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Supplemental material
Current-induced electrical self-oscillations across out-of-plane
threshold switches based on VO2 layers integrated in crossbars
geometry
A. Beaumont1, J. Leroy1, J.- C. Orlianges2 and A. Crunteanu1,a)
1
XLIM Research Institute UMR 7252, CNRS/ University of Limoges, 123 avenue Albert
Thomas, 87060 Limoges, FRANCE
2
SPCTS UMR 7513, CNRS/ University of Limoges, 12 rue Atlantis, 87068 Limoges,
FRANCE
1. STUDY OF THE DISPERSION OF ELECTRICAL PARAMETERS
The study of the variability of the activation bias is described in the paper but the
dispersion of other parameters was also assessed and a brief description is presented
hereafter. Low bias resistivity was determined for every device of the batch by performing a
linear regression on the experimental data measured for biases inferior to 0.1 V (Figure S1).
It may be observed that both the median low bias resistivity and the dispersion of low bias
resistivity are decreased when the size of capacitors is diminished.
The high bias resistivity was also determined for every device of the batch with a linear
regression on the experimental data measured for biases superior to the transition bias
(Figure S2). Similarly as in the case of low bias resistivity, both the median high bias
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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resistivity and the dispersion of high bias resistivity are decreased when the size of capacitors
is diminished.
FIG. S1. : Histograms representing the dispersion of the resistivity measured before MIT at
low bias (0.1V) of (a) large (9x9 µm2), (b) medium (5x5 µm2) and (c) small (3x3 µm2) VO2
capacitors.
FIG. S2.: Histograms representing the dispersion of the resistivity measured after MIT at
high bias (the bias used depended on the activation bias but was chosen to ensure that the
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VO2 was in conductive state) of (a) large (9x9 µm2), (b) medium (5x5 µm2) and (c) small
(3x3 µm2) VO2 capacitors.
2. MODELLING
A. Choice of the model type
In this work, an analytical modelling of the self-oscillations in the circuit presented in the
Figure 3a of the paper. Several physical models have been published and we attempted to use
several of them to simulate our devices. However they all failed to reproduce the
experimental data presented in this work, certainly because all the pertinent physical
phenomena are not taken into account. An example of how these models fail to reproduce our
data is given in Figure S3 where the dotted blue line shows the results obtained with the
model of Pickett et al.S1, a recent and advanced model based on the thermal triggering of MIT
in oxide materials.
FIG. S3.: DC experimental I-V characteristics of the device L11 described in the paper
(black circles) and bias simulated with the Pickett et al.’s model [S1] (blue dashed line) as a
function of the current injected in the MIM device.
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This model assumes that when small currents are injected in a metal-oxide-metal structure, a
conductive filament inside the oxide is created. When the current is increased, the
surrounding oxide material is transformed to a metallic phase, thus gradually increasing the
diameter of the filament. Since the part of metallic oxide is growing, the out-of-plane
resistivity of the layer between the metallic electrodes becomes lower, as observed in the
experimental data. For the simulation of our VO2-based devices, the following parameters
were used: a resistivity in the metallic phase of 10-2 Ω.m, a resistivity in the insulating phase
of 0.255 Ω.m, a channel diameter of 6.06 µm, a VO2 thickness of 130 nm, a thermal
conductivity of 7.2 W.m-1.K-1, an external temperature of 298 K, a transition temperature of
341 K and a total series resistance of 130 Ω. The thermodynamic parameters were found not
relevant given the static measurements that were to simulate. As shown on Figure S3,
although both the low bias insulating state and the high bias metallic state are well simulated
by this thermalmodel, the part of the insulating state I-V curve for V>0.5 V and the negative
differential region are not fitted and this situation persists independent of the parameters used
to carry out the simulations.
B. Derivation of the oscillation frequency
As described in the paper, a simple model defining three points (named as A, B and C as
shown in Figure 2a of the paper) is used to model the static I-V characteristic when the MOM
device is biased by a current sweep. The main assumption of the model is that the I-V
characteristic is purely linear between each of the three points. The complete derivation
leading to the final equations is presented hereafter. When considering the equivalent circuit
presented in the Figure 3a of the paper, one may write the value of the total current I0 as a
function of the other currents:
I 0  iC  iDUT ,
(S1)
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where iC is the current through the external capacitor and iDUT is the current flowing through
the MIM device. Outside of the NDR part of the I-V characteristic, the current iDUT may be
expressed as a function of the bias vDUT across the MIM device:
iDUT  v DUT   ,
(S2)
where σ is the conductance of VO2 and is worth σi or σm depending on whether the VO2 is in
insulating or metallic phase and β is the value of the current when the voltage is null (βi=0 in
insulating phase but βm≠0 in metallic phase). Similarly, the current ic depends on the
capacitance of the external capacitor (CEXT) and on the bias vC across it:
iC  C EXT
dv C
.
dt
(S3)
In order to take the parasitic inductance of cables into account, the bias vL which develops
across the equivalent inductor L is calculated. It depends on the current iL flowing through it
according to the following relation:
vL  L
di L
.
dt
(S4)
Then one may derive a differential equation by noting that vC = vDUT + vL and iL=iDUT:
d 2v C C EXT dv C
I 
.
LC EXT

vC  0
2
 dt

dt
(S5)
Equation S5 is a second order differential equation with a constant right hand member
therefore its characteristic equation has to be written as follows:
LC EXT 2 
C EXT
 1 0,

