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Observation of dynamic charge stripes in Tm0.19Yb0.81B12 at the metal-insulator transition

Higher accuracy low temperature charge transport measurements in combination with precise X-ray diffraction experiment have allowed detecting the symmetry lowering in the single domain Tm0.19Yb0.81B12 crystals of the family of dodecaborides with metal-insulator transition. Basing on the fine structure analysis we discover formation of dynamic charge stripes within the semiconducting matrix of Tm0.19Yb0.81B12. The charge dynamics in these metallic nano-size conducting channels is characterized by broad-band optical spectroscopy that allowed estimating the frequency (~2.4 10^11 Hz) of quantum motion of the charge carriers. It is suggested that caused by cooperative Jahn-Teller effect in the boron sub-lattice, the large amplitude rattling modes of the Tm and Yb ions are responsible for modulation of the conduction band along [110] direction through the variation of 5d-2p hybridization of electron states.

During last twenty years a special attention was paid to search for an answer on two main questions: (1) Is there any universal scenario of the dynamic charge and spin stripe formation in the strongly correlated electron systems? and (2) How can the dynamic stripes be detected experimentally? The origin of conducting stripes is discussed usually from a strong-coupling perspective and various approaches were developed to describe (i) frustrated phase separation (or micro phase separation), (ii) spin-charge "topological" properties, and (iii) valence-bond solid formation [14]. Although the existence of both charge stripes and nematic order in strongly correlated electron systems is now a well-established experimental fact, the origin of these inhomogeneities remains controversial. Taking into account that, from the strong-coupling perspective, stripes are a real-space pattern of micro phase separation [8], the nanoscale visualization of the filamentary structure could be realized with the help of scanning tunneling microscopy and spectroscopy (STM and STS). However, STS and STM are essentially static and surface sensitive techniques and consequently dynamic stripes can only be detected if pinned by impurities. Among indirect methods, anisotropic charge transport measurements on single domain crystals are considered as most effective technique enable to detect the symmetry lowering below the transition to fluctuating stripe phase [14]. Among other experiments that recently have been successfully used for observation of dynamic charge stripes, both inelastic neutron scattering [30] and ultrafast and equilibrium mid-infrared spectroscopy [31] should be mentioned. In these studies the lattice fluctuations associated with the dynamic charge stripes were detected in the model compounds -doped nickelates La 2-x Sr x NiO 4 (see also [32]).
Very recently, fluctuating charge stripes have been detected in non-magnetic dodecaboride LuB 12 [29,33] and their origin was explained in terms of the cooperative dynamic Jahn-Teller effect in the boron B 12 cubooctahedra. It was suggested [29,33], that this kind of boron lattice instability induces large amplitude vibrations of rare earth ions (rattling modes) resulting in (i) dramatic changes of 5d − 2p hybridization of electron states along the unique crystallographic axis and (ii) formation of highly conductive channels-dynamic charge stripes with a strong charge carrier scattering on the filamentary structure in the metallic matrix of RB 12 . Taking into account that these metallic channels in the rare earth dodecaborides could be better visualized when formed within lower-conducting, dielectric matrix, it looks promising to investigate the stripe signatures in the crystals with composition close to YbB 12 that is a narrow-gap semiconductor, namely, the solid solutions of the Tm 1-x Yb x B 12 family [34,35]. That is why we have chosen for such studies crystals of Tm 1-x Yb x B 12 with x=0.81 and with isotopically pure boron ( 11 B) to avoid any 10 B-11 B substitutional disorder. Below we present the results of low temperature resistivity, magnetoresistance and transverse even effect measurements in combination with both precise X-ray diffraction and broad-band room-temperature optical studies. It is shown that dynamic charge stripes in Tm 1-x Yb x B 12 may be reliably detected in the difference Fourier maps resulting from the accurate analysis of the crystal structure fine details.
The lower symmetry electron-density distribution is confirmed by the maximal entropy method (MEM) [36]. At low temperatures the filamentary structure is also detected in the measurements of anisotropy of both the magnetoresistance and transverse even effect [35]. Moreover, about an order of magnitude different DC-and far-infrared AC-conductivities observed at room temperature are also indicative of the dynamic stripe formation in the Tm 0.19 Yb 0.81 B 12 dodecaboride matrix.
Experimental details. The details of growth of Tm 1-x Yb x B 12 single crystals, preparation of samples and the technique of magnetotransport measurements on the installation with a sample rotation in a magnetic field were described in [35,37]. In the present study we have used the same X-rays diffractometers and special software to extract fine details of the crystal structure as it was done recently with LuB 12 [29,33]. Broad-band reflectivity measurements of Tm 0.19 Yb 0.81 B 12 were carried out on the high quality Tm 0.19 Yb 0.81 B 12 single crystal ≈6 mm in diameter using a set of apparatus, including Vertex 80V Fourier-transform spectrometer, J.A.
Woollam V-VASE ellipsometer, cw backward-wave oscillator [38] and pulsed TeraView THz spectrometers (see [39] for more details) and the data from [40] were used to extend the reflection coefficient spectra up to ≈400000 cm -1 .
