An accurate locally-active memristor model for S-type NDR in NbOx
Gary A. Gibson,1 Srinitya Musunuru,1 Jiaming Zhang,1 Ken Vandenberghe,2
James Lee,1 Cheng-Chih Hsieh,1 Warren Jackson,1 Yoocharn Jeon,1
Dick Henze,1 Zhiyong Li,1 and R. Stanley Williams1
1Hewlett-Packard
2PTD-PPS,
Laboratories, 1501 Page Mill Road, Palo Alto, California 94304
Hewlett-Packard Company, 1070 NE Circle Boulevard, Corvallis, Oregon 97330
Supplementary Material
Electroforming Process
3
1.0E-04
1.0E-02
2 (Reduction)
Type I
Type II
4
Current (A)
1.0E-05
1
Log Current
Current (A)
1.0E-03
1.0E-06
(a)
0
Tetragonal NbO2
d(440)=0.34 nm
1.0E-05
1.0E-06
0
0.5
1
Bias (V)
Time
1.5
1
2
1.0E-04
(b)
0
1.0E-07
3
1.0E-07
0
0.4
0.8
Bias (V)
Fig. S1. Sequence of V-I curves recorded during formation process that consists of logarithmic current sweeps of
increasing amplitude. Numbers indicate order of sweeps; arrows indicate time evolution. For clarity only a subset
of curves are shown. (a) Example of Type I forming, which results in increasing currents as the initially amorphous
Nb2O5 is reduced through interaction with the TiN electrodes. (b) Example of Type II forming, which includes
crystallization to a more resistive tetragonal NbO2 state after the initial reduction. The slope of curve 5 in (b) is
positive at high currents due to a ~100 ohm resistance in series with the selector.
Figure S1 displays I-V characteristics that illustrate typical sequences of forming steps for both
Type I and Type II forming. Cross-sectional TEM characterization of similar samples reveals that
Type I forming consists of reduction of the initially amorphous NbOx by the neighboring TiN
electrodes, resulting in higher currents. Type II forming also begins with reduction of the NbOx
but is followed by crystallization to a more resistive tetragonal NbO2 state. The data in Fig. S1a
comes from a 32 nm diameter device whereas the device of Fig. S1b is 195 nm in diameter. The
deposited NbOx thickness is 15 nm in both cases, which is an intermediate thickness at which both
Type I and Type II forming have been observed. The somewhat larger forming voltage required
for the device of Fig. S1a is due primarily to its smaller device area and the ensuing lower currents.
These lower currents result in a significantly lower dissipated power at a given applied bias. This
lower power is partially, but not fully, compensated by a higher thermal isolation value R th, as
indicated in Fig. 3. Because the dissipated power decreases by a larger factor than the increase in
1
Rth, the internal temperature is somewhat lower for the Fig. S1a device than the Fig. S1b device at
the same applied bias. Hence, in the first case the higher formation voltage does not result in
crystallization because the internal temperature is not quite as high.
Only devices with NbOx thicknesses thicker than ~ 8 nm exhibit Type II forming. Low-bias
temperature-dependent V-I characteristics for the virgin state of these thicker devices are well fit
by Eqs. 1 and 2 up to the initiation of the electroforming. A better fit to the low bias virgin state
conduction is obtained for the 8 nm devices if a series resistance due to Schottky emission is added.
In both cases, the electroforming process appears to occur due to the onset of thermal runaway,
after which all the tested devices are governed by self-heating 3DmodPF conduction.
When starting with thinner a-Nb2O5 layers, crystallization is likely suppressed in part because they
are better thermally anchored to the electrodes and don’t get as hot. Their crystallization may also
be inhibited due to strain and interaction with the electrodes. In other cases, thinner layers may be
rapidly reduced to a composition that is less oxygen rich than NbO2, retarding crystallization. For
these initially oxygen rich thin films, however, the composition may be stabilized at a composition
with only slightly less oxygen than a-NbO2 because oxygen diffusion into the TiN electrodes slows
as their surfaces become oxidized. In this case NDR may still be observed. Films with a starting
composition near a-NbO2 on the other hand are rapidly reduced, before crystallization can occur,
to a composition that is too close to metallic to exhibit NDR. These hypotheses are still under
investigation.
Model Fits to Devices with Various Geometries
0.2
0.6
0
0.2
0.4
0.6
0.8
0
1.0E+08
1.0E+07
1.0E+07
1.0E+07
1.0E+06
1.0E+05
8 nm thick NbOx
Type I, radius = 26 nm
1.0E+03
0.5
1.0E+05
13 nm thick NbOx
Type II, radius = 26 nm
1.0E+04
1.0E+03
Bias (V)
0
1.0E+06
1
1.5
Current (A/cm2)
1.0E+08
1.0E+04
0.2
0.4
1.0E+07
1.0E+07
1.0E+07
1.0E+05
1.0E+04
1.0E+03
26.5 nm thick NbOx
Type II, radius = 26 nm
Bias (V)
Current (A/cm2)
1.0E+08
1.0E+06
1.0E+05
1.0E+04
1.0E+03
42 nm thick NbOx
Type I, radius = 113 nm
Bias (V)
Bias (V)
0
0.8
1.0E+08
1.0E+06
1.5
1.0E+05
1.0E+03
0.6
1
1.0E+04
Bias (V)
0
0.5
15 nm thick NbOx
Type II, radius = 16 nm
1.0E+06
1.0E+08
Current (A/cm2)
Current (A/cm2)
0.4
Current (A/cm2)
Current (A/cm2)
0
1.0E+08
0.5
1.0E+06
1.0E+05
1.0E+04
1.0E+03
1
T(K)
450
425
400
375
350
325
300
275
250
42 nm thick NbOx
Type II, radius = 113 nm
Bias (V)
Figure S2. Electrothermal data (solid curves) and compact model (dashed curves) fits for devices with different
NbOx layer thicknesses, device areas, and type of formation. Ambient temperatures are color coded.
