Temperature-dependent thermoelectric properties of individual silver nanowires

Individual highly pure single-crystalline silver nanowires (Ag NWs) were investigated with regard to the electrical conductivity $\sigma$, the thermal conductivity $\lambda$, and the Seebeck coefficient $S$ as a function of the temperature $T$ between $1.4\mathrm{K}$ and room temperature (RT). Transmission electron microscopy was performed subsequently to the thermoelectric characterization of the Ag NWs, so that their transport properties can be correlated with the structural data. The crystal structure, surface morphology and the rare occurrence of kinks and twinning were identified. The thermoelectric properties of the Ag NWs are discussed in comparison to the bulk: $S_{\mathrm{Ag},\mathrm{Pt}}\left(T\right)$ was measured with respect to platinum and is in agreement with the bulk, while $\sigma \left(T\right)$ and $\lambda \left(T\right)$ showed reduced values with respect to the bulk. The latter are both notably dominated by surface scattering caused by an increased surface-to-volume ratio. By lowering $T$, the electron mean free path strongly exceeds the NW's diameter of 150 nm so that the transition from diffusive transport to quasiballistic one-dimensional transport is observed. An important result of this work is that the Lorenz number $L\left(T\right)$ turns out to be independent of surface scattering. Instead, the characteristic of $L\left(T\right)$ is determined by the material's purity. Moreover, $\sigma \left(T\right)$ and $L\left(T\right)$ can be described by the bulk Debye temperature of silver. A detailed discussion of the temperature dependence of $L\left(T\right)$ and the scattering mechanisms is given.

Figure 1

(a) Sketch of the thermoelectric nanowire characterization platform (TNCP) cantilever tips (black: Pt lines, bright gray: ${\mathrm{SiO}}_2$, dark gray: NW, white: vacuum). (b) SEM image of a tilted TNCP with an EBID contacted Ag NW (Ag NW3).

Figure 2

(a) HAADF STEM micrograph showing Ag NWs with the rare case of kinks. The inset shows the electron diffraction pattern obtained from a bunch of NWs (see the region marked by a dashed line in the STEM image). (b) HAADF STEM image showing a residual diffraction contrast due to the presence of a twin boundary along the growth direction of an Ag NW.

Figure 3

(a) TEM micrograph showing an amorphous shell on the NW surface. The interface between the shell and the vacuum is indicated by a dotted line. (b) EDX spectrum indicating the presence of C in the surface layer. A HAADF STEM image in the inset shows the NW surface with indicated beam position during the measurement.

Figure 4

(a) HAADF STEM image of the NW surface showing the presence of round-shaped particles. (b) EDX line scan across the surface (see the arrow in the HAADF STEM inset) proving the presence of Ag and S in the surface particles.

Figure 5

(a) HAADF STEM micrograph of a top view of Ag NW2 on the TNCP. The inset shows the corresponding TEM image. (b) HAADF STEM image of a segment of Ag NW2. (c) TEM image of a kink at the NW2. The insets show selected area electron diffraction (SAED) patterns of the corresponding NW segments.

Figure 6

Four point measured $I-V$ curves of an individual Ag NW (Ag NW4) at different bath temperatures. The inset shows the measured resistances for the Ag NWs with diameters ranging from 107 to 150 nm.

Figure 7

Measured resistance of Ag NW4 fitted by Eqs. (1) and (2). Best fit to the measurement data is given by Eq. (2) with ${\Theta }_{\mathrm{D}}=215\mathrm{K},R_0\to 1.62\Omega ,T_{\mathrm{SS}}\to 49.4\mathrm{K},M\to 2.1\Omega$, and $R_{\mathrm{e}}-\mathrm{ph}\to 48\Omega$ shown by the green solid line. The dashed red line shows the fit of Eq. (1) for fixed ${\Theta }_{\mathrm{D}}=215\mathrm{K}$ with $R_{\mathrm{e}}-\mathrm{ph}=48\Omega$ and $R_0=3.7\Omega$. The dotted blue line shows the fit for Eq. (1) with ${\Theta }_{\mathrm{D}}=86\mathrm{K}$ and $R_0\to 1.62\Omega$. Inset (a) shows the measured resistances and fits for lower temperatures. Inset (b) shows the TCR $\alpha$ of NW4 as a function of temperature.

