# Non-linear model of nanoscale devices for memory application

## Abstract

COMSOL multiphysics software based model has been developed for the mem-devices comprising of undoped and doped CdSe/starch quantum dots and CdS/PVA nanocomposites as active layer. The assembly of quantum dots/nanocomposites can be represented by an equivalent structure comprising of almost infinitely alternating repetition of building blocks, each block having conducting and non-conducting regions. The time-dependent inductance (L) along with time-dependent resistance (R) and capacitance (C) are used as model input and the solutions are obtained using semiconductor, electric circuit and ordinary differential equation module. From this study it is clear that the mem-behaviour of the as-fabricated nanodevices having \(\frac{{R_{OFF} }}{{R_{ON} }} > 10\) can be well explained by the time-dependent R, C and L features of the nanoparticle assembly adopting COMSOL Multiphysics software. However, for devices with \(\frac{{R_{OFF} }}{{R_{ON} }}\) < 10, hysteresis behavior is governed by only time-dependent R and C features. As higher (> 10) \(\frac{{R_{OFF} }}{{R_{ON} }}\) values enhance efficiency of memory units, the present model incorporating time-dependent L in addition to time-dependent R and C will be useful for optimization in the device design for application as memory units.

## Keywords

Quantum dot Time-dependent inductance R_{OFF}/R

_{ON}ratio Memory devices Modeling and simulation

## PACS Nos.

85.30 De 73.21 La 73.43 cd 02.70-c 07.05.Tp## 1 Introduction

For over 150 years resistors, capacitors and inductors were believed to be the only three fundamental passive circuit elements. But, in 1971, Prof. Leon Chua [1] of the department of Electrical Engineering and Computer Science, University of California, USA, reasoned from symmetry arguments that there should be a fourth fundamental circuit element which he christened as “memristor” (acronym for memory resistor). Resistance of the memristor called memristance relates charge (*q*) and flux (*φ*) given by \(M = d\varphi /dq\). In case of linear elements in which memristance (\(M\)) is a constant, memristor is identical to conventional resistor. However, when M itself is a function of charge \(q\) yielding a nonlinear circuit element, then no combination of resistive, capacitive and inductive circuit elements can duplicate the properties of a memristor. In an attempt to make possible the continuation of Moore’s law [2] for at least another decade, Strukov et al. of HP Labs [3, 4] in 2008 observed memristive behavior in TiO_{2} thin films. Thus, it took about four decades for practical realization of this basic element. The reason behind silence of about four decades for practical realisation is that the researchers were searching memristive characteristics in some wrong places, as the fact that nanostructures are the only possible candidates having potential of memristive behaviour was yet to be established then.

The idea originated by Strukov et al. for observing memristive behavior in semiconductor thin films was extended further towards magnetic nanoparticles by Kim et al. in the year 2009 [5]. Kim and co-researchers investigated the electrical properties of Quantum Dot (QD) assemblies consisting of an infinite number of monodispersed, crystalline magnetite (Fe_{3}O_{4}) particles. They observed that assemblies of Fe_{3}O_{4} nanoparticles with sizes below 10 nm exhibit a voltage-current hysteresis with an abrupt and large bipolar resistance switching characteristics (\(R_{OFF} /R _{ON} \approx 20\)). According to them, this type of behaviour could be interpreted by adopting an extended memristor model that combines a time-dependent resistance and a time-dependent capacitance. They also observed that such behaviour is not only restricted to magnetites, but also to other nanoparticle assemblies. The observations along with supporting modeling and simulations presented by Kim et al. regarding the switching behaviour of nanoparticle assemblies can be regarded as the framework for devising new applications to a wide range of electronic devices. Recent in-depth theoretical works on hybrid systems comprising of Quantum Dots coupled to metallic nanoparticles encourage one to carry out investigation on systems having arrays of metal doped Quantum Dots for achieving some novel characteristics like Mem – behaviours [6, 7, 8, 9, 10].

Many physical devices have been presented so far by various researchers exhibiting qualitative characteristics that point to a memristor (e.g. nonlinearity, memory, hysteresis, zero crossing etc.), using nanocomposites [3, 4, 5, 11, 12, 13, 14, 15]. Apart from the memristive behavior, mem-capacitive as well as mem-inductive behaviors have also been experimentally observed recently [13]. Experimental findings reveal that by changing different parameters (e.g. active layer material, doping material and concentration, pH value, molarity etc.) and environment (e.g. capping agent, temperature etc.) of the nanocomposites or semiconductor quantum dots (QDs), there are possibilities for obtaining prominent changes in their mem-behavior.

