Nonlinear dynamics of conduction electrons in organic conductors

Nonlinear conductivity of organic conductors is explained quantitatively starting from a phenomenological energy-balance equation. Experimental results of ${\left(\text{TMET-TTP}\right)}_4{\text{PF}}_6$ are presented, where TMET-TTP is 2-[4, 5-bis(methylthio)-1,3-dithiol-2-ylidene]-5-[4,5-ethylenedithio-1,3-dithiol-2-ylidene]-1,3,4,6-tetrathiapentalene, and the simulation reproduces the nonlinear properties very well, including the voltage-current characteristics, power-law excess conductivity, and temperature dependence of the threshold field and current. The obtained heat capacity is much smaller than the lattice heat capacity, indicating unimportance of the Joule heating. Since the organic conductors show very little hysteretic switching, the time dependence of the energy balance is traced explicitly, defining nonlinear dynamics of hypothetical electron temperature, which reproduces oscillating states similar to that of the organic thyristor and even chaotic states under certain circumstances. It is expected that transient phenomena such as oscillatory and random outputs in some sort of organic and inorganic nonlinear conductors are quantitatively explained by this method, which in turn affords a playground of nonlinear dynamics.

Figure 1

(Color online) Temperature dependence of conductivity in ${\left(\text{TMET-TTP}\right)}_4{\text{PF}}_6$ with the molecular and crystal structures (Ref. 23). The conductivity is well represented by $\sigma ={\sigma }_1\text{exp}\left(-T_g/T\right)$, where $T_g=660\text{K}$, ${\sigma }_1=450\Omega \text{cm}$ for the in-plane conduction, and ${\sigma }_1=0.9\Omega \text{cm}$ for the out-of-plane conduction.

Figure 2

(Color online) Observed (a)–(c) and simulated (d)–(f) nonlinear conductivities in ${\left(\text{TMET-TTP}\right)}_4{\text{PF}}_6$. (a) $E\text{−}J$ characteristics measured by the four-probe method, applying a 5 ms current pulse parallel to the stacking direction $\left(\parallel c\right)$. The fitting curves are derived from Eq. (1). (b) $E\text{−}J$ characteristics measured by the two-probe method with application of a 5 ms voltage pulse parallel to the stacking direction $\left(\parallel c\right)$. (c) $E\text{−}J$ characteristics measured by the two-probe method with application of a voltage pulse perpendicular to the conducting sheet $\left(\parallel a\right)$. (d) In-plane four-probe $E\text{−}J$ characteristics simulated from Eq. (3) so as to reproduce (a). The dotted curve is based on smaller $\alpha =50\text{W}{\text{K}}^{-1}{\text{cm}}^{-3}$ at 60 K, and the other parameters are unchanged. (e) In-plane two-probe measurements at 60, 70, 80, 90, 100, and 120 K. ${\rho }_L=0.8\Omega \text{cm}$. The dotted curve is based on larger $nC=1.0\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$ at 70 K. (f) Out-of-plane two-probe measurements $\left(\parallel a\right)$ at 60, 70, 80, 90, 100, and 120 K. ${\rho }_L=860\Omega \text{cm}$.

Figure 3

(Color online) Temperature dependence of observed and simulated threshold currents, fields, and power in ${\left(\text{TMET-TTP}\right)}_4{\text{PF}}_6$. (a) Observed temperature dependence of $J_{\text{th}}$ extracted from Fig. 2a and of $E_{\text{th}}$ from Figs. 2b, 2c. (b) Observed temperature dependence of the applied power $J_{\text{th}}\times E_{\text{th}}$ at the threshold, taken from Figs. 2a, 2b, 2c. (c) Simulated $J_{\text{th}}$ taken from Fig. 2d, and $E_{\text{th}}$ from Figs. 2e, 2f. (d) Simulated threshold power taken from Figs. 2d, 2e, 2f. The triangles are based on larger $nC=1.0\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$.

Figure 4

(Color online) (a) Time profile in 4P measurement, $E$, $\sigma$, and $T_e$; and (b) $P$ term, $K$ term, and $C$ term in Eq. (3) at $J=100\text{A}{\text{cm}}^{-2}$ and $T=60\text{K}$. (c) $J$ dependence of equilibrium $T_e$ and the $P$ term at 70 K. The solid curves are for $\alpha =200\text{W}{\text{K}}^{-1}{\text{cm}}^{-3}$ and the dotted ones for $\alpha =50\text{W}{\text{K}}^{-1}{\text{cm}}^{-3}$.

Figure 5

(Color online) (a) $\rho \text{−}J$ plot of the 4P measurement. (b) $\alpha$ dependence of the peak height $E_{\text{th}}$ in Fig. 2d and $m$ in Eq. (1) at 70 K. The solid curves are for $nC=0.1\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$, and the dotted curves for $nC=1.0\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$.

Figure 6

(Color online) (a) Simulated time profile of a voltage pulse in 2P measurement with various electron heat capacities. The vertical axis was shifted appropriately. (b) Chaos appears at $nC=0.02\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$.

Figure 7

(Color online) Schematic representation of energy flow from the electron system to the lattice and to the external environment.

Figure 8

(Color online) (a) $\Delta T_e$ given by $T_e$ for 4P (red curve at $J=200\text{A}{\text{cm}}^{-2}$) and 2P (blue curve at $E=100\text{V}{\text{cm}}^{-1}$) measurements at 60 K and $nC=0.1\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$. (b) Return map, where $\Delta T_e+T_e$ is given by $T_e$ for 2P measurement with a load resistor ${\rho }_L=0.8\Omega \text{cm}$ at $T=60\text{K}$. Blue $\left(E_0=400\text{V}{\text{cm}}^{-1}\right)$ and green $\left(E_0=100\text{V}{\text{cm}}^{-1}\right)$ curves at $nC=0.1\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$, and red curve $\left(E_0=400\text{V}{\text{cm}}^{-1}\right)$ at $nC=0.025\text{J}{\text{K}}^{-1}{\text{cm}}^{-3}$.

Figure 9

Bifurcation diagram at $E_0=400\text{V}{\text{cm}}^{-1}$ and $T=60\text{K}$.