Figure 1

Schematic of the three-terminal geometry. A 1D system (blue region) consisting of a number of LSs is connected to three terminals: the left and right ones (green) are electronic leads which have their own temperatures and chemical potentials, the terminal below (brown) is a thermal terminal (a phonon bath in this work) at temperature $T_P$.Reuse & Permissions

Figure 2

1D NNH. (a) Schematic of the three-site model in the three-terminal geometry. The three red dots stand for the three LSs. The arrows represent the hopping transitions in the dominant hopping path. The vertical (horizontal) direction stands for the energy (position). The color densities in the left and right sides represent the electronic distribution in the left and right leads, respectively. In the figure, the left lead has higher chemical potential and temperature. (b) The effect of decreasing the conductances in the middle of the system by a factor $r_g$ on the conductance and thermopower of a 1D NNH system. $X=G$ or $S$ represents the conductance or the thermopower, $X^{\prime }/X$ denotes the ratio of the conductance or thermopower after modifying the middle part of the system to its original value. Solid (dashed) curves are for $X=S$ ($X=G$) for a three-site model. The red dots (blue triangles) denote $X=S$ ($X=G$) for a 1D NNH model with 31 LSs. Note that for both models $S^{\prime }/S=1$. The parameters for the 31 LSs 1D NNH model are $k_{B}T=$20, $\mu =0$, and $\xi =0.001$. The LSs are located at the sites of a 1D lattice with a periodicity of $0.008$. The energy is uniformly distributed in the energy window $(-E_c,E_c)$ with $E_c=60$. The Mott hopping length is $\sqrt{\xi /\rho k_{B}T}/2=0.004$. This value, half of the nearest-neighbor distance, makes all the hopping processes except the NN one sufficiently small. By “decreasing the conductances in the middle of the system” we mean decreasing the conductances of the connections among the 1st, 2nd, and 3rd LSs, while for the 31 LSs 1D NNH model it means decreasing the conductance of all bonds between the 11th and 21st LSs by a factor of $r_g$.Reuse & Permissions

Figure 3

Three-terminal hopping transport in a 2D NNH system. (a) Illustration of the situation when the two electronic terminals are geometrically sharp. Each terminal is strongly coupled only with a single LS with energy ${\varepsilon }_{\ell }$ and ${\varepsilon }_r$ on the left and right, respectively. A possible hopping path is illustrated in the figure by the arrows. (b) A 2D system consisting of a series of parallel hopping chains between the two electronic leads. The hopping between different chains is negligible.Reuse & Permissions

Figure 4

The effect of decreasing the conductances in the middle of the system by a factor of $r_g$ on the conductance and thermopower, for a 2D NNH single realization. $X=G$ or $S$ represents the conductance or the thermopower, and $X^{\prime }/X$ denotes the ratio of the conductance or thermopower after changing the middle to the original value. The curves with $▵$ are for the thermopower while the ones with dots are for the conductance. The parameters are $k_{B}T=20$, $\mu =0$, $\xi =0.01$, $E_c=20$, and $\rho =40$. The Mott hopping length is ${[\xi /\left(\rho k_{B}T\right)]}^{1/3}/2=0.012$. The LSs are put on a 2D lattice with periodicity $0.025$. The number of lattice sites perpendicular to the transport direction is $N_y=20$. For the longer system (a), the number of sites along the transport direction is $N_x=31$. For the shorter system (b), $N_x=15$. Each lattice site is denoted by two indices $(i_x,i_y)$ with $i_x\in [-(N_x-1)/2,(N_x-1)/2]$ and $i_y\in [1,N_y]$. The term “conductance in the middle” stands for the conductance between two LSs $(i_{x1},i_{y1})$ and $(i_{x2},i_{y2})$ where $i_{x1}\in [-5,5]$ and $i_{x2}\in [-5,5]$.Reuse & Permissions

