Monolayer MoS₂ metal insulator transition based memcapacitor modeling with extension to a ternary device

Memristor was first introduced by Leon Chua in 1971 on the basis of his famous symmetry argument to conceive it theoretically.¹ Even though the first experiments on resistive memory have been proposed as early as in 1960^2–5 but they could not attract the attention of scientific community until a missing memristor was announced by Hewlett-Packard in 2008.⁶ Furthermore following the same lines, two other types of devices: memory capacitor (memcapacitor) and meminductor were theoretically defined.^7–11 Like memristor which is a memory resistor, memcapacitor is a memory capacitor, the capacitive state of which can be changed via external stimuli like electric field or voltage. Nano features and ionic transport mechanism inherited in Memristive systems introduce new challenges such as modeling, characterization, and system architectures. This interest is extended to other mem-elements¹² like memcapacitors and meminductors to widen their application areas to memory devices and circuit design. Many groups presented the mathematical and Spice based modeling of memcapacitor. All these models are based on general equations without addressing the specific implementation of memcapacitor except very few.^13–17 We propose a model which is based on the specific physical implementation with an edge of fast transition of the state over the previously realized models. The previously realized model uses slow polarization of metamaterial which limits the transition time in the range of usec. Our proposed physically realized model uses the Metal Insulator Transition material which makes the transition of state possible in the range of psec. The model is then used to provide a new framework for ternary device by simple cascading the proposed structure.

The proposed device structure shown in figure 1a which consists of an MIT material layer and a dielectric layer embedded between two external metal plates. The MIT layer can have two possible states, the insulating and the metallic, on the basis of which the device equivalent capacitance is changed accordingly. The MIT material which we used for our modeling and simulation is Molybdenum disulphide Monolayer (MoS₂) which is a transition metal dichalcogenide semiconductor with indirect band gap in bulk form and direct band gap in monolayer. The individual layers are weakly bound through van der waals forces and single atomic layers can be extracted through micromechanical cleavage technique or by the process of liquid-phase exfoliation. The Electronic properties of monolayer are much different than bulk as it is easy to control the conductivity in Monolayer due to its atomic thickness of (0.65 nm) by electrostatic control.¹⁸ The carrier concentration “n” is increased via external voltage to a level where MoS₂ can be treated as metal instead of insulator. In this letter, MoS₂ and MIT layers are used interchangeably.

In figure 1a, c_g is the gate capacitance introduced by a dielectric material and c_mit is the capacitance introduced by MIT material, .i.e. by depletion and misalignment of charge puddles in monolayer MoS₂. It should be noted that the process of percolation (explained later in text) makes the top electrode (metal plate) act like a gate with a voltage applied to it as v_g to control the percolation of charge puddles in MoS₂ layer. An Ioffe Regel Mott criterion is used to determine the state of MIT material and its boundary.¹⁸ The detail about the Ioffe Regal Transition modeling, the critical value of carrier concentration in monolayer MoS₂ and the gate voltage at the time of transition is given in the supplementary material S1.^19,20

We used percolation based modeling approach for monolayer MoS₂ MIT which refers to the creation of metallic charge puddles and its percolation. For a finite system, the percolation can be defined as the appearance of large cluster that spans through the whole system and connect the top and bottom of the system.^21–23 According to percolation model, the charge puddles percolates to form a conducting channel which then in turn transits the monolayer from insulating phase to metallic phase. Nanoscale metallic regions emerge from the insulating host, increasing in number and size to form percolative nature.^24–26 Let’s p be the percolation probabilty of monolayer MoS₂ varies between zero and one i.e. $0≤p≤1$⁠. We use the following equation to build its relation with the gate voltage and its transition value.

Where x = 1 and $k≈1010$ are the constant parameters and v_gt is the gate transition voltage at which the MIT transition occurs. The reason we used these constant parameters “x” and “k” specific values is because the MIT is ultra-fast process in the range of psec range^27,28 and these values gave an approximate fit for it. When the carrier concentration in MoS₂ reaches to 8 × 10¹²/cm² which is the critical value given in S1, the charge puddles start to percolate and the percolation probabilty drastically increases following a relatively sharp transition shown in figure 1b. We utilized the extracted value of gate transition voltage from S1 into equation (1) to determine the percolation probabilty and its critical value. Initially the percolation probabilty is near to zero, which represents the MoS₂ monolayer to be in insulating/semiconducting state. The percolation value near to one implies a conductive state of MIT material. The middle region shows transition from one state to another with a critical value of p_c = 0.4. The critical percolation value is the percolation value at the point of MIT transitioning gate voltage as shown in figure 1b. It should be noted that MIT phenomenon shows hysteresis and that is why we incorporated the effect of hysteresis in our modeling equation which gives hysteresis effect to the percolation probabilty shown in figure 1b.^29–31 

The function 1 – (2p – 1)^2X is used in (1) for smooth transition between two stable states that is to ensure zero time rate of change of percolation such that $ddt(p)≈$ 0 for p $≈$ 0,1 as shown in figure 1b. After we have determined the critical percolation probability at which the transition takes place, now we will analyze the conductivity of monolayer MoS₂ in terms of conductivity power law as follows.

Here “t” and “s” are the critical exponents³² and their values are chosen to be 1.3 where $u=tt+s$⁠.

