A Transition in Mechanisms of Size Dependent Electrical Transport at
Nanoscale Metal-Oxide Interfaces
Jiechang Hou, Stephen S. Nonnenmann, Wei Qin, and Dawn A. Bonnell*
Supplemental Material
1. Al-SrTiO3 contact
To verify that the Al/SrTiO3 back electrode junction is Ohmic a measurement was made on an
Al/SrTiO3/Al structure. The 0.02at% Nb-doped SrTiO3 single crystal substrate was also annealed
in air at 1000oC for 1 hour before electrode fabrication. On top of the smooth surface, Al was
deposited through a mask by thermal evaporation and electrodes of 9 sizes were fabricated, with
diameters ranging from 500 microns to 20 microns. Then a 100 nm Al thin film was deposited
on the back side of the substrate by thermal evaporation. The electronic properties were
measured on a commercially-available Probe Station (Signatone S1160) at room temperature.
The current-voltage (I-V) curves of the smallest two electrodes are shown in Figure S1. They
both show almost perfect linear Ohmic properties. The resistance of the 20 μm-patch is 490 Ω
and that of the 30 μm-patch is 310 Ω.
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Figure S1. Current-voltage characteristics of two Al/SrTiO3/Al structures verifying that the
back electrode is Ohmic.
2. Area calculation
The Au nanoparticle sizes were determined from topographic height images. The interface
orientation can be determined from the lateral 2D shape, as shown in Fig. S2. The equilibrium
shape of an unsupported Au nanoparticle is a truncated octahedron, with six (100) surfaces and
eight (111) surfaces. The area of the interfaces can be determined based on the geometry of the
truncated octahedron.
For the unsupported (100) interface, as shown in Fig. S2 (a), the
relationship between the particle height H and the edge length a is:
H  2 2a
(1)
Therefore, the square interface area can be calculated as:
A  a2 
H2
8
(2)
Similarly, for (111) interface, the edge length a is related to H as:
H  6a
(3)
2
And, accordingly, the hexagon interface area is:
A
3 3 2
3 2
a 
H
2
4
(4)
For nanoparticles supported on a substrate, the Au/substrate interface will affect the particle
shape according to energy considerations represented in the relevant Wulff Plot.1 In the absence
of reliable interfacial energy data, we consider here the maximum probable range given the
wetting properties of gold on oxide compounds. For a supported (100) interface, the interfacial
area can be approximately three times larger than the free surface. For a supported (111)
interface, the actual contact area can potentially vary approximately 1.2 times that of the
unsupported area. In the calculation of the Schottky barrier height, the difference in interfacial
area (30% uncertainty2) only has an almost negligible effect on the derived values (less than 5%),
included in the y error in Fig. 3 (a, b).
Figure S2. Schematics of a free-standing Au nanoparticle for (100) interfacial orientation (a) and
(111) interfacial orientation (b). The horizontal lines indicate the possible truncation.
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3. Fitting
As mentioned in the main text, the current response at very small voltages is not accessed due to
a combination of the small interface size and amplifier limitations. To verify that properties
determined from data in this voltage range, a higher conductivity substrate sample was
measured. Figure S3(a) shows the current density-voltage (J-V) curves of a 150 nm particle on
0.2 at% Nb-doped STO with higher (black) and lower (red) current limitation. The plateaus of
both curves result from the sensitivity of each amplifier. Excluding these plateaus, these two
curves superimpose on each other closely and show explicit continuity (guided by the blue
dashed line).
The fitting curves are omitted for neatness in Figure 2 (a, b) of the main text. Here, a fitting
result is shown as a representative. This is a (100)-oriented 100 nm nanoparticle on 0.02 at%
Nb-doped STO. With fitting values n= 3.3, ΦB=0.97 eV plugged back to Eqn. (1) in the main
text, a black dashed line shows a satisfactory match with the J-V curve. All other fitting results
have high coefficients of determination (R2 > 0.99).
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Figure S3. J-V curves obtained by higher (black) and lower (red) current limitation (a). The
continuity between lower and higher voltage regimes is shown. A fitting example of a 100 nm
particle, with the black dashed straight line as the fitting result (b).
4. Tunneling and thermionic current calculations
Figure S4 shows the transport mechanism ratio for interfaces with a range of relevant depletion
widths. The depletion widths used in the calculation are much smaller than the fully developed
macroscopic value. This is to account for the fact that the triangular barrier assumption in the
model overestimates the width that occurs in an exponentially decaying depletion width.
Further, for a macroscopic Au/STO contact, a barrier of 9 nm has been reported.3 Also, as the
electric field increases, the dielectric constant of STO will decrease, 4,5 resulting in a further
suppression of the depletion region that is not accounted for in the model. Finally, at high
voltages, the tunneling occurs through the top of the barrier where the width is smaller.
Therefore, we calculate the two contributions to the current over a range of widths to
demonstrate the effect, noting that the absolute values used in the calculations are 'apparent'
widths.
Thermionic emission was calculated by Eqn. (1) in the main text. Assuming a triangular
barrier, tunneling current was determined by the Fowler-Nordheim model6:
J tunneling 
 8 2m 3/2

q3 E 2
B
exp  

8 h B
3hqE


(5)
where E is the electric field across the tunneling barrier, calculated as E  V / W . V is the
applied bias and W is the depletion width. The depletion width can be calculated by the
traditional Schottky model, W  2 s 0Vbi / (qN D ) , with εs=310 from the vendor (Princeton
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Scientific). Vbi is the built-in potential of Au/STO Schottky barrier, with a value as 1 V. h is
planck constant. ΦB is Schottky barrier height. The current densities for these two mechanisms
and the ratio   J tunneling / J thermionic were determined. This model will underestimate the tunneling
contribution to the current because it relies on a triangular barrier rather than the exponential
decay in a depletion region and it does not account for field dependent dielectric constant
suppression in the depletion region. Measurements of depletion widths as small as 9 nm have
been reported on macroscopic interfaces.3
Therefore, results with a range of widths are
compared.
(The I-V curve measurement at different temperatures can distinguish different electron transport
mechanisms. But mostly, atomic force microscopy is difficult to use for variable-temperature
experiment.)
Figure S4. The effects of depletion width on the ratio of tunneling and thermionic emission at a
Au-STO interface.
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References:
1
N. Lopez, J. K. Nørskov, T. V. W. Janssens, A. Carlsson, A. Puig-Molina, B. S. Clausen, and
J. D. Grunwaldt, J. Catal. 225, 86 (2004).
2
H. Iddir, V. Komanicky, S. Öǧüt, H. You and P. Zapol, J. Phys. Chem. C 111, 14782 (2007).
3
H. Hasegawa, and T. Nishino, J. Appl. Phys. 69, 1501 (1991).
4
R. A. Berg, P. W. M. Blom, J. F. M. Cillessen, and R. M. Wolf, Appl. Phys. Lett. 66, 697
(1995).
5
D. A. Bonnell, and S. V. Kalinin, ZEITSCHRIFT FUR METALLKUNDE 94, 188 (2003).
6
M. Lenzlinger, and E. H. Snow, J. Appl. Phys. 40, 278 (1969).
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