The current analysis of a single electron transistor based on double graphene nanoscroll island

https://doi.org/10.1016/j.ssc.2021.114234Get rights and content

Highlights

  • A scheme for Single Electron Transistor (SET) utilizing double graphene nanoscroll island is proposed.

  • The impact of the number of islands and island material on coulomb diamonds are studied.

  • Impact of double graphene nanoscroll on coulomb diamonds is compared with other counterparts.

  • The mathematical model is proposed for SET current with double graphene nanoscroll island.

  • The impact of key parameters on the SET current are investigated.

Abstract

The single electron transistor (SET) is nanoscale device which operates fast based on tunneling of single electrons in potential barriers. It has low energy consumption, small size and simplified equivalent circuit. This transistor contains source, gate, drain electrodes and an island which is surrounded with three electrodes. The material of SET island affects on operation of this device therefore graphene as a famous carbon based material is selected for the island. Two islands comprised of graphene nanoribbon (GNR) and graphene nanoscroll (GNS) are chosen for SET. The GNS-SET has lower coulomb blockade region and coulomb diamond area than GNR-SET. Moreover the double GNR-SET and double GNS-SET are simulated and their charge stability diagrams are compared together. The result shows that double GNS-SET has lower zero current region and better operation than double GNR-SET. Therefore double GNS is selected for SET island and impact of important parameters on its current are comprehensively investigated. The increasing applied gate voltage, temperature, GNS length and GNS spiral length lead to current increase in double GNS-SET. Moreover, decreasing the number of turns for GNS increases the current in double GNS-SET. These parameters can be utilized for controlling and tunable current in double GNS SET.

Introduction

The transistor is an electronic device that used for current switching in integrated circuits [1]. The aggressive scaling of transistors based on Moore's law has enabled technology to fabricate nanoscale devices [2,3]. The single electron transistor with nanoscale dimensions works with tunneling of an electron through tunnel junctions [4]. This electron moves from source electrode to the island and then to drain electrode in SET [5,6]. The tunneling phenomena occurs with overcoming coulomb blockade (CB) conditions [[7], [8], [9]]. The speed of electron tunneling depends on mobility of material [4,[10], [11], [12]]. A carbon based materials is graphene that good candidates for SET island [[13], [14], [15], [16]]. The graphene nanoribbon (GNR) is a two-dimensional layer of carbon atoms arranged in honeycomb lattice as shown in Fig. 1. a [17]. Graphene nanoscroll (GNS) is rolled up graphene sheet which can be utilized for SET island as shown in Fig. 1. b [1,18].

The nanostructure of the GNS as well as some parameters are marked and specified on Fig. 1. b where "Nturn" is the number of turns, "L" is GNS length as SET island length [1]. The length of graphene sheet that is rolled up for GNS structure is called GNS spiral length.

The SET can be made from more than one island [19], which SET with double GNS is investigated in this paper. It's charge stability diagrams with GNR, GNS, double GNR and double GNS are simulated with Atomistix ToolKit (ATK) software [18]. Then, current of double GNS-SET is modelled and simulated with MATLAB code. The performance metrics of this SET for different parameters are evaluated in the following.

The single electron transistors have coulomb blockade regions [20]. These regions as coulomb diamonds can be simulated with Atomistix ToolKit software [11,18,21]. Four different islands are designed and simulated for SET under study. The graphene nanoribbon and graphene nanoscroll with 36 carbon atoms are designed with software for single island SET. The double GNS and double GNR are designed for SET islands as well. The charging energy of each island is calculated and presented in Table .1.

The charge stability diagrams of GNR-SET, GNS-SET, Double GNR-SET and Double-GNS SET are plotted in Fig. 2. The colors in Fig. 2-a and 2-b show the number of charge states: blue (0), light blue (1), green (2), and red (3). Also, the colors in Fig. 2-c and 2-d indicate the number of charge states: blue (0), light blue (1), green (2), orange (3), and red (4).

Some parameters are extracted from stability diagrams in Fig. 2 and are presented in Table .2. These data are compared together and the results are written in the following.

The comparison of data in Table .2 shows that SET island significantly affects on coulomb diamond areas and coulomb blockade range. It shows that GNS island has the lower area of coulomb diamonds than GNR island. Double GNS island has smaller coulomb diamonds than single GNS island because increasing number of islands can decrease area of coulomb diamonds and thus improves SET performance [21]. Moreover, double GNR island has smaller coulomb diamond area than GNR island. Therefore double GNS is a good choice for SET island. The impact of important parameters on the current of a double GNS-SET is investigated in the following.

The single electron transistor operates by transfer of single electrons between tunnel barriers. This transfer is based on the quantum mechanical effects. The wave crosses some regions and then electron wave function changes in different regions of SET. The Double GNS-SET is shown in Fig. 3. a and it's tunnel barriers are determined in Fig. 3. b.

