Power-law dependence of the optical conductivity observed in the quantum spin-liquid compound $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$

The Mott-insulator $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ is the prime candidate of a quantum spin-liquid with puzzling magnetic properties. Our terahertz and infrared investigations reveal that also the charge dynamics does not follow the expectations for a Mott insulator. We observe a large in-gap absorption where the excess conductivity exhibits a power-law behavior ${\sigma }_1^{\mathrm{exc}}\left(\omega \right)\propto {\omega }^n$ that grows stronger as the temperature decreases and extends all the way through the far infrared. With $n\approx 0.8$ to 1.5, the exponent is significantly smaller than predicted by Ng and Lee [T.-K. Ng and P. A. Lee, Phys. Rev. Lett. 99, 156402 (2007)] for spinon contributions to the optical conductivity. We suggest fluctuations become important in the spin-liquid state and couple to the electrodynamic properties differently compared to the antiferromagnetic Mott insulator $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl. We discuss the various possibilities of how charge fluctuations are influenced by the presence or absence of magnetic order.

Figure 1

(a) Sketch of the BEDT-TTF molecule. (b) For $\kappa$-(BEDT-TTF)${}_{2}X$, the molecules are arranged in dimers, which constitute an anisotropic triangular lattice within the conduction layer. In the case of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$, the plane is labeled as $bc$, indicated by the arrows, while it is the $ca$ plane in the case of $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl. The inter-dimer transfer integrals are labeled by $t$ and $t^{\prime }$ and can be calculated by tight-binding studies of molecular orbitals or ab initio calculations.[12, 13, 21] With the intradimer transfer integral $t_d\approx 0.2$ eV[16, 22] and the onsite Coulomb repulsion $U\approx 2t_d$,[23] one obtains at ambient conditions $U/t=5.5$ with the ratio of the two inter-dimer transfer integrals $t^{\prime }/t\approx 0.44$ in the case of the Mott insulator $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl. For the spin-liquid compound $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$, the effective Hubbard $U$ is larger ($U/t=7.3$) and most important the transfer integrals $t^{\prime }/t=0.83$ are very close to equality.

Figure 2

(a) The temperature dependence of the in-plane dc resistivity of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ (bold blue line) and $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl (light red line) evidences an insulating behavior. (b) The low-temperature optical conductivity ($\kappa$-CN: $T=13$ K; $\kappa$-Cl: $T=20$ K) does not show a Mott gap, but also reveals important differences between both compounds.

Figure 3

Evolvement of the optical reflectivity and conductivity of $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl as the temperatures is varied. For both polarizations of the light, a gap opens upon cooling and the spectral weight shifts to higher energies.

Figure 4

Optical reflectivity and conductivity of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ measured at different temperatures along the two in-plane polarizations as indicated. As the temperature is lowered the low-frequency reflectivity and conductivity rises, very much in contrast to $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl displayed in Fig. 2.

Figure 5

Optical conductivity of (a) and (b) $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Cl and (c) and (d) $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ for the two in-plane polarizations. The temperature evolution is completely different for the two compounds, as indicated by the red arrows. (e) and (f) The low-frequency conductivity of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ plotted in a double logarithmic fashion in order to demonstrate the power-law behavior. The blue dashed lines correspond to ${\sigma }_1\left(\omega \right)\propto {\omega }^{1/2}$ and ${\sigma }_1\left(\omega \right)\propto {\omega }^2$, respectively.

Figure 6

Optical conductivity of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ ($E\parallel b$, $T=12.6$ K) and background due to molecular and lattice vibrations and interband transitions, which is subtracted in order to retrieve the possible electronic and spinon contributions.

Figure 7

Frequency-dependent conductivity of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$ after the background due to vibrational and interband excitations has been subtracted. The experiments were performed for the polarization $E\parallel b$ at different temperatures as indicated. A power-law behavior ${\sigma }_1\left(\omega \right)-{\sigma }_1^{\mathrm{background}}\left(\omega \right)={\sigma }_1^{\mathrm{exc}}\left(\omega \right)\propto {\omega }^n$ can be extracted over a large frequency range (indicated by red lines). A change in slope may be identified for higher frequencies (blue lines) that becomes in particular obvious at low temperatures. The temperature dependence of the exponents are summarized in Fig. 8 for both polarizations.

Figure 8

Exponents $n$ (red dots) and $m$ (blue squares) extracted from the power law ${\sigma }_1^{\mathrm{exc}}=A{\omega }^n{{|}_{\omega <{\omega }_{\mathrm{c}}}+B{\omega }^m|}_{\omega >{\omega }_{\mathrm{c}}}$ measured at different temperatures for the polarizations $E\parallel b$ and $E\parallel c$. The lines correspond to spline interpolations.