Controlling the thermoelectric effect by mechanical manipulation of the electron’s quantum phase in atomic junctions

The thermoelectric voltage developed across an atomic metal junction (i.e., a nanostructure in which one or a few atoms connect two metal electrodes) in response to a temperature difference between the electrodes, results from the quantum interference of electrons that pass through the junction multiple times after being scattered by the surrounding defects. Here we report successfully tuning this quantum interference and thus controlling the magnitude and sign of the thermoelectric voltage by applying a mechanical force that deforms the junction. The observed switching of the thermoelectric voltage is reversible and can be cycled many times. Our ab initio and semi-empirical calculations elucidate the detailed mechanism by which the quantum interference is tuned. We show that the applied strain alters the quantum phases of electrons passing through the narrowest part of the junction and hence modifies the electronic quantum interference in the device. Tuning the quantum interference causes the energies of electronic transport resonances to shift, which affects the thermoelectric voltage. These experimental and theoretical studies reveal that Au atomic junctions can be made to exhibit both positive and negative thermoelectric voltages on demand, and demonstrate the importance and tunability of the quantum interference effect in the atomic-scale metal nanostructures.


S1. Conductance and thermoelectric voltage measurement
The measurements were performed using the mechanically controllable break junction (MCBJ) technique. Figure S1a shows the MCBJ sample measured in the present study.  Figure S1b shows the conductance change of the Au contact during stretching of the contact, and Fig. S1c shows the corresponding conductance histogram.
The last step at 1 G0 (2e 2 /h) in the conductance trace and the 1 G0 peak in the conductance histogram correspond to the formation of the Au atomic junction. There is a temperature difference between the junction and each side of the contact, where the thermometer is attached, caused by the heat flow through the glass tape, conducting wires attached to the 3 sample, and conducting wires attached to the thermometer, heat flow between the stem part of the electrode and atomic junction, and heat dissipation in the atomic junction 1,2 .
The estimation of these heat flows and heat dissipation is very difficult, and thus, it is hard to evaluate the actual temperature difference over the junction. A previously reported study determined the actual temperature difference over the contact to be a fraction, 0.3~0.5, of the measured temperature difference using the bulk thermopower of pure gold or scanning thermal microscopy [3][4][5] . The present experimental setup is the same as in this previously study. So, the actual temperature difference over the contact can be obtained by multiplying the experimentally obtained temperature difference by a factor of 0.3~0.5.
We calibrated the displacement by the following process. In the MCBJ, the setup itself acts as a reduction gear for the motion of the piezo element (x) with respect to the relative displacement of the two electrodes (y). For the ideal case of homogeneous strain in the bending beam, the displacement ratio (r) between y and x is given by 2 where t, u and l are the thickness of the bending beam, the distance between the epoxy droplets and the distance between the two counter supports, respectively. The displacement ratio was 2×10 -3 for the present setup, with l=20mm, t=1mm, u=0.1 mm.
We also calibrated the displacement ratio based on the length histogram of the clean Au atomic chain. A sequence of peaks was observed in the length histogram. The interval of the peaks was defined as 0.255 nm, which corresponds to the Au-Au distance of a clean Au atomic wire. On integration Eq. 1, the thermoelectric voltage is given by Where SJunction and Sbulk-Au are the thermopower of the Au atomic junction and bulk Au, respectively. Upon adding Eq. 2-4: From which the junction thermopower is obtained by The thermopower of bulk Au is known to be for the temperature regime of 20 K to 40 K 4,5 . The thermopower of the Au atomic junction is obtained in this study based on Eq (6) and (7).
where SJunction and Sbulk-Au are the thermopower of the Au atomic junction and bulk Au.
The Sbulk-Au is 0.7 μV/K around 25 K 4,5 . Within the small temperature difference regime (∆ <10 K), the thermopower is regarded as having a constant value; the thermoelectric voltage increases with the temperature difference. From the slope, the thermopower of the Au atomic junction is evaluated to be -0.7 μV/K, whose absolute value is much smaller than that of a single molecular junction and close to zero 6 . The standard deviation of thermoelectric voltage increases with the temperature difference (Fig. S3b).  Figure S4 shows examples of thermoelectric switching of the Au atomic junction. Figure   S4c and d present results of mono Au atomic junctions whose conductances are below 1

S5. Origin of the change in the thermoelectric voltage due to mechanical strain
The phase differences Δφ developed across the junction by the wave functions are expected to influence the electronic quantum interference patterns throughout both electrodes as well as in the junction and therefore to influence the energies at which the transmission resonances in Fig.3e and f occur. An appropriate acoustic analogy is that adding to the length of an organ pipe or guitar string lowers the resonant frequencies of the instrument.
By setting for real  and , we estimate the phase differences Δφ developed across the junction by the wave functions shown in the insets of Fig.3c and d to be 2.23π, and 2.36π respectively. Thus Δφ is higher while the energies of the resonances α and β are lower for the structure in Fig.3d than for the structure in Fig.3c.
We can account qualitatively for the above finding that the resonance energies are lower for higher Δφ if we note that gold, at the simplest level of modeling, can be regarded as a metal with a positive effective mass at the Fermi energy. For such a material the electron de Broglie wavelength is larger at lower energies. A resonant electronic state is characterized by the phase differences of its wave function between locations in the two electrodes. If, for a particular resonant state, the junction between the two electrodes changes in such a way that the electron phase difference Δφ across the junction increases, then for the system to remain on resonance, phase differences between the junction's