Results and discussion. Magnetotransport anisotropy. Fig. 1a shows the temperature dependences of the resistivity ρ(T) and the Hall coefficient R H (T, H 0 ) of the Tm 0.19 Yb 0.81 B 12 samples. The Arrhenius plot of the data (see [35] for details) reveals two activation regimes on the R H (T, H 0 ) curve at T> 10 K, corresponding to the energy gap E g ≈ 200 K≈ 17.8 meV (I) and the binding energy of the manybody states (resonance at E F ) in the gap E p ≈ 75 K (II), respectively. At temperatures below 10 K a transition to a coherent regime of charge transport is observed in the low field region that is suppressed by an external magnetic field of 80 kOe ( where schematic arrangement of these angular measurements is shown). It is worth noting that in addition to the ordinary Hall effect signal ρ H (T, H 0 ) in Tm 0.19 Yb 0.81 B 12 a transverse even effect ρ TE (φ, H 0 ) is also recorded from the Hall probes, and this second harmonic term ρ TE (φ, H 0 )= ρ H2 cos2φ is directly related to the formation of the filamentary structure of the conducting channels in the semiconducting matrix of the dodecaboride [35]. The procedure of separating the isotropic odd Hall component ρ H (φ, H 0 ) and the anisotropic transverse even contribution ρ TE is described in detail in [35]. The families of both resistivity curves ρ anis /ρ 0 (φ,H 0 ) and ρ TE (φ, H 0 ) separated within the analysis (see also [39], for details) are presented in polar coordinates in figs.1c and 1d, respectively (color shows the amplitude of the resistivity components). It is clearly seen that these two carrier scattering diagrams demonstrate uniaxial anisotropy and the selected axis is located predominantly along the H//[110] direction for both the resistivity (Fig.1c) and the transverse even effect (Fig.1d).
Crystal anisotropy. The compound Tm 0.19 Yb 0.81 B 12 crystallizes in a cubic lattice with the period a = 7.4679(3). We refined the structure in the high-symmetry Fm-3m group using room temperature X-ray data with low residual factor R = 0.49%. At the same time, accurately measured periods of the crystal lattice appeared to be slightly different: a = 7.4686(5), b = 7.4684(4), c = 7.4668(4). Similar 'tetragonal' distortion a ≈ b > c was observed in LuB 12 [29]. Difference Fourier synthesis of electron density (ED) [39], which in fact needs no data on crystal symmetry, is used to reveal fine structural details. Difference Fourier maps (Fig. 2a) show a low-symmetry distribution of difference ED in the faces (100), (010), (001) of the unit cell.
The asymmetry of ED distribution is confirmed by processing the structural data using the maximal entropy method (MEM) [36], which does not need atomic coordinates and chemical composition to restore ED from observed structure factors F obs . MEM maps in the three faces are presented in Fig. 2b for comparison. MEM reconstructs 'normal' but not difference ED, so light boron atoms are as clearly seen as heavy R atoms. As is clear from Fig. 2, the traces of conducting channels are observed along the face diagonals [011], , [110], but the most pronounced charge stripes are detected along  in the (010) plane.
Dynamic conductivity. The broad-band reflectivity spectrum of Tm 0.19 Yb 0.81 B 12 single crystal as presented in fig.3a (see [39] for measurement details). With the Kramers-Kronig analysis, the spectrum of dynamical conductivity was calculated, as shown in Fig.3b. Rich set of peaks in the spectrum was modeled with the sum of Lorentzian expressions where Δε is the dielectric contribution, ν 0 is resonance frequency and is the damping constant.
The details of the conductivity spectrum analysis and parameters of corresponding excitations will be published elsewhere (see also Table 1 in [39]). Among the most important results, two issues should be mentioned here: (i) the value of conductivity at terahertz frequencies, (30-40 cm -1 ) ≈ 1500 Ω -1 cm -1 , is by about an order of magnitude below the measured DC conductivity  DC ≈13000 Ω -1 cm -1 and (ii) modeling the mismatch with the Drude conductivity term (dashed line in Fig.3b)  provides the scattering rate of carriers 8 cm -1 which coincides very well with the damping of two quasilocal vibrations of the heavy R-ions (R-Tm and Yb) located at 107 cm -1 (~154 K) and 132 cm -1 (~190 K) (see Table 1 in [39]). For comparison, similar values of Einstein temperature Θ E =160-206 K have been detected previously in EXAFS [41], heat capacity [42] and inelastic neutron scattering [43] studies of the dodecaborides. Taking into account that these rattling modes induce variation of 5d-2p hybridization in RB 12 [29], we suggest that this kind of "modulation" of the conduction band is responsible for the dynamic charge stripes formation in the dodecaboride matrix. Besides, the frequency ~2.4•10 11 Hz (8 cm -1 ) may be deduced from the conductivity spectra as the characteristic of the quantum motion of charges in the dynamic stripes. As a result, the DC conductivity in Tm 0.19 Yb 0.81 B 12 at room temperature should be attributed to the charge transport in dynamic stripes that are more conductive than surrounding semiconducting matrix.
Consequently, the value of the DC conductivity is determined by the stripes that percolate through the crystal while relatively smaller AC conductivity is provided by the THz-FIR reflectivity of the 'whole' sample (conducting stripes + semiconducting matrix).