1
Supplementary Material
Monolayer MoS2 Metal Insulator Transition based Memcapacitor Modeling with
extension to a ternary device
S1: Ioffe Regel Modeling and the critical MIT Transition values:
From electronic phase point of view, the MIT material can be either in bound or non-bound state 1.
The bound state refers to an insulating phase where the outer electrons are tightly bound to the inner
positive nucleus. The Non-bounding state refers to a metallic phase where the outer electrons are free
to move and are not under the influence of nucleus. The gate voltage increases the carrier
concentration n of MoS2 layer to a level nc (critical carrier concentration) when no more outer
electrons remain bound to the nucleus because of the shielding of outer electrons and screening of
coulomb interaction.
The carrier concentration n in MoS2 monolayer at any given gate voltage is given below 2.
n=
V
Where Vs = ∫0 g (1 −
Ct0tal
) dVg
Cg
Cg (Vg − Vs − VT )
,
e
(1, 𝑆)
is the surface potential of MoS2 layer calculated by applying
poisson equation to the vertical heterostructure of figure 1a in quasi-quantitative manner 2.
Ctotal
is the total capacitance of the device given later in this letter, VT the threshold voltage for MIT
Transition of MoS2 monolayer and e represents the electronic charge 2. The MIT state can be
determined on the basis of carrier concentration in MoS2 layer by Ioffe Regel Criterion as given below
3
.
k F × le ≈ 1 Shows the critical point of Transition,
k F × le ≪ 1 Shows the Insulating state,
(2a, S)
(2b, S)
2
k F × le ≫ 1 Shows the Metallic state,
Where k F = √2πn is the Fermi wave vector and le =
hkF μ
e
(2c, S)
is the mean free path of electrons with
h for planks constant and μ for electron mobility. At certain gate voltage say “vgt ” the gate
transition voltage, the carrier concentration n in MoS2 monolayer becomes equal to the critical value,
i.e. n ≈ nc
referred as critical carrier concentration, thus at this point the transition from insulating
to metallic phase occurs. The critical carrier concentration value can be found using Ioffe Regel
Criterion of equation (2,S). Initially in figure 1a in the main text, the gate capacitance Cg ≈ εg ⁄dg, the
MoS2 layer capacitance CMoS2 ≈ εMoS2 ⁄dMoS2 , and the total capacitance of the device
−1
((1⁄ CMoS2 ) + (1⁄Cg ))
Ctotal =
with εg gate dielectric, dg gate dielectric thickness, εMoS2 monolayer
MoS2 dielectric and dMoS2 monolayer thickness are defined. The gate voltage then induces the IMT
(Insulator to Metal Transition) in monolayer and consequently creates charge puddles which percolate
to form a conducting channel from one end to the other one. The electron mean free path le and
Fermi wave vector k F are continuously calculated to check the value of Ioffe Regel according to
concentration and gate voltage which satisfies k F × le ≈ 1 is
equation (2, S). The value of carrier
termed as the critical carrier concentration
nc and gate transition voltage vg t respectively as
shown in figure 1,S . It should be noted that the critical value and the transition voltage depends upon
the initial parameters values which define the individual capacitance components in our proposed
device. We extract the critical carrier concentration for MIT of monolayer MoS2 to be 8 ×
1012 /cm2 and the gate transition voltage to be 1.65V using Ioffe Regel for the initial parameters
2
of 𝑑𝑀𝑜𝑠 = 0.65 𝑛𝑚, 𝜀𝑀𝑜𝑠 = 5𝜀0 , 𝜀𝑔 = 30𝜀0 , 𝑑𝑔 = 30 𝑛𝑚 𝑎𝑛𝑑 𝜇 = 168
𝑐𝑚
𝑣.𝑠
as shown in figure 1S.
