Thermoelectric properties of atomically thin silicene and germanene nanostructures

The thermoelectric properties in one- and two-dimensional silicon and germanium structures have been investigated using first-principles density functional techniques and linear response for the thermal and electrical transport. We have considered here the two-dimensional silicene and germanene, together with nanoribbons of different widths. For the nano ribbons, we have also investigated the possibility of nano structuring these systems by mixing silicon and germanium. We found that the figure of merit at room temperature of these systems is remarkably high, up to 2.5.

Figure 1

Geometrical structures of one-layer $3\times 3$ (a) silicene and (d) germanene on top of five-layer $4\times 4$ Ag(111). (b), (e) Distorted silicene and germanene obtained by removing the silver substrate from (a) and (d), respectively. (c), (f) Electronic energy bands corresponding to the distorted silicene and germanene grown on Ag(111), respectively, where the dotted line denotes the Fermi energy.

Figure 2

(a), (b) Dimensionless electronic figure of merit $Z{T}_e$ at room temperature as a function of chemical potential $\mu$ corresponding to the unsupported distorted silicene and germanene, respectively, evaluated in the constant relaxation time approximation (black continuous line) and for $1/{\tau }_e\propto E$ (red dashed line).

Figure 3

Electron energy bands of freestanding (a) silicene and (b) germanene, respectively, where the dotted line denotes the Fermi energy.

Figure 4

(a), (b) Dimensionless electronic figure of merit $Z{T}_e$ at room temperature as a function of chemical potential $\mu$ for the freestanding silicene and germanene, respectively, evaluated in the constant relaxation time approximation (black continuous line) and for $1/{\tau }_e\propto E$ (red dashed line).

Figure 5

(a), (b) Optimized geometrical structures of $Z$-GeNRs and $A$-GeNRs and their lateral views. For the atoms at the edges, we passivate the unsaturated bonds with hydrogen atoms. $W_Z$ and $W_A$ denote the width of the nanoribbons for the zigzag- and armchair-terminated nanoribbons, respectively. (c) Electron energy band for $Z$-GeNRs with $W_Z=6$ for the AFM state. Notice the presence of a small electronic gap. (d) Electron energy band for $Z$-GeNRs with $W_Z=6$ for the FM state. (e) Electron energy band of $A$-GeNRs with $W_A=6$ corresponding to the NM state. In (c)–(e) the Fermi energy is chosen as the reference energy and set to 0. (f), (g) Phonon energy dispersions for $Z$-GeNRs with $W_Z=6$ and $A$-GeNRs with $W_A=6$, respectively.

Figure 6

(a), (b) Electron transmission coefficient as a function of energy for $Z$-GeNRs and $A$-GeNRs with various ribbon width, respectively. (c), (d) Band gap of $Z$-GeNRs and $A$-GeNRs as a function of the ribbon widths $W_Z$ and $W_A$, respectively. (e), (f) Electrical conductance. (g), (h) Seebeck coefficient. (i), (j) Electron and phonon thermal conductances for $Z$-GeNRs and $A$-GeNRs versus chemical potential $\mu$, where the temperature is set at 300 K.

Figure 7

Phonon thermal conductance ${\kappa }_p$ of (a) $Z$-GeNRs and (b) $A$-GeNRs with different ribbon widths as a function of temperature. (c) The logarithm of ${\kappa }_p$ for $A$-GeNRs as a function of the logarithm of $T$, where the linear behavior is shown as we expect according to Eq. (10). Each curve corresponding to a specified ribbon width has the same meaning as in Fig. 6.

Figure 8

Figure of merit $ZT$ at room temperature for (a) $Z$-GeNRs and (b) $A$-GeNRs as a function of ribbon widths $W_Z$ and $W_A$, respectively. In black (square hollow points) we report the peak value of $ZT$ at negative values of the chemical potential $\mu$ associated with the hole transport, and in red (square full points) the peak value of $ZT$ associated with the electron transport (positive $\mu$).

Figure 9

Phonon thermal conductance ${\kappa }_p$ of (a) $Z$-SiNRs and (b) $A$-SiNRs as a function of temperature, respectively, where each curve corresponds to a specified ribbon width as shown in Fig. 6.

Figure 10

Figure of merit $ZT$ at room temperature for (a) $Z$-SiNRs and (b) $A$-SiNRs as a function of ribbon widths $W_Z$ and $W_A$, where the black square hollow points and the red square full points correspond to the hole and electron transport, respectively.

Figure 11

Geometrical structures of (a) $Z$-SiGeNRs and (b) $A$-SiGeNRs, where the line encloses a supercell along the ribbon axis and $L_{\text{Si}}$ and $L_{\text{Ge}}$ are the lengths of silicene and germanene stripes in the supercell, respectively. Here we have chosen $L_{\text{Si}}=L_{\text{Ge}}=3$ and used the hydrogen to passivate the ribbon edges.

Figure 12

(a), (b) Electron transmission coefficient as a function of energy for $Z$-SiGeNRs and $A$-SiGeNRs, respectively. (c), (d) Band gap as a function of ribbon widths $W_Z$ and $W_A$, respectively. (e), (f) Electrical conductance. (g), (h) Seebeck coefficient. (i), (j) Electron and phonon thermal conductances as a function of chemical potential $\mu$ for $Z$-SiGeNRs and $A$-SiGeNRs, respectively, where we have set the temperature $T=300$ K. Each curve in (a), (b), and (e)–(j) corresponding to a specified ribbon width endows the same meaning as in Fig. 6.

Figure 13

Figure of merit $ZT$ at room temperature for (a) $Z$-SiGeNRs and (b) $A$-SiGeNRs as a function of nanoribbons with widths $W_Z$ and $W_A$, where the hollow and full points correspond to the hole and electron transport, respectively.

Figure 14

Figure of merit $ZT$ at $T=300$ K for $Z$-SiGeNRs and $A$-SiGeNRs as a function of ribbon widths $W_Z$ and $W_A$ under different component lengths of silicene and germanene stripes: [(a), (b)] $L_{\text{Si}}=L_{\text{Ge}}=2$, [(c), (d)] $L_{\text{Si}}=L_{\text{Ge}}=3$, and [(e), (f)] $L_{\text{Si}}=L_{\text{Ge}}=4$, respectively. (g), (h) Figure of merit as a function of temperature for $Z$-SiGeNRs and $A$-SiGeNRs with the corresponding ribbon widths 3 and 4, where the lengths of the silicene and germanene stripes in the supercell are $L_{\text{Si}}=L_{\text{Ge}}=3$. The hollow and full points correspond to the hole and electron transport, respectively

Figure 15

Figure of merit $ZT$ at $T=300$ K for $A$-GeNRs, $A$-SiNRs, $A$-SiGeNRs, and disordered $A$-SiGeNRs with ribbon widths (a) $W_A=3$ and (b) $W_A=4$ as a function of chemical potential $\mu$, where we have taken the supercell length $L_S=L_{\text{Ge}}+L_{\text{Si}}=6$, respectively.

Figure 16

Phonon thermal conductance of various armchair nanoribbons with the ribbon width $W_A=3$, where the supercell length $L_S$ for $A$-SiGeNRs and disordered $A$-SiGeNRs is (a) $L_S=6$ and (b) $L_S=20$, respectively. The curves in color are calculated from $ab$ initio and the other curves in gray are calculated from tight binding. The different curves share the same meaning as those in Fig. 15.

Figure 17

Phonon thermal conductance ${\kappa }_p$ at $T=300$ K calculated from tight binding as a function of supercell length $L_S$ corresponding to the ribbon widths (a) $W_A=3$ and (b) $W_A=4$, respectively.