Supplementary Material for Observation of the Charged Defect
Migration that Causes the Degradation of Double-Schottky Barriers
Using a Nondestructive Quantitative Profiling Technique
Chenlu Cheng, Jinliang He, Jun Hu
The State Key Lab of Power System, Department of Electrical Engineering, Tsinghua University,
Beijing 100084, China
We provide additional information on prerequisite for PEA measurement, on theoretical analysis of
acoustic attenuation effect during PEA measurement, on additional simulation results based on
acoustic attenuation model and on the reasons for excluding the directly-bonded bicrystal.
We offer supplementary information here to facilitate the comprehension of the work presented in the
manuscript. The cross-section of fabricated bicrystal is shown in FIG. S1, in which the thickness of grain
boundary layer is uniformly smaller than 1 μm to well imitate the individual grain boundary of actual situation
inside ZnO electroceramics. Moreover, the double-Schottky barrier model based on positively charged donor
ions and negative interfacial states, i.e. Gupta’s instability modelS1, is also constructed.
FIG. S1. The cross-section of fabricated bicrystal (as discussed in the manuscript) observed by optical high
magnification microscope (FS-70, Mitutoyo Corporation, Kawasaki-shi, Kanagawa, Japan), based on which Gupta’s
instability modelS1 is presented. Nix (x=l for left side while x=r for right side) is the density of states at interface and Ndx
represents the donor concentrations of depletion layer in ZnO which all can be corresponded to the amplitudes of
waveforms in FIG. 2 from manuscript.
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I. Prerequisite for PEA measurement
Space charge measurement was first carried out on undoped ZnO single crystals with high resistivity
of size 20 mm (W) × 20 mm (L) × 3 mm (H) (orientation <0001>), fabricated by hydro-thermal method,
before testing on ZnO bicrystal. This was aimed at the validation of the effectiveness of PEA method
performing on ZnO material. Measurement result is presented in FIG. S2, in which the waves representing the
sheet charges induced at electrodes are included while the signal describing the ZnO single crystal bulk is flat
since no charge is injected into or accumulated inside the material.
FIG. S2. Unprocessed output waveform of measurement on ZnO single crystal. The “Amplitude” of derived signal
can be converted to “Charge Density” while the “Time” converted to “Position” by convolution techniques as exemplified
in FIG. 2 in manuscript. Besides, the noise can be further filtered.
Before performing test on bicrystal, we should give the assumption that negative interfacial states and
the uniformly distributed positive donor charges are indispensable to the formation of double-Schottky barrier
(DSB). For as-fabricated bicrystal in the manuscript, i.e. the two ZnO crystals are separated by a thin dopants
layer rather than directly bonded to each other, the theoretical distribution of space charge is shown as the
“theoretic” curve in FIG. 3. If the electric pulses of Dirac function are applied on the bicrystal, the space
charges will generate acoustic waves of exactly identical shape as the distribution form of space charge.
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However Dirac pulse source is contemporarily unavailable. By employing electric pulse of certain width (such
as 3 ns in the manuscript), the negative interfacial state of Dirac function form will generate acoustic wave
much like the waveform of the applied pulse (see FIG. 1(b) in the manuscript). Therefore the waveforms
containing the information of DSB before propagation in the bicrystal are given as “original” curve in FIG. 3.
In addition to the simulation method, using the acoustic velocity given in manuscript, we have also
predicted the position the waveforms of DSB may locate themselves at, and the experimental results coincide
with the predicted ones which proves the validity, i.e. the distance between the waveforms representing the
respective lower and upper electrodes is ~4 mm and the waveforms describing the DSB locate themselves at
the middle of the whole waveforms recorded. And we have successfully performed the PEA methods on tens
of samples. Besides, the primary importance is that the variation with time of the waveforms of the DSB
(presented in FIG.2 in the manuscript) is self-evident, proving its validity by itself.
II. Theoretical analysis of acoustic attenuation
Based on basic knowledge of Fourier Transformation, it is known that a square waves becoming more
and more Gaussian-likeS2 during propagation is due to its high-frequency components are more easily to
attenuate. Hence, based on formula (3) in manuscript, we present further theoretical analysis to gain an insight
into the acoustic attenuation effect.
We take Laplace transformation for both sides of formula (3) as
  2T
L  2
 t

L

 2  2T
 3T 
V


 a

2
x 2t 
 x
d 2T
d 2T
( j ) 2 T  Va2 2   j 2
dx
dx
(S1)
and arrive at
d 2T
2

T 0
dx 2 Va2  j
3
(S2)
where

is angular frequency and we define the factor
wave number k  k  j such that
of frequency

