Two-Dimensional Material Memristor (Atomristor and Memtransistor)

1. Theory

Definition: A 2D material memristor is a resistive switching device whose active layer is a monolayer or a few layers of a layered material such as $Mo S_{2}$ , h-BN or $WS e_{2}$ .
Atomristor (vertical): A metal / 2D layer / metal sandwich only a few atoms thick that switches between high and low resistance.
Memtransistor (lateral): A gated 2D channel between two electrodes whose resistance switches, with the gate as a third control terminal.
Placement: It sits in the resistive folder because the state is stored as a resistance change from moving ions or defects.
Proposed mechanisms: Metal-atom filaments through the layer (ECM-like), field-driven vacancy motion (VCM-like), or vacancy migration along grain boundaries in polycrystalline films.
Contact barrier: Metal contacts on 2D semiconductors are strongly Fermi-level pinned; van der Waals contacts approach the Schottky-Mott limit.

2. Working Principle

Vertical switching: A voltage across the sub-nanometre layer moves atoms or vacancies, opening or closing a conductive path at low switching voltage.
Off-state conduction: Current tunnels through the few-layer film, falling exponentially with the number of layers.
Lateral switching: A drain voltage moves defects along grain boundaries or near contacts, changing the barrier and channel resistance.
Gate tuning: In memtransistors the gate sets the carrier density, tuning both resistance states and the switching voltage.
Barrier-controlled on/off ratio: A barrier change $Δ φ_{B}$ changes the current by about $exp (q Δ φ_{B} / k_{B} T)$ .
Variability: Non-uniform thickness, defect density and grain size give large device-to-device and cycle-to-cycle variation.

3. Applications

Current status: Research devices; uniform large-area growth and CMOS integration are not yet solved.
Neuromorphic computing (research): Multi-terminal memtransistors as gate-tunable artificial synapses.
Ultra-thin stacked memory (target): Low-voltage, sub-nanometre-thick resistive cells for dense 3D integration.

