Nature | Vol 585 | 24 September 2020 | 519We also searched for spectro-microscopy signatures of a Mott transi-
tion by applying a modified in operando synchrotron X-ray absorption
technique^25 specifically within the hysteresis, by removing signals from
operating regions outside of the hysteresis. We observed lowering of
the π* band and disappearance of a d⁎ band at higher temperatures,
consistent with both an electronic Mott insulator-to-metal transition
and also the associated changes to the structural phase ordering in
NbO 2 (Fig. 1g, Supplementary Fig. 8)^26. To induce the desired
higher-order Mott transition dynamics, the geometric structure of the
element was optimized for both electrical and thermal properties^27 ,
and the material composition was also carefully tuned (Fig. 1h). Neu-
romorphic functions occurring only within a narrow subset of material
compositions that support NDR is predicted by Chua’s theory of local
activity (see Supplementary Information section 1, Supplemen-
tary Fig. 1)^2.
The element was powered by connecting a tunable constant voltage
across the terminals that could access different parts of its current–
voltage curve via load-lines determined by the internal resistor and
the applied voltage (Fig. 2a). When biased just below the hysteresis
(vext = 1.8 V), we observed self-sustained sinusoidal oscillations, but
when biased within the hysteresis (vext = 1.95 V), we observed peri-
odic two-spike bursting, which changed into periodic single spikes
at a higher voltage (vext = 2.05 V), similar to a neuron’s action potential
(Fig. 2b, c). At even higher voltages (vext = 2.1 V, load-line biasing just
above the hysteresis), the spikes abruptly transitioned to low-intensity
periodic features similar to the super-threshold damped spiking of
neurons^5. In an additional experiment, we slowed down the dynamics
using an external capacitor and a resistor along with the element to
observe the entire extent of behaviours noted above within a single
time series by tuning the voltage across the terminals. Including the
behaviours noted here, we identified a total of 15 different neuromor-
phic responses of our third-order element that originate by tuning the
voltage across the element (Supplementary Information section 5,
Supplementary Figs. 9–19).
We constructed a simple compact model for the element to explore
the dynamics in simulation. We used a Schottky equation (equation ( 1 ))
for the state-dependent conduction model, because any sufficiently
nonlinear thermally activated transport can produce NDR. We repre-
sented the dynamics of the temperature state variable T using New-
ton’s law of cooling (equation ( 2 )), and included the dynamics of the
capacitor via the Kirchhoff current and voltage laws (equation ( 3 )). To
represent the Mott transition, we included a switchable Ohmic conduc-
tor (Rmet) in parallel with the Schottky transport when the NbO 2 is in its
metallic state at higher power levels (equation ( 4 )). Equation ( 5 ) models
a hysteresis in the switchable Ohmic resistor.
iAT qv
kTqφ
kT=1−exp−exp−
(1)qv
dκ
Sch^20 m
B0
B0mT
tiv
CTT
RCd
d =−−
oxm (2)
thamb
thth
v
tCRvvid
d=11
m (−)− (3)
Sextmoxiiv
R
ox=+Sch m (4)
met
RR αii
=−tanh +sgn t II βd
met0 ox d −+ (5)ox
12In equations ( 1 )–( 5 ), iSch is the Schottky current; A, q 0 , κ, d, φ, kB, α,
R 0 and β are constants; vm is the voltage across NbO 2 ; iox is the current
through NbO 2 ; t is time; T is the absolute temperature of NbO 2 and Tamb
is the ambient temperature; Cth and Rth are the thermal capacitance
and resistance, respectively; RS is the series resistor (RS = Rint when530 540 55050Data
FitXAS in OD (×^10–3)Energy (eV)–101Nb 4d*
Nb 5s*Si 1 -Si 2d||*WSiNxSiO 2
TiNNbO 2TiNPt0.0 0.2 0.40.00.30.60.9i (mA)mNDRvext or iext RintCintRNbO 2 (T, iNbO 2 )imvdevIntegrated circuit element[001]acdgefQuasi-static Mott
transition: the
missing dynamicsvmMott dynamicsStatic Mott
NDR
No NDRh1.0 1.5 2.0 2.50.1110Vm (V) x in NbOx0ΔOD(×^1–3 0
)π* σ*U(×^10–3Ωm)SS
TNTbFig. 1 | Element construction and static measurements. a, Circuit model of
the integrated element. b, Schematic illustration of the structure, indicating
the presence of the different electronic constituents. c, Cross-sectional
transmission electron micrograph displaying a part of the structure of the
element. Scale bar is 30 nm. d, Quasistatic current–voltage behaviour of the
element. e, High-resolution transmission electron micrograph of a region
within an element (indicated by dotted rectangle in c) that was operated past
the box-shaped hysteresis in d. Dashed green box contains obvious crystal
ordering. Scale bar is 8 nm. f, Electron diffraction pattern of the crystalline
region indicating monoclinic [001] ordering. g, In operando X-ray absorption
spectrum (XAS) (measured in optical density, OD) of a prototypical component
identical in material stack and behaviour to the element under study, along
with the spectral difference corresponding to two current levels (i 1 = 0.6 mA
and i 2 = 1 mA) on either side of the hysteresis in the element’s current–voltage
behaviour. The constituent bands are marked. The spectral difference (Si 1 − Si 2 )
(in change in OD, ΔOD) indicates the occurrence of a Mott transition
specifically within the hysteresis (see Supplementary Information sections 2–4
for additional details). h, Resistivity plotted against the stoichiometry of NbOx,
with corresponding electrical behaviours: (I) NDR, (II) hysteresis in the
current–voltage behaviour, (III) dynamical neuromorphic properties when
operated within the hysteresis, and (IV) no NDR. The blue (NDR) region exhibits
only behaviour (I), green (Static Mott) exhibits only behaviours (I) and (II), red
(Mott dynamics) exhibits behaviours (I)–(III), and the grey region exhibits only
behaviour (IV).