2
Meyer-Nedel Behavior
14
Type I
Type II
ln(σp)
13
12
11
10
9
0
0.1
0.2
0.3
E
Figure S3. Meyer-Neldel plot of the low bias conductivity prefactor versus the effective hopping energy, as
determined from fitting Eqs. 1 and 2 to temperature dependent V-I curves. Orange and blue dots correspond to
devices that underwent Type I and II forming, respectively.
Equations Governing Schottky-Emission Based NDR
The I-V curve for an NDR device that is self-heated by a Schottky interface is determined from
the following parametric equation:
𝜂𝜐1/2 −1
𝐴
𝑖 = ( 2 ) (𝑡 + 𝑖𝑣)2 𝑒 𝑡+𝑖𝑣
𝑑
Where,
𝑞∅𝐵 𝑑 2
𝐼 ≡ 𝐴𝑅 (
) 𝑖
𝐾𝐵
𝐾𝐵
𝑉≡
𝜈
𝑞∅𝐵 𝑅𝑡ℎ𝑒𝑟𝑚 𝐴𝑅 𝑑 2
1⁄
2
𝑞𝐾𝐵
𝜂≡(
)
4𝜋𝜖𝑖 𝑅𝑡ℎ𝑒𝑟𝑚 𝐴𝑅 𝑑 3 ∅3𝐵
𝑇 = 𝑇𝑎𝑚𝑏 + 𝑅𝑡ℎ𝑒𝑟𝑚 𝐼𝑉
𝑞∅𝐵
𝑇𝑎𝑚𝑏 ≡
𝑡
𝐾𝐵
𝐴𝑅 = 𝑅𝑖𝑐ℎ𝑎𝑟𝑑𝑠𝑜𝑛′ 𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑞 = 𝑓𝑢𝑛𝑑𝑎𝑚𝑒𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑟𝑔𝑒
𝐾𝐵 = 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛′ 𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜖𝑖 = 𝑜𝑝𝑡𝑖𝑐𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
3
∅𝐵 = 𝑆𝑐ℎ𝑜𝑡𝑡𝑘𝑦 𝑏𝑎𝑟𝑟𝑖𝑒𝑟 ℎ𝑒𝑖𝑔ℎ𝑡
𝑑 = 𝑉/(𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑖𝑒𝑙𝑑 𝑎𝑐𝑟𝑜𝑠𝑠 𝑖𝑛𝑠𝑢𝑙𝑎𝑡𝑜𝑟)
𝐴 = 𝑑𝑒𝑣𝑖𝑐𝑒 𝑎𝑟𝑒𝑎
𝑇𝑎𝑚𝑏 = 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
𝑅𝑡ℎ𝑒𝑟𝑚
= 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑆𝑐ℎ𝑜𝑡𝑡𝑘𝑦 𝑏𝑎𝑟𝑟𝑖𝑒𝑟 𝑎𝑛𝑑 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑔𝑟𝑜𝑢𝑛𝑑 𝑎𝑡 𝑇𝑎𝑚𝑏
Thermally-Controlled Dynamics
1.4E-09
1.2E-09
Rth∙Cth (sec)
140
26.5 nm thick
15 nm thick
8 nm thick
1.0E-09
120
100
8.0E-10
80
6.0E-10
60
4.0E-10
40
2.0E-10
20
0.0E+00
0
25
52
Power @NDR Onset (W)
In addition to accurately describing the quasi-static behavior of the NbOx selectors, our compact
model also elucidates the impact of heating and cooling on their dynamic behavior. These thermal
effects must be considered in the future design of high speed selectors or other S-type NDR devices
based on self-heating. To this end, our NbOx devices can be viewed as locally active memristors
with the temperature of the active region, TN, as the dynamical state variable, as governed by Eqs.
(1) and (2). An effective thermal time constant Rth∙Cth can be estimated by using the value of Rth
determined by fitting the static I-V curves and a value for Cth determined from a numerical
simulation. To obtain the latter we constructed a full three dimensional simulation of our devices
using adjustable spatial dimensions and literature values for the specific heats of the component
materials. Cth could then be determined for a specific geometry by monitoring the temperature
rise that results from Joule heating within the active region of the device. The time constants
determined in this fashion are summarized in Figure S4.
100
Diameter (nm)
Figure S4. Plots of thermal time constant RthCth (solid curves) and power required to hold a selector at the onset of
NDR (dashed curves) as a function of device diameter for three different NbO x layer thicknesses.
Although these thermal time constants are on the order of nanoseconds, dynamic simulations
4
show that when the current or voltage are swept up and then down linearly, significant hysteretic
effects are observed in their V-I characteristics even for scans on the order of hundreds of
nanoseconds. Such dynamic effects must be accounted for when considering, for example, the
impact of a given write pulse on memory cells that employ these selectors, particularly in cases
where the state of the cell’s memory element is capable of changing faster than the selectors’
thermal response. For example, understanding of these interactions should guide efforts to lower
bit error rates and increase bandwidth when writing memristive memory through the utilization
of tailored pulse shapes.
Figure S4 also plots the DC power required to bias NbOx selectors of different dimensions at the
onset of NDR. These powers can be used to derive rough estimates for the energies required to
turn these selectors on when addressing memory cells that employ them. These energies are
typically in the pJ range for 25 nm diameter selectors and 100 nsec ‘on’ times.
5