Crystal structure, physical properties and superconductivity in $A_{x}$Fe$_2$Se$_2$ single crystals

We studied the correlation among structure and transport properties and superconductivity in the different $A_x$Fe$_2$Se$_2$ single crystals ($A$ = K, Rb, and Cs). Two sets of (00$l$) reflections are observed in the X-ray single crystal diffraction patterns, and arise from the intrinsic inhomogeneous distribution of the intercalated alkali atoms. The occurrence of superconductivity is closely related to the {\sl c}-axis lattice constant, and the $A$ content is crucial to superconductivity. The hump observed in resistivity seems to be irrelevant to superconductivity. There exist many deficiencies within the FeSe layers in $A_x$Fe$_2$Se$_2$, while their $T_{\rm c}$ does not change so much. In this sense, superconductivity is robust to the Fe and Se vacancies. Very high resistivity in the normal state should arise from such defects in the conducting FeSe layers. $A_x$Fe$_2$Se$_2$ ($A$ = K, Rb, and Cs) single crystals show the same susceptibility behavior in the normal state, and no anomaly is observed in susceptibility at the hump temperature in resistivity. The clear jump in specific heat for Rb$_x$Fe$_2$Se$_2$ and K$_x$Fe$_2$Se$_2$ single crystals shows the good bulk superconductivity in these crystals.

and it is much more complicated than that in the Fe-Se systems. The intercalated alkali atoms could be crucial to the superconductivity. The normal-state resistivity should be influenced by these vacancies seriously. But how these vacancies affect the physical properties still remains unresolved.
In this article, we systematically studied the effect of the starting materials and the heating process on the single crystal growth for A x Fe 2 Se 2 (A = K, Rb, and Cs), and measured the physical properties of these single crystals and determined their crystal structures. It is found that two sets of (00l) reflections exist in all the crystals, and superconductivity is closely related to the c-axis lattice constant, indicating that the A content is crucial to the superconductivity. The hump in resistivity arises from the defects within the conducting FeSe layers and is irrelevant to superconductivity. No anomaly is observed in magnetic susceptibility at the temperature of hump in resistivity. The clear jump in specific heat for superconducting K x Fe 2 Se 2 and Rb x Fe 2 Se 2 single crystals indicates the good bulk superconductivity in these crystals.
Single crystals A x Fe 2 Se 2 (A = K, Rb, and Cs) were grown by Bridgeman method as described elsewhere [12,13]. The starting materials and the heating process are very important to get superconducting single crystal, and even is slightly changed to dramatically affect its physical properties. The three different batches of Rb x Fe 2 Se 2 and three batches of K x Fe 2 Se 2 single crystals were gotten by the slightly change of the heating temperatures and starting materials. The single crystals were characterized by X-ray single crystal diffraction, magnetic susceptibility, and electrical transport measurements. X-ray single crystal diffraction was performed on a TTRAX3 theta/theta rotating anode X-ray Diffractometer (Japan) with Cu Kα radiation and a fixed graphite monochroma- tor. Magnetic susceptibility measurements were carried out using the Quantum Design MPMS-SQUID. The measurement of resistivity and magnetoresistance were done on the Quantum Design PPMS-9.