Name	Formula	Related Formulae	Variables/Constants
Off-state tunnelling through a few-layer film	$J_{o f f} \approx J_{1} exp (- β_{L} N)$	$β_{L} = 2 κ t_{L} = \frac{2 t _{L} 2 m ^{*} φ}{ℏ}$ $E = \frac{V}{N t _{L}}$	$J_{o f f} : c u r r e n t d e n s i t y o f t h e hi g h - r es i s t an ce s t a t e c a r r i e d b y d i r ec t t u nn e l l in g t h r o ug h N l a y er s o f t h e tw o - d im e n s i o na l ma t er ia l$ $J_{1} : c u r r e n t d e n s i t y e x t r a p o l a t e d t o o n e l a y er$ $N : n u mb er o f l a y er s$ $β_{L} : d ec a y o f t h e c u r r e n t p er l a y er$ $t_{L} : t hi c k n ess o f o n e l a y er in c l u d in g t h e v an d er W aa l s g a p (ab o u t a f r a c t i o n o f a nan o m e t r e)$ $κ : d ec a y co n s t an t o f t h e w a v e f u n c t i o n in t h e f i l m$ $m^{*}, φ : t u nn e l l in g e f f ec t i v e ma ss an d ba r r i er h e i g h t$ $E : a v er a g e f i e l d in t h e f i l m a t t h e v o l t a g e V (f or e x am pl e, 1 V a cr oss t h r ee l a y er s o f ab o u t 0.65 nm e a c h g i v es a f i e l d o f ab o u t 5 M V / c m)$
Gate-tunable lateral channel (memtransistor)	$R = R_{c} + \frac{L}{W} \frac{1}{q μ n _{s}}$	$n_{s} = \frac{C _{o x} ( V _{G} - V _{T} )}{q}$	$R : tw o - t er mina l r es i s t an ce o f a l a t er a l c hann e l o f t h e tw o - d im e n s i o na l se mi co n d u c t or in t h e l in e a r r e g im e$ $R_{c} : t o t a l co n t a c t r es i s t an ce o f t h e tw o e l ec t r o d es$ $L, W : c hann e l l e n g t h an d w i d t h$ $q : e l e m e n t a r y c ha r g e$ $μ : c a r r i er m o bi l i t y (w hi c h in a p o l y cr y s t a l l in e f i l m i s r e d u ce d b y t h e g r ain b o u n d a r i es an d c han g es w i t h t h e s w i t c hin g s t a t e)$ $n_{s} : s h ee t d e n s i t y o f t h e c a r r i er s in d u ce d b y t h e g a t e$ $C_{o x} : g a t e c a p a c i t an ce p er u ni t a r e a$ $V_{G}, V_{T} : g a t e v o l t a g e an d t h r es h o l d v o l t a g e (t h e g a t e v o l t a g e t u n es t h e r es i s t an ce o f b o t h t h e hi g h an d t h e l o w s t a t e)$
Schottky barrier at a metal contact on a 2D material	$φ_{B} = S (Φ_{M} - χ) + (1 - S) φ_{C N L}$	$S = \frac{d φ _{B}}{d Φ _{M}}$	$φ_{B} : S c h o tt k y ba r r i er h e i g h t f or e l ec t r o n s a t t h e m e t a l /2 D in t er f a ce (a s an e n er g y d i v i d e d b y q)$ $S : p innin g f a c t or, e q u a l t o o n e f or t h e i d e a l S c h o tt k y - M o tt l imi t an d z er o f or co m pl e t e F er mi - l e v e l p innin g$ $Φ_{M} : w or k f u n c t i o n o f t h e m e t a l$ $χ : e l ec t r o n a f f ini t y o f t h e tw o - d im e n s i o na l se mi co n d u c t or$ $φ_{C N L} : c ha r g e n e u t r a l i t y l e v e l o f t h e in t er f a ce m e a s u r e d f r o m t h e co n d u c t i o n ban d e d g e$ $(a v an d er W aa l s co n t a c t w i t h a t r an s f er r e d m e t a l c an g i v e S c l ose t o o n e, w h er e a s an e v a p or a t e d m e t a l p in s t h e l e v e l b ec a u se o f t h e d ama g e an d t h e in t er f a ce s t a t es)$
Thermionic emission over the barrier (2D)	$J = A_{2 D}^{*} T^{3/2} exp (- \frac{q φ _{B}}{k _{B} T}) [exp (\frac{q V}{n k _{B} T}) - 1]$	$ln (\frac{J}{T ^{3/2}}) = ln A_{2 D}^{*} - \frac{q φ _{B}}{k _{B} T}$	$J : c u r r e n t p er u ni t co n t a c t w i d t h (A / m) o v er t h e ba r r i er f or a tw o - d im e n s i o na l c hann e l, w h er e t h e c a r r i er s a r e co n f in e d in o n e d i r ec t i o n$ $A_{2 D}^{*} : tw o - d im e n s i o na l R i c ha r d so n co n s t an t, w hi c h d i f f er s f r o m t h e t h r ee - d im e n s i o na l o n e$ $T : t e m p er a t u r e$ $φ_{B} : S c h o tt k y ba r r i er h e i g h t$ $V, n : v o l t a g e a cr oss t h e ba r r i er an d i d e a l i t y f a c t or$ $k_{B} : B o l t z mann co n s t an t$ $(t h e seco n d r e l a t i o n i s t h e A r r h e ni u s pl o t u se d t o e x t r a c t t h e ba r r i er h e i g h t f r o m t h e t e m p er a t u r e d e p e n d e n ce)$ $(a ppl i es t o inj ec t i o n in t o t h e pl an e o f t h e c hann e l, i . e . t h e co n t a c t s o f a l a t er a l m e m t r an s i s t or; in a v er t i c a l a t o m r i s t or t h e c u r r e n t cr osses t h e l a y er, an d t h e t h r ee - d im e n s i o na l T^{2} f or m or t h e t u nn e l l in g r o w i s u se d in s t e a d)$
On/off ratio from a barrier change	$\frac{I _{O N}}{I _{O F F}} \approx exp (\frac{q Δ φ _{B}}{k _{B} T})$	$Δ φ_{B} = φ_{B, O F F} - φ_{B, O N}$	$I_{O N}, I_{O F F} : c u r r e n t s o f t h e l o w - r es i s t an ce an d t h e hi g h - r es i s t an ce s t a t e a t t h e s am e v o l t a g e$ $Δ φ_{B} : c han g e o f t h e S c h o tt k y ba r r i er h e i g h t b e tw ee n t h e tw o s t a t es (f or e x am pl e b y t h e m o v e m e n t o f v a c an c i es or i o n s n e a r t h e co n t a c t)$ $φ_{B, O N}, φ_{B, O F F} : ba r r i er h e i g h t s o f t h e tw o s t a t es$ $q, k_{B}, T : e l e m e n t a r y c ha r g e, B o l t z mann co n s t an t an d t e m p er a t u r e$ $(t h e r a t i o i s f or t h er mi o ni c e mi ss i o n w i t h e q u a l p r e f a c t or s; a ba r r i er c han g e o f 0.3 e V g i v es exp (0.3/0.02585) \approx 1.1 \times 1 0^{5} a t 300 K)$

Important Questions

Ques: What is a two-dimensional (2D) material memristor?

