Nanometrology: Absolute Seebeck coefficient of individual silver nanowires

Thermoelectric phenomena can be strongly modified in nanomaterials. The determination of the absolute Seebeck coefficient is a major challenge for metrology with respect to micro- and nanostructures due to the fact that the transport properties of the bulk material are no more valid. Here, we demonstrate a method to determine the absolute Seebeck coefficient S of individual metallic nanowires. For highly pure and single crystalline silver nanowires, we show the influence of nanopatterning on S in the temperature range between 16 K and 300 K. At room temperature, a nanowire diameter below 200 nm suppresses S by 50% compared to the bulk material to less than S = 1 μVK−1, which is attributed to the reduced electron mean free path. The temperature dependence of the absolute Seebeck coefficient depends on size effects. Thermodiffusion and phonon drag are reduced with respect to the bulk material and the ratio of electron-phonon to phonon-phonon interaction is significantly increased.

The formula (1) was applied on the temperature-dependent absolute Seebeck coefficient of bulk silver 1 . The fit parameters are given in table I and are discussed in the main text.
The uncertainty of the electrical conductivity σ of the silver nanowires primarily comes from the determination of the geometry parameters. The diameter d of the silver nanowires was measured by scanning (SEM) and transmission electron microscopy (TEM) at several points along each nanowire. The uncertainty of the diameter results from the resolution limitation of the SEM and TEM investigations and from the nanowire diameter variation and is between 5 nm and 20 nm. The length l of the silver nanowires was measured by scanning electron microscopy. The uncertainty of the length l mainly comes from the size of contact area that is defined by the electron beam-induced deposition contacts. The uncertainty of the length varies between 0.4 µm and 1.0 µm. The four-terminal resistance R was determined by linear fits of corresponding I-V curves. The relative uncertainty of R is less than 1 %. Overall, the average relative uncertainty of σ at room temperature is about 12.5 %.
The uncertainty of the electron mean free path Λ of the silver nanowire (NW 3) is mainly set by the uncertainty of the electrical conductivities coming from the silver nanowires and from bulk silver. The electrical conductivity of the silver nanowire was measured from room temperature down to T = 140 K and extrapolated from the electrical resistance by the Bloch-Grüneisen formula for T ≤ 130 K. The relative uncertainty of Λ is about 24 %.

Seebeck coefficient.
The relative Seebeck coefficient is determined by the thermovoltage U S and the temperature difference δT that is created by a micro heater. The temperature difference is increased stepwise by applying a heating current I H from zero to −I H,max and from zero to +I H,max in equidistant steps. At each step, the thermovoltage is measured ten times and then arithmetically averaged. The uncertainty of the thermovoltage is given by the confidence interval of the measurement results. The relative Seebeck coefficient is given by the mean of the three slopes of the fit lines to the U S (0... − I H,max ) versus δT (0... − I H,max ), U S (0... + I H,max ) versus δT (0... + I H,max ) and U S (0...−, +I H,max ) versus δT (0...−, +I H,max ) plots, respectively. The uncertainty of the relative Seebeck coefficient is determined by the modulus of the largest deviation of the mean value. The average temperature increase by the micro heater determines the uncertainty of the bath temperature T of the temperature-dependent Seebeck coefficient S(T ), which is typically less than 5 % of the bath temperature.
The absolute Seebeck coefficient of a calibrated platinum conduction line was used to determine the absolute Seebeck coefficient of the silver nanowires by the following equation, S Ag = S Ag,Pt + S Pt . (2) Only data points of S Ag,Pt and S Pt at equal bath temperatures (no interpolation) were used to calculate S Ag . The resulting uncertainty of the absolute Seebeck coefficient was determined by propagation of uncertainty.

Seebeck model fit.
Applying formula 1 on our measurement data yields the parameters F diff , F ph and F τ . A best, a maximum and a minimum fit line to the measurement data was used to determine the arithmetic mean of each parameter. The uncertainty was derived from the largest deviation of the mean value. The parameters of the silver nanowires and of the bulk material are given in table I.  We ensured that the platinum conduction lines were equivalent by using the same purity of the sputtering material, by the measure of the thickness, by the same heat treatment, by the same residual resistance ratio and by the same temperature coefficient of the resistance. For these reasons, the conduction lines that were used in the present work as Seebeck reference material for the silver nanowires and the platinum conduction lines that were used in the separate experiment 2 can be seen as identical from a thermoelectric point of view. Figure 2 shows the temperature coefficient of the resistance of the platinum conduction lines α Pt that were used for the relative Seebeck measurements of the silver nanowires (samples NW 1 -NW 3) as a function of the bath temperature T . Furthermore, the temperature coefficient of the reference platinum conduction line, whose absolute Seebeck coefficient was used to determine the absolute Seebeck coefficient of the silver nanowires, is given. The absolute Seebeck coefficient of this platinum conduction line was determined in a separate experiment 2 . These temperature coefficients are all in agreement with each other. In addition, the temperature coefficient of a platinum thin film with the same thickness but without additional heat treatment is added. The lack of the heat treatment leads to a temperature coefficient that is clearly reduced compared to the reference platinum conduction line. This in turn leads to a reduced absolute Seebeck coefficient compared to the thin film with heat treatment. A detailed discussion of the absolute Seebeck coefficient of thin platinum films and the effects of heat treatment on the transport properties is given in reference 2 .