The memristor model of Chua [1] was constructed by Biolek et al. in MATLAB environment providing the solution of Ordinary Differential Equation [16]. Modifications of the initial equations of HP model were done by adding a parameter η which indicates the polarity of the memristor [17]. Panayiotis et al. [18], in their theoretical study, identified the conditions under which ideal memristors comply with reciprocity theorem of classical circuit theory. A hardware correlated model developed by Pino et al. showed a very good fit to the I–V characteristics of the practical device developed at Boise State University [19]. Biolek et al. showed that the voltage and current of ideal memristors are under all circumstances governed by an ordinary differential equation of first order which can be of Bernoulli, Riccati, Abel type or of different first order type [20].

It is worth-noting that, most of the models put forward till date are based on the constitutive differential equations based on voltage and resistance and current and capacitance. Regarding theoretical works on mem-behavior of nanocomposites, although there are adequate as well as concrete works (both theoretical and experimental) on frequency-dependent R, L and C behaviour of nanocomposites [21], only time-dependent R and C features and parasitic (i.e. time-independent) inductance [22] of nanocomposites have been taken into account till date for mem-behaviour modeling of nanocomposites. Thus, for a complete physical description of the mem-system one can design a differential equation incorporating time-dependent inductance along with time-dependent resistance and capacitance as governing equation of mem-model.

In this paper, modeling and simulation adopting COMSOL Multiphysics software for the observed mem-behavior of the undoped and doped CdSe/Starch QD devices fabricated by the present group and of CdS/PVA nanocomposite devices [13] are presented. Single doping as well as co-doping of varying concentrations is taken as controlling factors for observing significant mem-behavior in the present investigation. Time-dependent inductance in addition to time-dependent resistance and capacitance of the systems are incorporated in the modeling and simulation presented here.

## 2 Materials and experimental method

### 2.1 Chemicals and materials

All the chemicals and materials purchased are of high purity (99%). Cadmium Acetate (Cd(CH_{3}COO)_{2}), Selenium metal Powder (black 99%), Sodium Sulphite(Na_{2}SO_{3}), Manganous Chloride (MnCl_{2}, 4H_{2}O), Cupric Sulphate (CuSO_{4}, 5H_{2}O), Ammonia Solution and Distilled water are used as chemicals for the synthesis of undoped CdSe and Mn/Cu/Mn:Cu doped CdSe nanoparticles. Starch is used as the capping agent. All chemicals are used directly without any purification.

### 2.2 Preparation of undoped and doped CdSe nanoparticles

The samples consisting of undoped and doped CdSe (Mn/Cu/Mn:Cu-codoped) nanocomposites are synthesized by the chemical bath deposition technique. For the formation of undoped CdSe nanoparticles, 3.7812 g (0.1 M) of sodium sulphite is dissolved in 300 ml of distilled water. 1.5 gm of selenium metal powder is added to this solution and the whole solution is stirred for 8 h at a temperature of 80°C and then cooled down to room temperature. In an another beaker, 20 ml of 3% starch solution is taken and cadmium acetate (0.534 g, 0.1 M) is mixed with it by stirring at 40 °C. Ammonia solution is then added until the PH value becomes > 9.0. Finally, 20 ml of freshly prepared Na_{2}SeSO_{3} solution is added drop by drop to this solution under constant stirring rate at a temperature of 65 °C. After 20 min of stirring, the solution becomes chocolate brown in colour, which indicates the formation of CdSe nanoparticles. This sample is coded as S1.

In order to obtain Mn/Cu doped CdSe 0.004gm, 0.008, 0.016, 0.0267, 0.0534, and 0.08 g of MnCl_{2}/CuSO_{4} are added to cadmium acetate (0.534gm, 30 ml) solution to prepare two sets of 0.5, 1, 2, 5, 10 and 15% Mn/Cu doping samples (S2/S8,S3/S9,S4/S10,S5/S11,S6/S12,S7/S13, respectively). Then the steps same as adopted to prepare undoped CdSe samples are followed to get the final solutions which are of reddish brown colour for Mn doped and grayish blue for Cu doped solutions. It is observed that the colours of the samples get slightly changed with change in doping percentage.