It is a resistive switching device in which the switching layer is a monolayer or a few layers of a layered material such as MoS $_{2}$ , h-BN or WSe $_{2}$ . The vertical form is a metal / 2D layer / metal sandwich, a few atoms thick, in which the resistance changes between a high and a low state under voltage (an atomristor). The lateral form is a channel of the 2D material between two electrodes with a gate (a memtransistor), in which the switching is related to the grain boundaries or defects in the film. The thickness of the active layer is below a nanometre in a monolayer, so the switching voltage is low and the cell is very thin, which is the main attraction. Reference: R. Ge et al., “Atomristor: nonvolatile resistance switching in atomic sheets of transition metal dichalcogenides,” Nano Lett. 18, 434 (2018); M. Wang et al., “Robust memristors based on layered two-dimensional materials,” Nat. Electron. 1, 130 (2018).

Ques: What mechanisms are proposed for the switching?

Three are discussed. In the first, metal atoms (such as Ag or Cu from an active electrode) migrate through the 2D layer or along its defects and form a conductive filament, as in an electrochemical metallization cell; the filament length is then the layer thickness. In the second, vacancies (sulfur vacancies in MoS $_{2}$ , boron or nitrogen vacancies in h-BN) are moved by the field and change the local barrier or the conductance, as in a valence change cell. In the third, in a polycrystalline lateral film the vacancies move along the grain boundaries and modulate a barrier there. Which mechanism applies in a given device depends on the electrodes, the layer number and the defect density, and in several reports the mechanism was inferred and not directly observed. Reference: V.K. Sangwan et al., “Gate-tunable memristive phenomena mediated by grain boundaries in single-layer MoS $_{2}$ ,” Nat. Nanotechnol. 10, 403 (2015); R. Ge et al., Nano Lett. 18, 434 (2018).

Ques: How does the gate change a memtransistor?

The channel is a transistor, so the gate sets the carrier density $n_{s} = C_{o x} (V_{G} - V_{T}) / q$ and the resistance of both states according to the channel row. The switching between the states is by a drain voltage that moves the defects, and the gate voltage tunes the resistance of each state and the switching voltage, which gives a third terminal that controls the memory state, for example in a multi-terminal synapse where the gate sets the weight update. The price is the larger area, and the dependence of the resistance on the contact and the grain boundaries. Reference: V.K. Sangwan et al., “Multi-terminal memtransistors from polycrystalline monolayer molybdenum disulfide,” Nature 554, 500 (2018); V.K. Sangwan et al., Nat. Nanotechnol. 10, 403 (2015).

Ques: Why is the contact important in a 2D device?

The contact barrier is a large part of the resistance. A metal evaporated on a 2D semiconductor typically pins the Fermi level (a small pinning factor $S$ ), so the barrier height is almost independent of the metal work function, and a van der Waals contact formed by transferring the metal can reach a pinning factor close to one. The current over the barrier follows the thermionic row, with the $T^{3/2}$ dependence of injection into a two-dimensional channel (a lateral device; a vertical cell crosses the layer instead), and a change in the barrier height $Δ φ_{B}$ by the movement of defects near the contact changes the current by $exp (q Δ φ_{B} / k_{B} T)$ , which can be a large part of the on/off ratio in a lateral device. Reference: Y. Liu et al., “Approaching the Schottky-Mott limit in van der Waals metal-semiconductor junctions,” Nature 557, 696 (2018); A. Allain, J. Kang, K. Banerjee and A. Kis, “Electrical contacts to two-dimensional semiconductors,” Nat. Mater. 14, 1195 (2015).

Ques: What are the open problems of 2D memristors?

The films are made by chemical vapor deposition or by exfoliation and transfer, and the thickness, the defect density and the grain size are not uniform, so the device-to-device and cycle-to-cycle variations are large. Integration with a CMOS process and the uniform growth of a large area are not solved, and the switching mechanism is not agreed. The endurance and retention reported for devices vary from a few cycles to a large number, and they are covered on the reliability pages. The potential advantages are the thin active layer, a low switching voltage, and the possibility to stack the devices. Reference: K. Zhu et al., “The development of integrated circuits based on two-dimensional materials,” Nat. Electron. 4, 775 (2021); M. Wang et al., Nat. Electron. 1, 130 (2018).

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