(S6)
where λ is an arbitrary variable. The discriminant of this equation relation is
2
C

   EXT   4LC EXT
  
(S7)
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and its sign has to be studied. It is positive when L, CEXT and σ fulfil the following
conditions:
C EXT
 4L .
2
(S8)
This condition is expected to correspond to the charging of CEXT because usually, σi is low
when the VO2 is in insulating phase and the solution of Equation S5 is then:
  1

 
I0   
1 1
 1 
V

exp





i 
 0
t 
  2 L
2  2 i L i
 i 


i
 

  1

 
I0   
1  1
 
 1 
V

exp





t  ,
0
i

  2 L
2  2 i L i
 i 


i
 

I 
 0
v C (t )  
(S9)
i
where V0 is the initial voltage for t=0 and δi is a constant homogeneous to a frequency
defined by
2
i 

2LC EXT
 C EXT 

  4LC EXT
 i 

.
2LC EXT
(S10)
To get an analytical expression for the duration of a charge or a discharge of the external
capacitor, one may approximate Equation S9. Since the two terms multiplied by the time in
the exponential are commonly of the same order of magnitude, it is possible to write that
  1
 
exp  
  i t 
 2 L

 
 i
  1
 
exp  
  i t  .
 2 L

 
 i
(S11)
This enables one to rewrite Equation S9 as:
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
I 
1 1
 1  V 0  0
2  2 i L i


v C (t )  
  1


  i t
 exp  


  2 i L
 I0  
.
 
i

(S12)
Then it is straightforward to derive the time needed to charge the external capacitor so that
the voltage across it varies from V1 to V2:
tV0V
1
2

  iV 1

 1 


 I 
1
1 1
.

log  
 1   0

2  2 i L i
 1

   iV 2  1  






i 

 2 i L

 I0  


(S13)
The sign of the discriminant in Equation S7 may also be negative corresponding to an
opposite condition to Equation S8 on L, CEXT and σ. This usually corresponds to the discharge
of the capacitor CEXT because for the VO2 in metallic state, σm is high. Then the solution of
Equation S5 is:

v C (t )  V 0 


 t  I 0  
I 0   
1
,
cos

t

sin

t
exp







 
m
m

 m 
2

L

2

L


m

 m 
m
(S14)
where V0 is the initial voltage for t=0 and δm is defined by
m 

2LC EXT
4LC EXT

C

  EXT 
 m 
2LC EXT
2
.
(S15)
To get an analytical expression for the duration of a charge or a discharge of the external
capacitor, one may approximate Equation S14 by pointing out that the term containing the
sine is negligible beside the one containing the cosine and that the exponential term may be
approximated to 1 in practical circuits. Then Equation S14 may be approximated as follows:
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
v C (t )  V 0 

I0   
I0  
.
 cos  mt  
m 
m
(S16)
Here again it is possible to derive the time needed to charge the external capacitor so that the
voltage across it varies from V1 to V2:
tV0V
1
2
  mV 2

1 

I 
1
.

arccos  0
  mV1

m
 1 

 I0  

(S17)
It is finally possible to derive the duration of a full oscillation period consisting of a discharge
from point A to point B and a charge from point B to point A. The expression of the
frequency of the oscillations is:
f

t
0
VB V A
1
 tV0V
A
. (S18)
B


  iV B

  mV B



1



1




  I 0  i
I


1
1

 1 1

 
m


log  
 1 
arccos  0

  mV A

2  2 i L i


   iV A  1    m
  1  


1

 


i

  2 i L
I


I


m
 0
 

i
 0



1
It is worth noting that this relation has to be adapted as a function of the parameters of the
experimental set-up and the devices, concerning the sign of the discriminant given in
Equation S7 in particular.
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FIG. S4. Simulation of the voltage across the MIM device as a function of time for the device
L11 with a current bias of 0.8 mA. The simulations made with the numerical simulator (black
line) are compared with the approximate analytical expression for the charge (Equation S12,
red line) and for the discharge (Equation S16, blue line).
C. Comparison with a numerical simulator
In order to confirm the values of frequency oscillations obtained with Equation S18, a
numerical simulator resolving the basic equation of circuits (Kirchhoff laws) as a function of
the time has been scripted with Python programming languageS2 and the numpy libraryS3. For
each time step, the simulator uses a dichotomy algorithm to solve the system of equations
defined by Equations S1, S2, S3 and S4 if an inductor is inserted in series with the MOM
device. The capability of the expressions derived here above was assessed by comparing
them to the results of simulations performed with the numerical simulator. The Figure S4
shows an example of the good agreement between the simulations and the approximate
expressions of the bias as a function of the time.
REFERENCES
S1
M.D. Pickett and R.S. Williams, Nanotechnology, 23 215202 (2012).
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S2
G. Van Rossum, The python language reference,
http://docs.python.org/release/2.7.6/reference/.
S3
P.F. Dubois, K. Hinsen and J. Hugunin, Computers in Physics, 10, 262 (1996).
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