the inverse of the peak value of S66,cr as a function of the distance from the QCP concentration x − xC . The curve of Tsc is a visual guide. Source: These figures were taken from Ref. [25]. © 2011, Journal of the Physical Society of Japan.
compounds. This coincidence strongly suggests the intimate relationship between superconductivity and QCP in this system as discussed in Section 6.2. It should be noted that the quantum criticality appears in structural fluctuations. On this point, we would like to name the structural quantum criticality. 6.2. Correlation between elastic anomaly and superconductivity We now discuss the relationship between the elastic anomaly and superconductivity. As seen in Fig. 10(b), the amount of 1/S66,cr is proportional to x − xC , where xC = 0.07 for this system. Such behavior on the χ as a function of the distance from QCP has been predicted theoretically for the itinerant electron magnetic QCP system [53]. It is surprising that such well-known behavior holds in this system with the respect of S66 instead of χ for the magnetic system. As shown in the same figure, Tsc decreases with increasing x − xC . So we can recognize an apparent correlation between Tsc and 1/S66,cr that Tsc is a function of 1/S66,cr . The explanation for this interesting fact is speculated to be as follows. As shown in Fig. 11, the underdoped sample exhibits a small anomaly at Tsc , while a large upturn at Tsc is seen in the overdoped samples. Once the system enters the orthorhombic phase from the tetragonal phase, structural fluctuations are suppressed in the ordered phase. In the overdoped samples, however, structural fluctuations still
survive even at Tsc . The amount of the anomaly at Tsc correlates with the peak height of S66,cr , which is the measure of structural fluctuation. The large anomaly at Tsc for the overdoped samples suggests strong coupling between the structural fluctuations and superconductivity. These facts suggest that the origin of S66,cr is deeply related to the emergence of superconductivity. According to the NMR measurement of Ba(Fe1−x Cox )2 As2 , remarkable enhancements of the spin susceptibility in 75 As NMR Knight sift and (1/T1 T )0.5 were observed [54]. These enhancements also show similar quantum critical behavior to the S66 in the elastic measurement. There is no doubt that the QCP behavior observed in NMR and elastic measurements would be expected to be ascribed to the same origin, because the elastic constant is no sensitive probe for magnetism, but is sensitive to orbital (quadrupole), contrary to NMR. In that sense, our measurement preferably observes the orbital (quadrupole) side of the quantum criticality. 7. Elastic anomaly associated with superconducting transition and large Grüneisen parameter The elastic investigation showed strong electron–lattice coupling with respect to the structural transition. The strong electron–lattice coupling in iron-based compound was suggested from another point of view [24]. Fig. 12 shows the temperature dependence of CL = 12 (C11 + C12 + 2C66 ) in the magnetic field, which was applied parallel to the propagation direction of the sound [110]. CL shows a softening with decreasing temperature in the region above 26 K, which is caused by the elastic anomaly of C66 . A remarkable step-like anomaly was observed at Tsc = 24.5 K at 0T . The Tsc tends to decrease as applying the field, which confirms that the step-wise anomaly is originated from the superconductivity. The shape of the elastic anomaly in CL suggests that the coupling between the superconducting order parameter η and elastic strain ε is −ξ η2 ε , which is the same as so called magneto–elastic coupling. This type of coupling is originated from the elastic strain dependence of Tsc . We consider that the CL anomaly at Tsc is not originated from C66 , but from 12 (C11 + C12 ), which comes from the coupling between εXX +εYY (see Fig. 2) and the superconducting order parameter, because the anomaly in C66 for this sample is small as shown in Fig. 11. The magneto–elastic coupling constant ξ is related to Grüneisen Tsc parameter Ωsc for Tsc , which is defined as Ω ≡ − T1 ∂∂ε . By sc
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However, irrespective of whether the spin or orbital mediate superconductivity, the results of the present study suggest that the structural fluctuation must be actively incorporated into their role, and that the orbital degrees of Fe–3d must also be taken into consideration in order to understand the full picture of superconductivity in iron-based compounds.
CL (GPa)
115
Acknowledgments
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113
112 15
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20
25
30
35
The authors wish to thank H. Fukuyama, H. Hosono, Y. Õno, H. Kontani, Y. Yanagi, J. Schmalian, T. Goto, P. Thalmeier and B. Lüthi for their valuable discussions. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas, ‘‘Heavy Electrons’’ (No. 20102007), of The Ministry of Education, Culture, Sports, Science and Technology, Japan, and the Transformative Research-project on Iron Pnictides of the Japan Science and Technology Agency.
Temperature (K)
References Fig. 12. (Color online) Temperature dependence of (C11 + C12 + 2C66 )/2 for 6.0% sample in the magnetic field of 0, 3 and 5T is displayed with an expanded scale near Tsc . Source: This figure was taken from Ref. [24]. © 2011, Institute of Physics.
employing the scaling Ansatz, we can get the formula of ∆C = −Ωsc2 ∆CV Tsc [55]. Here, ∆C and ∆CV are jumps of the elastic constant and the specific heat at Tsc , respectively. By using ∆CL = 1 GPa and ∆CV = 0.67 J/mol · K [56], we get |Ωsc | = 62. This value is comparable to Ωsc = 15 and −40 for a and c axes, respectively, which were evaluated from thermal expansion data, [56] and Ehrenfest’s relation. Large Ωsc has been reported in heavy fermion systems [57]. The large value of the iron-based superconductor leads us to infer that this system should be categorized into strongly correlated electron system [58]. It would be interesting that the oxide superconductor La2−x Srx CuO4 also shows a large Ωsc [59]. We suppose that the large Grüneisen parameter may be related to the fact that this system is located closely to the quantum critical point. 8. Summaries and conclusion Iron-based superconductor Ba(Fe1−x Cox )2 As2 shows a remarkable elastic softening in C66 . This softening is a precursor for the tetragonal-orthorhombic transition. Similar elastic softening has been observed in A15 compounds, Laves phase compounds and La1−x Srx CuO4 . The correlation between the superconductivity and the elastic anomaly has attracted much attention, but has not been clarified so far. In most cases showing the elastic anomalies, the lattice instability comes from the phonon anomaly. Such a case is not so interesting from the view of superconductivity, because the energy scale of phonon is not large and cannot act a relevant role in the emergence of high-Tc superconductivity. This consideration leads us to misunderstand the relevance of the signature of the elastic anomaly. We have confidence that our study on iron-based superconductor is spotlighted on the relevance of structural fluctuation through the systematic investigation. If the elastic anomaly is caused by the origin possessing high-energy scale; for example, band, orbital (quadrupole) fluctuations or magnetic fluctuations, it may lead to a high-Tc superconductivity. However, we infer that a key point is the strong electron–lattice coupling. In the case of Ba(Fe1−x Cox )2 As2 , the strong electron–lattice coupling is suggested both for the structural instability in C66 and the anomaly at Tsc in CL . We suppose that it would be a characteristic feature of multi-band systems. Further discussions will be necessary on the role of the spin and orbital in the mechanism of superconductivity.
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