Figure 5

1D VRH. (a) The boundary part of the hopping backbone, generalizing Fig. 2a. The gray region depicts the main part of the backbone. The connections (arrows) to the leads are through the LSs (red dots) near the boundaries. (b), (c) Length $(L-R_M)/R_M$ dependence of the average $〈\left|\ln \right(|X^{\prime }/X|\left)\right|〉$. In (b), the conductance $X=G$ (dashed curves) and the thermopower $X=S$ (solid curves). In (c), the three-terminal thermopower $X=S_p$. Results given for three different temperatures $k_{B}T=$15 (blue dots), 30 (red triangles), and 60 (green squares). $r_g=100$, $\mu =0$, $\xi =0.1$, and $\rho =0.03$, $k_B{T}_0=333$, $E_c=424$, 600, and 849 for the three temperatures, respectively. Along each curve from left to right, the number of LSs for the first, second$,...,$ fifth data point are $N=30$, 50, 70, 100, and 120, respectively. The Mott distances are $R_M=0.4,0.49,0.54,0.58,0.60(0.28,0.35,0.38,0.41,0.42)$ and the corresponding Mott energies are $E_M=120,147,161,174,180(170,208,227,246,254)$ for the five points at $k_{B}T=15\left(30\right),$ respectively. $R_M$ ($E_M$) at $k_{B}T=60$ is half (two times) of that at $k_{B}T=15$ for the five points, respectively. The average nearest-neighbor distances $2L/N$ for the three $T$'s are 0.039, 0.028, and 0.02, respectively. The results are averaged over ${10}^6$ random configurations. The curves in (b) and (c) are guides to the eye.Reuse & Permissions

Figure 6

The effect of decreasing the conductances in the middle part of a random 1D VRH system, by a factor of $r_g$, on the conductance and the thermopower. $X=G$ or $S$, with $X^{\prime }/X$ denoting the ratio of the conductance or thermopower after changing the middle to the original value. The curves with red triangles depict the thermopower while the ones with blue dots depict the conductance. The parameters are $k_{B}T=$15, $\mu =0$, $\xi =0.1$, $\rho =0.03$, $k_B{T}_0=333$, and $E_c=424$. The average distance between adjacent LSs is $0.039$. For the longer (a) system ($N=400$, $2L=22R_M$), the resistances in the region $(-2R_M,2R_M)$ are increased by a factor of $r_g$. For the shorter (b) system ($N=40$, $2L=3.5R_M$), the resistances in the region $(-R_M,R_M)$ are increased in the same way.Reuse & Permissions

Figure 7

Thermopower in 1D VRH. (a) The variance of the thermopower $\mathrm{Var}\left(eST\right)=\langle {(eST-\langle eST\rangle )}^2\rangle$ as a function of the length of the system $2L/R_M$ at $k_{B}T=$15 (blue dots), 30 (red triangles), and 60 (green squares). Along each curve from left to right, the number of LSs for the first, second$,...,$ seventh data points are $N=30$, 50, 70, 100, 120, 200, and 400, respectively. The other parameters are the same as in Fig. 5. (b) Probability distribution function (PDF) of the thermopower for 1D VRH systems, where $k_{B}T=15$, $\mu =0$, $\xi =0.1$, $\rho =0.03$, $E_c=424$, and $N=120$. The average distance is $2L/N=0.039$. The results are obtained from ${10}^6$ random configurations. The straight red line is an exponential fit to the tail $\sim \exp (-C|eS|T/E_0)$ with $C\simeq 5.5$. In the inset, we show how the thermopower evolves as a function of the system length for two random configurations. The parameters are $k_{B}T=30$, $\mu =0$, $\xi =0.1$, $\rho =0.03$, and $E_c=600$. The average distance between adjacent LSs is $0.028$. In (a), the curves are guides to the eye.Reuse & Permissions

Figure 8

(a) Schematic of the nonspanning hopping paths which may contribute significantly to the heat conductances. The wavy lines denote the phonons involved in the processes. (b), (c) Distance $(L-R_M)/R_M$ dependence of the average of (b) $\left|X\right|/\left(G{E}_c^2{e}^{-2}\right)$ with $X=K_e^0$ (blue dots), $X=L_3$ (red triangles), and $X=K_{pe}$ (green squares) for $k_{B}T=15$ (solid curves) and 30 (dashed curves), (c) $\left|\ln \right(|X^{\prime }/X|\left)\right|$ for $X=K_{pe}$ (solid curves), $X=K_e^0$ (dashed curves), and $X=L_3$ (dotted-dashed curves) for three different temperatures with $k_{B}T=$15 (blue dots), 30 (red triangles), and 60 (green squares). The other parameters are the same as those in Fig. 5. The results are obtained by averaging over ${10}^6$ random configurations. Note that in (c) the curves for $K_e^0$ are quite indistinguishable from those of $K_{pe}$ since $K_{pe}=4K_e^0$ when the nonspanning paths determine the heat conduction. The curves in (b) and (c) are guides to the eye.Reuse & Permissions