The calculated critical values for these initial parameters in our model are in close approximation with
previous experimental observation
4, 2
. It should be noted that MIT can only be observed ideally at
absolute zero temperature at which metal conducts and non-mental doesn’t conduct 1. MIT is
3
distinguished more accurately at low temperature therefore; we used the mobility of μ =
2
168 cm ⁄v. s at 4k 4 for making our model more accurate.
Figure 1S: Extraction of Critical Carrier Concentration and Gate Transition Voltage of MIT for
monolayer MoS2 in our proposed device structure of memcapacitor
S2: Percolation based MoS2 Conductivity:
Percolation based conductivity of MoS2 monolayer can be divided into three main regions on the basis
of percolation probabilty and its critical value as shown in figure 2S and its inset. For simulating the
percolation based conductivity, we chose experimental values of the base line conductivities such as
1 𝜇𝑆 for insulating phase and 250 𝜇𝑆 for metallic phase at 4K for monolayer MoS2 5. The quantum
conductivity σMIT of MIT layer which is the conductivity at crossover point should be nearly equal
e2 ⁄h where e is electronic charge and h is plank constant 5. In our simulation in figure 2, S, we show
the quantum conductivity of 39.3 𝜇𝑆 or e2 ⁄h at the critical percolation probabilty 0.4, which is in
close approximation to the experimental value in 5. The inset log scale plot in figure 2, S shows that
when p < pc , there is no infinite cluster of charge puddles so that MoS2 monolayer is in insulating
state. The term infinite cluster is used for a cluster of charge puddles that runs from one end of the
layer to the other end without any vacant gap. At
p ≈ pc there might be some infinite clusters with
a very small size. The green color round balls shows the infinite cluster that spans from bottom to the
4
top of the layer while the blue balls show a finite cluster which doesn’t take part in making any
complete conducting channel. For p > pc the size of infinite cluster become very large by combining
with the finite cluster and the whole area becomes a conducting channel to turn state of monolayer
MoS2 from insulating to metallic 2.
Figure 2S: Linear scale conductivity of monolayer MoS2 as a function of percolation probabilty. Inset
Log scale conductivity of monolayer MoS2 as a function of percolation probabilty showing three
distinct regions having different percolation of charge puddles.
S3: Memcapacitive based ternary device
In order to determine the individual percolations states of both the top and bottom MoS 2 layers, we
need to calculate the carrier concentration in both MIT layers. Let’s n1 and n2 be the carrier
concentrations in MoS2-1 and MoS2-2 respectively. Then according to the principle of electrostatic
charge control, n1 and n2 can be expressed as follows.
n1 ∝ Cg−1 × Vg ,
(3a, S)
5
n2 ∝
Cmem−1 × Cg−2
× Vg ,
Cmem−1 + Cg−2
(3b, S)
In equation (3,S), it is clear that at any instant “t”, n2(t) will always be less than n1(t) for a given
voltage Vg . This implies that after applying Ioffe Regel criterion to both the MoS2 layers, the value
of critical carrier concentration of layer 2 will always be higher than that of layer 1 such that nc 2 >
nc 1 and vgt 2 > vgt 1 . The difference between these two transition voltages is termed as ∆𝑉𝑔𝑡
which is directly proportional to the term
C
×C
Cg−1 − Cmem−1+Cg−2 . Figure 3S shows the plots for both
mem−1
g−2
percolation states p1 and p2 and overall state of the device at any instant time “t” for a given gate
voltage.
Initially both MoS2 layers are in insulating/semiconducting state thus the percolation state values for
both the layers is zero. This state of cascaded device is termed as state 1 having p1, p2=0. At certain
gate voltage vgt 1, top MoS2 layer transits to metallic state but the bottom layer is still in insulating
state. This state of the device is termed as state 2 with p1=1 and p2=0. Finally at vgt 2 > vgt 1 , the
bottom MoS2 layer undergoes IMT transition too and making the device in state 3 with p1, p2=1.
6
Figure 3S: Individual percolation states of both MoS2 layers having different gate transition voltage
forming three specific states.
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