The conductivity of monolayer in figure 2a LHS follows the same hysteric behavior like percolation in figure 1b. The base line conductivities. i.e. $σmet$ and $σins$ ,for our simulation are extracted in S2.^33,34 According to (2), the conductivity of MIT material is a function of percolation probabilty which then in turn is a function of gate voltage by (1).

We used percolation as the state equation of memcapacitor and performed its DC and frequency analysis (later in the text). The memcapacitance as a function of percolation of our proposed device structure can be written as follows.

The capacitance thus by making it a function of percolation probabilty can be varied between two capacitive states in hysterical manner following the percolation as shown in figure 2a. For zero percolation, the memcapacitive state is LCS, i.e. C_total making the MoS₂ layer to be in insulating state. For percolation probabilty value equal to 1, the memcapacitive state is HCS, i.e. C_g making the MoS₂ layer to be in metallic state. Thus by applying gate voltage or sweeping it towards positive side will introduce metallic charge puddles in MoS₂ layer and the percolation of these charge puddles change the state of the layer from insulating phase to metallic phase thus raising the total device capacitance to gate capacitance value which is termed as high capacitive state HCS as shown in figure 2a, RHS. The capacitance or memcapacitance of the device follows the MoS₂ monolayer conductivity in hysterical way triggered by the external top electrode gate voltage. It should be noted that microscopically, MIT is an electronic phase transition accompanied by structural transition.¹⁸ The insulating/semiconducting state of MoS₂ is trigonal prismatic while metallic state is octahedral coordinated. This type of dual transition gives an opportunity of stable/metastable states with hysteresis and memory effect reversibly tuned by carrier concentration via gate voltage.^35–37 DC analysis of our model in figure 2a shows hysteresis in C-V (Capacitance vs Voltage) which proves it to be a memcapacitive system.

In our model, we estimate the time required to change the conductivity of monolayer MoS₂ from its insulating base line conductivity to metallic base line conductivity, i.e. from $1μS$ to $250μS$ (Calculated in S2) for simulation purposes shown in figure 2b. Equation (2) along with (1) is simulated to estimate the transition time for MIT of monolayer MoS₂ as shown in figure 2b. It should be noted that in our model, we have a good control over the MIT transition speed by simply changing the parameter “x” value in (1). If we increase the “x” value, the transition time can be decreased further which makes the model more flexible in terms of transition speed. Our main focus is to model the state equation of memcapacitor in such a way that it can reflect the MIT of monolayer MoS₂.

The total charge on the external plates of memcapacitor and its current can be formulated as follows.

After DC, we performed frequency analysis of our model by defining two important frequencies, transition frequency f_T and switching frequency f_sw in this regard. Transition frequency of 14 GHz is calculated using MIT transition time of 74 psec for x = 1 in our model. The switching frequency refers to the input signal frequency which is varied to determine its effect on the percolation state and memcapacitive behavior. Initially, the switching frequency is kept smaller than the MIT transition frequency, i.e. f_sw < f_T which changes the state completely thus producing highest % change in percolative state and giving pinched off hysteresis loop of Q-V curve as shown in figure 3a,b respectively. Irrespective of the underlying physical mechanisms that define the state of the system, at f_sw < f_T, the system has enough time to adjust its value of percolation to a momentary value of the control parameter (voltage) and to go from 0 infinite clusters to large size infinite cluster. On the other hand, at f_sw > f_T, there is not enough time for any kind of percolation change during a period of oscillations of the control parameter which makes the % change in percolative state start decreasing and become very small as shown in figure 3a. In figure 3b, switching frequency is increased such that f_sw > f_T which makes the Q-V pinched off hysteresis to collapse and makes the system a linear capacitive system. The MIT material doesn’t have much time to percolate and thus cannot form infinite cluster and the percolation state changes to a very small amount.

We further extend our model into three states device instead of two states by simply cascading the structure as shown in figure 4a. The individual memcapacitances of the top and bottom device structures are dependent on the respective percolations of MoS₂ layers.

The total memcapacitance C_mem of cascaded structure is equivalent to the series combination of C_mem–1 and C_mem–2.

Here p1 is the percolative state of the top MoS₂ layer and p2 is the percolative state of the bottom MoS₂ layer. These two percolative states can give rise to three overall states of the device. The respective gate voltage for each percolative state and its dynamic behavior is given in S3. These three specific combinations of percolative states are then reflected in the overall capacitance of the device making it a three states device is shown in figure 4b. It should be noted that vertical cascading reduces the dynamic range. We propose that by using horizontal cascading in combination with vertical cascading, there is a possibility of increasing the dynamic range and states of such device.

In summary, we proposed a new and different framework for memcapacitor specific implementation based on metal insulator transition of MoS₂ monolayer using site based percolation theory. Initially the stand alone MIT model of MoS₂ monolayer was formulated. The percolation of MoS₂ MIT is then used as a state for memcapacitor modeling. The extracted critical values for gate transition voltage, carrier concentration and percolation probabilty by our model are in close approximation of the experimental values for almost the same initial parameters. The model also proves the quantum conductivity of monolayer MoS₂ MIT. The transition speed can also be controlled by a single parameter which makes the model generic to be used for other kinds of MIT materials as well. DC and Transient analysis shows the basic foot prints of memcapacitor. We proposed to eliminate the high frequency limitation of previous models by using fast transition of MIT in psec.The same model is then used to formulate a ternary device by simple cascading.