_{2}and CuSO

_{4}(equal amount) are added to 20 ml of cadmium acetate (0.534gm) solution for obtaining 5% (2.5%Mn + 2.5%Cu), 10%(5%Mn + 5%Cu) and 15%(7.5%Mn + 7.5%Cu)co-doping respectively. The steps same as the previous ones are then followed to get the respective final CdSe co-doped samples (S14, S15, S16). Scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDX), transmission electron microscopy (TEM), high resolution TEM (HRTEM), selected area energy dispersion (SAED) images for a few as-synthesized samples are shown in Figs. 1, 2.

### 2.3 Device Fabrication and observed mem-behaviour

_{OFF}/R

_{ON}which are summarized in Table 1, I–V curves being given elsewhere [23]. R

_{ON}and R

_{OFF}are the limit values of the memristor resistance for thickness maximum i.e. maximum doped region and thickness minimum i.e. minimum doped region [24].

\(\frac{{R_{OFF} }}{{R_{ON} }}\) for all the CdSe samples of present work

Sample code | Device code | \(\frac{{R_{OFF} }}{{R_{ON} }}\) | Doping profile |
---|---|---|---|

S1 | M1 | 14 | Undoped |

S2 | M2 | 0.9 | 0.5% Mn doped |

S3 | M3 | 1.1 | 1% Mn doped |

S4 | M4 | 4.5 | 2% Mn doped |

S5 | M5 | 1.3 | 5% Mn doped |

S6 | M6 | 1.5 | 10% Mn doped |

S7 | M7 | 19.5 | 15% Mn doped |

S8 | M8 | 19 | 0.5% Cu doped |

S9 | M9 | 5.8 | 1% Cu doped |

S10 | M10 | 3.3 | 2% Cu doped |

S11 | M11 | 1.6 | 5% Cu doped |

S12 | M12 | 41 | 10% Cu doped |

S13 | M13 | 51.8 | 15% Cu doped |

S14 | M14 | 12 | 5%(Cu + Mn) |

S15 | M15 | 5 | 10%(Cu + Mn) |

S16 | M16 | 23 | 15%(Cu + Mn) |

The active layer of an as-fabricated device can be considered as an assembly of CdSe QDs separated by dielectric (PVA matrix). Current conduction through the QDs may be either Resonant Tunneling (RT) or Coulomb Blockade (CB). RT can be ruled out for the observed behavior as no negative differential resistance (NDR) feature is observed in the I–V characteristics. The CB effect supports the ON/OFF switching features of the characteristics on the basis of charge trapping in CdSe QDs. If the QD is charged with an electron, the tunneling path through it gets effectively shut off by the Coulomb charging energy. These trapped charges form space charges, which lower the external electric field due to induced internal field lead to blocking of further electron injection. As a result the devices make a transition from Low Resistance State (LRS) to High Resistance State (HRS).On reversing the bias, trapped charges are released and internal field disappears, resulting in transition from HRS to LRS. The observed asymmetry in the I–V charactistics may be attributed to the Schottky controlled conduction in the device [23].

Observed mem-behaviors of CdS nanocomposites are presented elsewhere [13].

## 3 Modeling and simulation

### 3.1 Theoretical model

- 1.
The Quantum Dot assembly is an 1D array of Quantum Dots having doped and undoped charged carrier regions with a moving boundary in between.

- 2.
The system, being comprised of almost infinitely alternating repetition of conducting and non-conducting regions, possesses time-dependent resistance

*R(t*), time-dependent capacitance*C(t)*and time-dependent inductance*L(t).* - 3.
Considering possibility of two types of charge carrier ions, an additional time-dependent capacitance

*ΔC (t)*across the particle boundaries is assumed to exist, as suggested by Kim et al. [5].

Finite Element Method based COMSOL Multiphysics software is used to model the memory device using the semiconductor module. The model is based on the devices adopted in the experimental investigations considered for comparison (present work and [13]).

### 3.2 Simulation

- 1.The module used in this study is
- (i)
Semiconductor

- (ii)
Electric Circuit

- (iii)
Ordinary Differential Equation (ODE)

- (i)
- 2.
The resistance and capacitance in the present model are incorporated as proposed by Kim et al [5].

- 3.R
_{ON}and R_{OFF}values calculated (Table 2) from the experimental I–V curves for different mem-devices are used as model input in our simulation.Table 2R

_{OFF}/R_{ON}values calculated from experimental I-V curves for different mem-devicesDevice code

Experimental group

Active material of the device

R

_{OFF}/R_{ON}\({\text{M}}_{{{\text{K}}0}}\)

Kim et al [5].