These wave functions for these regions can be expressed with solving Schrödinger equations:22md2ψI(x)dx2+(EV)ψI(x)=0x022md2ψII(x)dx2+EψII(x)=00<x<L22md2ψIII(x)dx2+(EV)ψIII(x)=0xLΨI=A1ek1x+B1ek1xو k1=2m(V0E)ΨII=A2eik2x+B2eik2xوk2=2mEΨIII=A3eik3xو k3=2m(V0E)

The boundary conditions is solved for x=0 to x=L as:ΨI(0)=ΨII(0)=A1+B1=A2+B2ΨI(0)=ΨII(0)=k1A1k1B1=ik2A2ik2B2ΨII(a)=ΨIII(a)=A2eik2L+B2eik2L=A3eik1LΨII(a)=ΨIII(a)=ik2A2eik2Lik2B2eik2L=ik1A3ek1L

The transmission coefficient of a GNS island is calculated with:TGNS(E)=11+KGNS2sinh2(k2L)KGNS=(2+tam)E2EgGNS2tamE(EEgGNS)

Then the transmission coefficient of double GNS island will be given by:T(E)=E)(E)(E)GNS)2T(E)=[(1+(4E3ta(t2n12a+3aL123tn12a2)2(t2n12a+3aL123tn12a2)2+4E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)224E3ta(t2n12a+3aL123tn12a2)2(t2n12a+3aL123tn12a2)24E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)2)2×sinh2(L4E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)2))1where "E" is band energy, "T=2.5eV" is the nearest neighbor C–C tight binding overlap energy, "n1","n2" are values of chirality index numbers, "a" is the starting value for θ where θ denotes the GNS the rolling angle. Furthermore "L" is GNS spiral length and "L" is island length. Then, quantum drain current based on the Landauer formalism is given by:I=0ηT(E).F(E)dEF(E)=1exp(EEFkBT)+1where "T(E)" is the transmission coefficient of SET and "F(E)" is the Fermi probability function. "kB" is Boltzmann's constant and "T" is the temperature.

The mathematical model is written for current of double GNS island and can be calculated as:Ids1=0η[(1+(4E3ta(t2n12a+3aL123tn12a2)2(t2n12a+3aL123tn12a2)2+4E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)224E3ta(t2n12a+3aL123tn12a2)2(t2n12a+3aL123tn12a2)24E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)2)2×sinh2(L4E3ta(t2n22a+3aL123tn22a2)2(t2n22a+3aL123tn22a2)2))1.dEexη+1where x=EEgKBT and η=EFEKBT that "E" is energy levels, "EF" is Fermi level of islands and "Eg" is the band gap energy. Other parameters were defined previously. The mathematical model for current of double island GNS-SET is implemented with MATLAB codes. The curves of current vresus gate voltage are ploted in Fig. 4.

The increasing gate voltage increases current of double GNS-SET. Because higher gate voltage decreases island's energy level and an unoccupied energy level locates in the transfer window. The single electron from source electrode tunnels to island and then moves to drain electrode therefore current flows in double GNS-SET. Moreover coulomb blockade range decreases in higher applied gate voltages. High applied gate voltage increases speed of electron transfer in SET. The temperature is another factor which has impact on the SET current which is investigated in Fig. 5.

The analysis of curves in Fig. 5 show that increasing surroundings temprature increases SET current and also decreases coulomb blockade range and zero current region. Rising of temprature increases speed of single electron transfer in double GNS-SET therefore current flows more easily. The impact of GNS length on the SET current is shown in Fig. 6.

The curves of Fig. 6 Indicate that increasing GNS length increases SET current and it decreases coulomb blockade range. Increasing GNS length decreases tunnel barriers length therefore an electron can tunnel to island rapidly. The impact of GNS spiral length on the current is plotted in Fig. 7.

Analysis of Fig. 7 show that increasing GNS spiral length increases SET current and decreases coulomb blockaed range. Because increasing GNS spiral length decreases island length therefore electron can easier tunnel into the potential well.

The analysis of Fig. 8 indicates that decreasing GNS number of turns increases SET current. Therefore more number of turns leads to increased coulomb blockade range. Decreasing number of turns decreases island length and potential well. Therefore an electron can tunnel faster in double GNS-SET.

Section snippets

Conclusion

The single electron transistor is electronic device that can be used in future integrated circuits. The island has direct impact on the SET current. Two islands were selected from graphene nanoribbon and graphene nanoscroll to construct single island and double islands. The charge stability diagrams of GNR, GNS, double GNR and double GNS were compared together. The double GNS-SET showed lowest coulomb diamond area and coulomb blockade regions. Therefore SET operation was improved with double

Statement

This is to certify that all authors have made equal contributions to all of the following: (1) the conception and design of the study, simulations, and analysis of presented data, (2) drafting the article and revising it critically for important intellectual content, (3) final approval of the version to be submitted.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

Authors would like to thank the Research Management Center (RMC) of Universiti Teknologi Malaysia (UTM) for providing an excellent research environment in which to simulate this research by Atomistix ToolKit and to complete this work.

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