The typical temperature dependence of resistivity is observed for three batches of Rb x Fe 2 Se 2 single crystals as shown in Fig.1. Among these crystals, Rb x Fe 2 Se 2 -1 was obtained with nominal composition as Rb 0.8 Fe 2 Se 1.96 by being melt at 1080 • C and turning off furnace at 950 • C; Rb x Fe 2 Se 2 -2 and Rb x Fe 2 Se 2 -3 came from the same batch with the nominal composition as Rb 0.8 Fe 2 Se 2 by being melt at 1030 • C and turning off furnace at 700 • C; Rb x Fe 2 Se 2 -4 was grown with nominal composition as Rb 0.8 Fe 2 Se 1.96 by being melt at 1030 • C and switching off furnace at 700 • C. The resistivity of Rb x Fe 2 Se 2 -1 shows a very small hump around 290 K and then becomes metallic below this temperature with the residual resistivity ratio RRR = R(300 K)/R(35 K) ≈ 37.2. Superconductivity appears below 32.4 K and zero resistance is reached at 31.9 K. The superconducting transition width of Rb x Fe 2 Se 2 -1 is as narrow as 0.5 K although the resistivity at room temperature is as large as 70 mΩ cm at 300 K. The hump temperature of resistivity (T hump ) shifts to 265 K and 225 K for Rb x Fe 2 Se 2 -2 and Rb x Fe 2 Se 2 -3, respectively. The RRR decreases to 26.1 and 17.3 for the two crystals, respectively. These results indicate that the metallicity of Rb x Fe 2 Se 2 -2 and Rb x Fe 2 Se 2 -3 is weaker compared to Rb x Fe 2 Se 2 -1. However, as we can see from the inset of Fig.1 and Table I, the superconducting transition temperature seems not to vary with the change in the resistivity behavior and their values. The onset and zero resistance temperature for Rb x Fe 2 Se 2 -2 and Rb x Fe 2 Se 2 -3 are 32.0, 32.4 and 31.5, 31.6 K, respectively. Thus one can see that the humps in resistivity seems to be irrelevant to superconductivity. The large magnitude of the normal-state resistivity compared to FeAs-base pnictides [19][20][21] and FeSe [22] (usually with ρ(300 K) much less than 1 mΩ cm) reflects the existence of many deficiencies within the conducting FeSe layers in these Rb x Fe 2 Se 2 single crystals. The hump in resistivity should arise from such large amount of defects within the conducting FeSe layers. In this sense, superconductivity is quite robust to the vacancies within the FeSe layers. For Rb x Fe 2 Se 2 -4, although the resistivity still shows a hump at around 170 K, no superconductivity can be observed, and a strong semiconducting/insulator-like behavior is observed below 70 K. For comparison, we measured the temperature dependence of the Hall coefficient on the exactly same pieces of Rb x Fe 2 Se 2 -2, Rb x Fe 2 Se 2 -3 and Rb x Fe 2 Se 2 -4 as shown in Fig.2. The Hall coefficient of the superconducting crystals Rb x Fe 2 Se 2 -2 and Rb x Fe 2 Se 2 -3 is positive at high temperature, and gradually decreases with decreasing temperature and then becomes negative at low temperature. The sign change of the Hall coefficient is also observed in superconducting Tl 0.58 Rb 0.42 Fe 1.72 Se 2 samples previously [23]. Actually from ARPES results in A x Fe 2 Se 2 (A = K and Cs) [24,25], the A x Fe 2 Se 2 are electron over-doped and only electron pockets can be observed. However, hole pockets could exist in the superconducting samples due to sign change of Hall coefficient based on the Hall effect measurements, suggesting a pos- sibly multi-band nature of the superconductivity. For the sample without superconductivity, the Hall coefficient is negative in the whole temperature range. It indicates that dominant carrier is electron for non-superconducting crystal. It seems that the hole pocket might be quite important for the superconductivity. Therefore, it needs to be further investigated. What's more, no anomaly in the Hall coefficient is observed at T hump . It suggests that the humps in resistivity for A x Fe 2 Se 2 crystals are not related to a structural or magnetic transition, being contrasting to the facts that the anomaly of resistivity in underdoped FeAs-based superconductor is always relevant to a structural/magnetic transition. As shown in Fig. 1, one can observe almost the same T c for the single crystals with the different T hump . The X-ray single crystal diffraction was carried out for the same four pieces Rb x Fe 2 Se 2 single crystals shown in Fig. 1 to find out the relationship of superconductivity and structure. The X-ray diffraction (XRD) patterns are shown in Fig.  3. Surprisingly, two sets of (00l) reflections are observed in all the four samples. The two c-axis lattice parameters c1 and c2 are obtained (listed in Table I). The c-axis lattice parameters c1 and c2 correspond to the two sets of reflections with weak and strong intensities, respectively. These two distinct sets of reflections could arise from the inhomogeneous distribution of the intercalated Rb atoms. Considering the fact that the superconducting crystals show nearly fully shielding fraction, the reflections with c2 should be responsible for the superconductivity. From the superconducting to non-superconducting crystal, c1 is reduced by 0.55% while c2 is enhanced by more than 0.14%. It is found that the insulator-like behavior is enhanced with losing superconductivity. It indicates that superconductivity may exist within a limited range of the c-axis lattice parameter. In other word, the Rb content is crucial to the occurrence of superconductivity because the c-axis lattice parameter strongly depends on the Rb content.