Fe

_{3}O_{4}nanoassembly (QDs of size 7,10,12 nm)20

M1

Present work

CdSe undoped (QDs of size 6 nm)

14

M5

Present work

CdSe: 5% Mn doped (QDs of size 6.8 nm)

1.3

M7

Present work

CdSe:15% Cu doped (QDs of size 6.84 nm)

19.5

M8

Present work

CdSe: 0.5% Cu doped (QD of size 9.38 nm)

19

M9

Present work

CdSe: 1% Cu doped (QD of size 9.25 nm)

5.8

M13

Present work

CdSe: 15% Cu doped (QD of size 6.9 nm)

51.8

\({\text{M}}_{{{\text{S}}0_{1} }}\)

Sarma et al. [13]

CdS: 0.2% Cu doped (QDs of size 4.7 nm)

49

\({\text{M}}_{{{\text{S}}0_{2} }}\)

Sarma et al. [13]

CdS undoped (QDs of size 2.3 nm)

6

- 4.
The capacitance at ON state (C

_{ON}) is calculated using the parallel plate capacitor formula. For CdSe Quantum Dot, relative permittivity \({{\upvarepsilon }}_{\text{r}}\) = 4.5 [25].*D*is taken to be 2 nm and the device length (*d*) is taken in the range of (10–300) nm. For*d*= 10 nm, Area*A*= 1e−7 cm^{2}and for*d*= 300 nm,*A*= 0.3e−7 cm^{2}. - 5.
Mobility \(\mu = 0.02e^{ - 8} {\text{cm}}^{2} /{\text{Vs}}\) [26].

- 6.
The source used is sine source with applied voltage

*V*=*V*_{0}*(sinωt)*or applied current*I*=*I*_{0}*(sinωt).* - 7.
Current and flux are the two constitutive variables that govern the mem inductance. Here,

*L(W, I, t)*>*0*. An ordinary differential equation module is incorporated and the inductance is calculated using the concept of a metallic plate. The length of the plate is the as-defined*d*. - 8.The doped region width (
*W*) varies with flow of charge. The dopant type is n type. The inductance in the ON condition (L_{ON}) of the device is given by [17]where; permeability of free space \({{\upmu }}_{0 } = 4{{\uppi }} \times 10^{ - 7} {\text{H}}/{\text{M}}\)$${\text{L}}_{\text{ON}} = \frac{{{{\upmu }}_{0 } * {\text{d}}}}{4}$$(1)For, d = 300 nm, \({\text{L}}_{\text{ON}} = 3.768{\text{e}}^{ - 7}\) H.

- (9)The width (
*W*) of the doped region gets modulated depending on the amount of electric charge passing through the device. With electric current passing in a given direction, the boundary between the two regions is moving in the same direction. The inductance of the nanoassembly which is a function of*(W,t)*can be obtained from the state equation given by,$$I\left( t \right) = u^{ - 1} (W,\varphi ,t)\varphi (t)$$(2)Solving Eq. (3)$$\frac{du}{dt} = - f\left( {u,\varphi ,t} \right)*\frac{(D - W(t))}{D}$$(3)$$\frac{D}{(D - W(t))}\left[ {\frac{\partial u}{\partial t} - u_{0} } \right] = - u(W,t)$$(4)$$or, u\left( {W,t} \right) = \frac{D}{(D - W(t))}\left[ {u_{0} - \frac{\partial u}{\partial t}} \right]$$(5)

This time-dependent *L* is incorporated in the model and an ODE is distributed in all the directions as specified boundary conditions. With increase of time, inductance increases linearly with time and admit a steady state solution, which agrees well with Ventra et al. [11].

## 4 Results and discussion

*R(t)*and

*C(t).*The results obtained are given in Fig. 5 along with Kim et al.’s [5] simulation result. The nice agreement between the two results reveals correctness of the basic steps followed in the present model.

_{K0}and M1 whereas only \(I - V\) curves are given for some of the devices of Table 2 in Figs. 8, 9, 10, 11. It is worth-noting that the I-V hysteresis behaviour is dependent on the size of the nanoparticles.