Figure 8

Electrical conductivity ${\sigma }_{\mathrm{NW}}\left(T\right)$ of individual Ag NWs (NW1: ; NW2: $•$; NW3: ; NW4: ) as function of the temperature $T$. Fitted curves of Eq. (3) are depicted as solid line. Reference values of Ag bulk are denoted by ($\circ$) [31, 34]. (a) Electrical conductivity of NW4 compared to the bulk reference ($\circ$) in the measured temperature region. (b) Fraction $f_r$ of NWs of this study () compared to literature values on NW ensembles () [29].

Figure 9

Thermovoltage as a function of the temperature difference $\delta T$ between the cantilevers and (inset) thermovoltage as a function of the heater current at RT.

Figure 10

Measured Seebeck coefficients $S_{\mathrm{Ag},\mathrm{Pt}}$ of Ag NW1 () as a function of $T$ in comparison to bulk values ($◯$) [37, 38]. The solid line shows bulk values from Ref. [39]. The shaded area indicates the absolute error of $S_{\mathrm{Ag},\mathrm{Pt}}$ by comparison with Refs. [37, 38, 39, 40, 41].

Figure 11

Measured $U_{3\omega }$ of Ag NW4 as a function of $U_{1\omega }^3$ for (a) the temperature range from 7 to $48\mathrm{K}$ and (b) the temperature range from 75 to $213\mathrm{K}$.

Figure 12

Measured thermal conductivity ${\lambda }_{\exp }$ () of Ag NW4 and the calculated thermal conductivity ${\lambda }_{\mathrm{e}}$ () from a constant bulk Lorenz number $L_{\mathrm{Ag}}$. Corrected thermal conductivity ${\lambda }_{\mathrm{c}}$ () for a thermal contact resistance arising from the condition ${\lambda }_c\left(\mathrm{RT}\right)={\lambda }_e\left(\mathrm{RT}\right)=L_{\mathrm{Ag}}\sigma \left(\mathrm{RT}\right)\mathrm{RT}$. The dashed arrow marks the return of ${\lambda }_{\exp }\to {\lambda }_{\mathrm{e}}$ for $T<2\mathrm{K}$, whereas the dotted arrow and () show the reverse characteristic of $\lambda$ due to Joule heating of the NW. The inset shows $U_{3\omega }\left(U_{1\omega }^3\right)$ at $T=1.4\mathrm{K}$. For a mean temperature increase $\Delta T>1.65\mathrm{K}$ additional values for $\lambda$ are evaluated which are shown by ().

Figure 13

Measured total thermal resistance $R_{\mathrm{th}},\exp$ of Ag NW4 compared to the thermal resistance of the EBID contacts $R_{\mathrm{th}},\mathrm{EBID}$ [47].

Figure 14

Measured Lorenz number $L$ as function of normalized temperature () and Lorenz numbers corrected for thermal contact resistance () for Ag NW4. Bulk reference values ($◯$) are shown [34]. Low-temperature values of Ag bulk samples with different $\text{RRR}$ () [49]. The dark gray solid line shows the Lorenz number of a defect free monovalent metal, whereas bright gray lines represent $L$ for an imperfect metal with different defect concentrations [30]. The inset shows noncorrected values that compare with the theory of high-purity material. Here () are determined for the Joule heated NW (compare Fig. 12).

Figure 15

The determined electron mean free path (EMFP: ) and the thermal mean free path (TMFP: ,) of Ag NW4 as a function of normalized temperature $T/{\Theta }_{\mathrm{D}},\mathrm{Ag}$. The bulk EMFP ${\Lambda }_{\mathrm{el}},\mathrm{B}$ is calculated by Eq. (4) and depicted as ($\circ$). The solid line shows the calculated EMFP from ${\Lambda }_{\mathrm{el}},\mathrm{B}\left(T\right){\alpha }_{\mathrm{NW}}\left(T\right)/{\alpha }_{\mathrm{B}}\left(T\right)$ which compares with ${\Lambda }_{\mathrm{el}},\mathrm{NW}$ for $T>T_{\mathrm{I}}-\mathrm{II}$ [34, 35]. For $T<T_{\mathrm{I}}-\mathrm{II}$, the determined EMFP along the direction of transport ${\Lambda }_{\mathrm{el}},\mathrm{d}$ is shown by (). The three regions I (diffusive transport), II (quasiballistic transport), and III (dominating impurity scattering) are marked. In the inset, the low-temperature region is shown in logarithmic scale.