We then carefully measured resistivity and the XRD patterns for three pieces of K x Fe 2 Se 2 crystals, as shown in Fig. 4 . It shows that the K x Fe 2 Se 2 crystals exhibit the obviously different resistivity behavior. K x Fe 2 Se 2 -1 was grown using K 0.8 (FeSe) 2 as starting materials and being melt at 1030 • C for 3 hours. K x Fe 2 Se 2 -2 and K x Fe 2 Se 2 -3 were grown by using K 0.8 (FeSe) 2 as starting materials and being melt at 1030 • C for 2 hours and 950 • C for 20 hours, respectively. In Fig.4a, two sets of (00l) reflections can be observed, and this behavior is the same with those in the Rb x Fe 2 Se 2 samples, suggesting that the inhomogeneous distribution of the intercalated alkali atoms in the crystals is common feature. For the non-superconducting sample, the c-axis lattice constant c1 is smaller by more than 0.64%, while c2 is larger by 0.15% than those in the superconducting samples. These results are consistent with the results observed in the Rb x Fe 2 Se 2 samples, indicating that the content of alkali atom plays a crucial role for the occurrence of superconductivity. K x Fe 2 Se 2 -1 shows a broad resistivity hump at about 220 K and superconductivity at 31.7 K. For the K x Fe 2 Se 2 -2, T hump shifts to 120 K and superconductivity shows up at 30.3 K. Although the superconductivity disappears in K x Fe 2 Se 2 -3, the T hump for K x Fe 2 Se 2 -3 is higher than that for the superconducting K x Fe 2 Se 2 -2, strongly demonstrating that  the superconductivity is not correlated to the hump in resistivity. The position of the hump in resistivity reflects the vacancy level within the conducting FeSe layers. It suggests that the vacancies within the FeSe layers have much weaker correlation to superconductivity than the content of intercalated alkali atoms does. Figure 5 shows the X-ray single crystal diffraction pattern and the temperature dependence of resistivity for Cs x Fe 2 Se 2 single crystal. Totally metallic resistivity can be observed below 300 K. Superconductivity was ob-  Fig.5a, very small reflections corresponding to c1 can still be found except for the main reflections with c2. It suggests the inhomogeneous distribution of alkali atoms is common for all the A x Fe 2 Se 2 single crystals. It is worthy to note that the superconductivity always shows up around 30 K for the crystals A x Fe 2 Se 2 with changing the intercalated alkali atom A from K, Rb to Cs. The T c seems not to depend on the ionic radii of the intercalated alkali atoms although the superconductivity strongly depends on the A content. As shown in Fig.1 and Fig.4b, the hump in resistivity changes pronouncedly for the same alkali atom case, while the T c is nearly the same (about 30 K). Very large normal-state resistivity is observed in all the above A x Fe 2 Se 2 single crystals, suggesting that large amount of deficiencies within the conducting FeSe layers for all of these crystals. Based on these observations, T c seems to be robust to the vacancies within the FeSe layers. Magnetic susceptibility was measured on the A x Fe 2 Se 2 single crystals to investigate the correlation between the normal-state resistivity and magnetism. Figure 6 shows the anisotropic magnetic susceptibility with the magnetic field of 5 T applied within the ab-plane and along the c-axis for Rb x Fe 2 Se 2 -1, Rb x Fe 2 Se 2 -2, Rb x Fe 2 Se 2 -4, respectively. Although the samples show very different resistivity behavior, such as the different T hump s, the magnitude of resistivity and T c , the normal-state susceptibility shows the quite similar behavior to each other. As the field is applied within the ab-plane, the magnitude of susceptibility varies within 20% in the normal state and the susceptibility itself shows a broad minimum. No anomaly can be found at T hump , suggesting that hump in resistivity cannot be ascribed to a magnetic transition. Fig.7 shows the susceptibility with the magnetic field of 5 T applied parallel and perpendicular to the c-axis for K x Fe 2 Se 2 -2 and Cs x Fe 2 Se 2 , respectively. The similar behavior with those of Rb x Fe 2 Se 2 crystals shown in Fig.6 is observed. These results indicate that although the electronic properties change dramatically from system to system and from crystal to crystal, the magnetic property does not change a lot.  