Figure 6d shows that Kim et al.’s experimental curve agrees more with the present model (which incorporates the additional parameter *L(t)*) than with Kim et al.’s model where *L(t)* is not considered. Moreover, the obtained characteristics for devices \(M_{K0}\), M1, M7, M8, M13 and \(M_{{S0_{1} }}\) (having \(\frac{{R_{OFF} }}{{R_{ON} }} > 10\)) can be well explained by the present model with *L(t),* whereas obtained characteristics for devices M5, M9 and \(M_{{S0_{2} }}\)(having \(\frac{{R_{OFF} }}{{R_{ON} }} < 10\)) are well explained by the present model without *L(t)*. This result reveals that for devices having low \(\frac{{R_{OFF} }}{{R_{ON} }}\) values (< 10), effect of time-varying inductance is not significant. As for efficient memory devices, \(\frac{{R_{OFF} }}{{R_{ON} }}\) values should be high enough (> 10) [27], so, the present model will be useful for optimization in the design parameters of mem-devices for application as memory units.

It is worth-mentioning that in order to model non-linear nanodevices in which unconventional and exotic quantum mechanical phenomena e.g. tunneling, Coloumb blockade, single-electron dynamics, Kondo effects etc. govern the electron transport properties, it is necessary to introduce some non-linear elementary circuit elements which play the same role as the set of basic vectors used to define a vector space [28]. Such specific circuit elements adopted in the present model are time-dependent R, C and L. Hence, these circuit elements can be regarded as the basic components of the equivalent circuit representation of non-linear nano devices.

## 5 Conclusion

COMSOL Multiphysics software based model have been developed for the mem-devices comprising of undoped and doped CdSe/Starch quantum dots and CdS/PVA nanocomposites as active layer. The time-dependent inductance along with time-dependent resistance and capacitance are used as model input and the solutions are obtained using semiconductor, electric circuit and ordinary differential equation module. From this study it is clear that the mem-behaviour of the as-fabricated nanodevices having \(\frac{{R_{OFF} }}{{R_{ON} }} > 10\) can be well explained by the time-dependent R, C and L features of the nanoparticle assembly adopting COMSOL Multiphysics software. However, for devices with \(\frac{{R_{OFF} }}{{R_{ON} }}\) < 10, hysteresis behavior is governed by only time-dependent R and C features. As higher (> 10) \(\frac{{R_{OFF} }}{{R_{ON} }}\) values enhance efficiency of memory units, the present model incorporating time-dependent \(L\) in addition to time-dependent \(R\) and \(C\) will be useful for optimization in the device design for application as memory units.

## Notes

### Acknowledgements

The First Author J. Devi would like to acknowledge Department of Science and Technology, Govt. of India and third author S. Sarma would like to acknowledge University of South Africa, South Africa.

### Funding

This study was funded by Department of Science and Technology, Govt. of India (Grant Number SR/WOS-A/ET-1102/2015).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- [1]
- [2]
- [3]
- [4]
- [5]T H Kim, E Y Jang, N J Lee, D J Choi, K-J Lee, J Jang, J Choi, S H Moon and J Cheon
*Nanoletter*s**9**2229(2009)ADSCrossRefGoogle Scholar - [6]
- [7]F Carreño, M A Anton, S Melle, O G Calderon, E Cabera-Granado, J Cox, M R Singh and A Eqatz-Gonez
*J. Appl. Phys.***115**064304 (2014)ADSCrossRefGoogle Scholar - [8]
- [9]M R Singh, M C Sekhar, S Balakrishnan and S Masood
*J. Appl. Phys.***122**034306 (2017)ADSCrossRefGoogle Scholar - [10]M R Singh, J Guo, J M J Cid and J E D H Martinez
*J. Appl. Phys.***121**094303(2017)ADSCrossRefGoogle Scholar - [11]
- [12]
- [13]
- [14]R Bhadra PhD thesis titled: Synthesis and characterization of some Semiconductor nanocystallites with emphasis on quantum dots for application in electronics (2009)Google Scholar
- [15]
- [16]
- [17]
- [18]P S Georgiou, M Barahona, S N Yaliraki and E M Drakakis
*Microelectron. J.***45**1363 (2014)CrossRefGoogle Scholar - [19]R E Pino, J W Bohl, N McDonald, B Wysocki, P Rozwood, K A Campbell, A Obela and ATimilsina et al.
*IEEE/ACM Int. Symp. Nanoscale Archit.***1**(2010)Google Scholar - [20]
- [21]
- [22]V Mladenov and S Kirilov ISTET 2013:
*International Symposium on Theoretical Electrical Engineering*: Pilsen, Czech Republic, p. II-13–II-14 24th–26th June 2013Google Scholar - [23]
- [24]
- [25]
- [26]D Yu, B L Wehrenberg, P Jha, J Ma and P Guyot-Sionnesta
*J. Appl. Phys.***99**104315 (2006)ADSCrossRefGoogle Scholar - [27]
- [28]