2
V  j
2
a
2
Va2  j
is equivalent to the square of complex
 k 2 . It is known that for a motion equation of plane waves
in lossy medium  T  k T  0 , its solution is T  T f e
2
2
j ( t  kx )
 Tf e
 x
e j (t kx ) where
we preferentially omit the backward component and only give the forward wave part. And the sound absorption
coefficient  gives the detailed description of attenuation effect. The absorption coefficient for formula (S2) is:
 
where A( ) 
Va4 A( ) 2   2 2 A( ) 2  Va2 A( )
2
(S3)
2
. Frequency dependence of  can be found in FIG. S3, which is discussed
Va4   2 2
in the frequency range we are interested in. Analysis on frequency dependence of  indicates
that high frequency components of wave propagating in the system described by formula (3) are
more apt to attenuate, proving its validity and effectiveness.
FIG. S3. The frequency dependence of acoustic absorption coefficient
III. Additional simulation results
Based on the acoustic attenuation model presented in the manuscript, experimental results in FIG. 2
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can be well simulated. Here, FIG. 2(b) is randomly selected and given its corresponding simulation results
(simulation for FIG. 2(a) and 2(c) should be straightforward). Prior to numerical simulation, the assumption for
simulation should be discussed, i.e. during degradation process only the density of space charge in depletion
layer or at interface decreases, and neither the distributive width nor distributive form of charged ion will vary.
For instance, it is generally accepted to assume charges distribute inside depletion layer of Schottky barrier as a
rectangular function; and it is further assumed here that during degradation only the value of rectangular
function decreases and the width of rectangular will be invariant. Calculated results, as seen in FIG. S4,
coincide well with the experimental ones, for instance the reverse-biased Schottky barrier decreases
significantly while the forward-biased one seems unchanged. Especially since the interfacial states of right side
decreases, the geometric center of negative part of simulated curves moves leftward gradually, in accordance
with the experimental results. This further validates the effectiveness of acoustic attenuation model presented in
the manuscript.
FIG. S4. Comparison between simulated PEA curves of DSB degradation considering acoustic attenuation effect
and experimental data presented in FIG. 2(b) in manuscript. Scatter points are drawn for simulation results while line
plots are for experimental values. Moreover, scatter points and line curve sharing identical color belong to the same group
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of aging time: “Green” color is for “0 &100 min”, “blue” for “400 min”, “purple” for “500 min”, “red” for “700 min” and
“yellow” for “800 min”.
IV. Reason for exclusion of directly bonded bicrystal
Based on the acoustic attenuation model in the manuscript, the full picture of wave propagation is
obtained. For instance, in lossless medium, the acoustic wave could retain its initial waveform irrespective of
the long distance it traverses; while propagating in very lossy medium, an initially square wave becomes more
and more Gaussian-like as it propagates away from its point of origin.
Here we further explain why we choose to fabricate bicrystal with an intergranular layer rather than
the one in which two crystals are directly bonded. The case of no intergranular layer is shown in FIG. S5. The
“original” curve in FIG. S5 is the before-propagation waveform of acoustic waves generated by the idealized
space charge distributions as “theoretic” curve under electrical pulse. During propagation, waveforms become
broader due to the effect of acoustic attenuation and the wave representing the negative interfacial states is
easily canceled by the waves describing the depletion layers. Thus the final recorded waveforms may be
incomprehensible, such as the “final” curve. Actually, praseodymium-doped bicrystal where two single crystals
bonded atomically is also prepared following Sato’s proceduresS3,S4 (not shown here), on which the PEA
measurement is performed with its results in accordance with the previous predictions, i.e. the waveform
representing the negative interfacial states disappears. This turns out to be a relief since the fabrication of
bismuth-doped bicrystal of excellent nonlinearity without intergranular layer remains extremely challenging.
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FIG. S5. The case considering the bicrystal without an intergranular layer (Multimedia view)
S1
T. K. Gupta, and W. G. Carlson, J. Mater. Sci. 20, 3487 (1985).
J. B. Bernstein, Ph. D. thesis, Massachusetts Institute of Technology, 1990.
S3
Y. Sato, J. P. Buban, T. Mizoguchi, N. Shibata, M. Yodogawa, T. Yamamoto, and Y. Ikuhara, Phys. Rev.
Lett. 97, 106802 (2006).
S4
Y. Sato, T. Yamamoto, and Y. Ikuhara, J. Am. Ceram. Soc. 90, 337 (2007).
S2
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