Fig.8a shows the temperature dependence of the specific heat (C/T ) around T c at the magnetic field of 0 T and 9 T for Rb x Fe 2 Se 2 -2 and K x Fe 2 Se 2 -1, respectively. At 0 T, one can see the clear specific heat anomaly at T c . Although 9 T is far lower than the upper critical field, which is estimated higher than 100 T, [12,13] the anomaly in specific heat is completely suppressed. The specific heat jump (C(0T)-C(9T))/T against T is plot-ted in Fig.8b for the two crystals. The heat capacity jumps for Rb x Fe 2 Se 2 -2 and K x Fe 2 Se 2 -1 crystals shows almost the same behavior. The clear heat capacity jump in the superconducting samples definitely indicates the good bulk superconductivity in these crystals.
The X-ray single crystal diffraction patterns reveals two sets of (00l) reflections existing in all the crystals of A x Fe 2 Se 2 . Such two sets of reflections strongly depend on the starting composition and heating process. Although the superconducting phase is dominant, the trace of second phase is still observed as shown in Fig.5a. These results indicate the existence of inhomogeneous distribution of the A atoms in all the crystals. It is found that superconductivity is closely related to the c-axis lattice constant, indicating that the A content is crucial to the superconductivity because the c-axis lattice parameter strongly depends on the A content. The A content in single crystals is quite sensitive to the nominal composition and condition of crystal growth. Therefore, it is not easy to grow the single crystal with superconductivity. The very large normal-state resistivity relative to other iron pnictide superconductors suggests the large amount of deficiencies within the conducting FeSe layers for all the A x Fe 2 Se 2 single crystals. The hump in resistivity should arise from such defects, and seems to be irrelevant to superconductivity. Despite of the existence of many deficiencies within the conducting FeSe layers in A x Fe 2 Se 2 , T c does not change a lot with varying A from K to Rb and Cs. Therefore, superconductivity seems robust to the such vacancies. No anomaly is observed in magnetic susceptibility at the hump temperature in resistivity. It suggests that the humps in resistivity in A x Fe 2 Se 2 are not related to a structural or magnetic transition, being contrasting to the facts that the anomaly of resistivity in the underdoped FeAs-based superconductor is always relevant to a structural/magnetic transition.
In conclusion, we systematically studied the structure by the X-ray single crystal diffraction and measured the transport properties in the A x Fe 2 Se 2 single crystals. All the samples show two sets of (00l) reflections in X-ray single crystal diffraction patterns, indicating the intrinsically inhomogeneous distribution of the intercalated A atoms. The occurrence of Superconductivity is closely related to the c-axis lattice parameter, indicating that the A content is crucial to the superconductivity in A x Fe 2 Se 2 . The very large magnitude of the normalstate resistivity reflects the large amount of deficiencies within the conducting FeSe layers. The hump in resistivity should originate from these defects and is found to be irrelevant to the superconductivity. In this sense, superconductivity is robust to the vacancies within the FeSe layers. No anomaly in susceptibility is observed to be associated with the hump in resistivity. The clear jump in specific heat for Rb x Fe 2 Se 2 and K x Fe 2 Se 2 superconducting single crystals indicates the good bulk superconductivity in these crystals.