Electrical Characteristics of the Carbon Nanotube Field-Effect Transistors With Extended Contacts Obtained Within ab-initio Based Model

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1 2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) Eletrial Charateristis of the Carbon Nanotube Field-Effet Transistors With Extended Contats Obtained Within ab-initio Based Model Artem Fediai, Dmitry A. Ryndyk, Gianaurelio Cuniberti Institute for Material Sienes and Max Bergmann Center of Biomaterials, Center for Advaning Eletronis Dresden, Dresden Center for Computational Material Siene, TU Dresden, Dresden, Germany. Abstrat In our previous work the model of the eletron transport in CNTFETs with extended ontats was developed for equilibrium ase using ombination of the density-funtional theory and equilibrium Green funtions formalism. Here we extrapolate it to the ase of the arbitrary biased CNTFET using simplest thinkable assumptions, whih allows us to alulate drain urrent as a funtion of the gate and drain potentials. For Al and Pd eletrodes we show qualitative agreement of our model with the existing experimental results both in terms of polarity of the devie (p- or n-type FET) and dependene on the ontat length. Most of the predition justified by experimental data were done at ab-initio level for the first time. Keywor arbone nanotube; Green funtion formalism; density funtional theory; ontat I. INTRODUCTION Carbon nanotube field effet transistors (CNTFETs) attrat the interest of the sientifi ommunity as a possible substitution of the onventional MOSFETs. In a CNTFET a hannel is onstituted by a single or multiple arbon nanotubes, whih are subjeted to the field of the gate eletrode and onneted to the drain and soure metalli eletrodes. The main expetations from the CNTFET tehnology are attributed to the superior eletron 5 2 mobility ( > 10 m Vs ) [1] of a stand-alone CNT. As a hannel of the CNTFET, however, CNT should neessarily be onneted to the soure and drain eletrodes. This ontat is the greatest obstale in reahing the theoretial maximum of eletron mobility in CNTs. As opposed to onventional eletronis, metal- CNT ontat resistane annot be eliminated by the doping of the hannel under the eletrode as the doping of the CNT is (at least up to now) very hallenging. Nevertheless, in 2003 the first CNT- FET with ohmi ontats was demonstrated. Generally, it is was not lear from the theoretial works, how the ohmi ontat an be established, sine it is usually not possible to reah that without the doping in onventional eletronis. The possible answer was given by Neme et.al. [2], [3], and in 4 years a remarkable onfirmation has been obtained within experimental works of Franklin et.al. [4]. The main point of the works [2], [3] is the onsideration of the realisti CNT-metal ontats, namely the ase when the CNT is resting on the metalli eletrode or is embedded into it. Within the toy model as well as the model partially based on DFT alulations it has been shown that the urrent an flow into the hannel along the onsiderable length of the CNT segment embedded into the metal, and not only in the plae, where the CNT hannel enters an eletrode. In ase of embedded geometry and Pd eletrode this length reahes an order of 100 nm. Our previous work [5] is a development of the work of Neme et.al. [2], [3], and allows to perform systemati treatment of the ontat between the CNT any an eletrode omposed of any metal with diret alulation of the ballisti eletron transport based on DFT Kohn-Sham Hamiltonian. By our model [5] for the first time all the relevant details of the metal-cnt ontats are taken into aount, namely doping of the CNT by the metal and the strength of the eletrial bonding. This, however, was done for equilibrium situation, i.e. all gate, soure and drain eletrode have the same potentials. This makes it possible to make some assumption about the non-equilibrium state, but not to ompare simulation results with the experimental data, whih are usually represented by the output and transfer harateristis, transondutane, ontat resistane and their dependenes on the ontat length. The aim of the urrent work is to use existing model of the CNT-metal ontat in equilibrium state to build the simplified model of the eletron transport in nonequilibrium state. The end goal is to obtain observable quantities and dependenes, whih in this ase are stati eletrial harateristis. Then we ompare obtained eletrial harateristis with the existing experimental results to show their qualitative agreement. This ould be viewed as a proof of onsisteny of the whole method, developed in [5]. Most of our onerns in the present work are related to the dependene of the CNTFET eletrial harateristis on the ontat length, and this is not aidentally. As it is mentioned in [6], the target ontat length of the ontat should be from 7 to 14 nm till the year 2020 to preserve the urrent node trend. It is thus important to be able to predit eletrial properties not only for the infinite ontat length, but also saling to the /15/$ IEEE 261

2 2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) target lengths mentioned. Till now our approah [5] is the only one, whih performs orrespondent alulation ompletely at the ab-initio level; we use no fitting parameter to agree with experiment. As we do not mean to study influene of the partiular geometry upon the ontat, but onentrate ourselves at the main tren related to the material of the ontat and its length, we aept the simplest approximation to take the biases "drain-soure" and "gate-soure" into aount. This also makes the alulation time for eah bias point reasonable, enabling alulation of the whole I-V harateristis. alulation we know Hamiltonian of the nanotube and souredrain eletrodes at zero voltage. We searh for the simplest way to take the fiel reated by the eletrodes under arbitrary voltages into aount. II. PREREQUISITES TO CALCULATE CURRENT-VOLTAGE CHARACTERISTICS: ELECTRON TRANSPORT IN EQUILIBRIUM In this setion we summarize the results of our method developed to alulate eletron transport in CNTFET in equilibrium [5]. The method is based on density funtional theory as implemented in module "Quikstep" of CP2K [7]. We have used DFT as a mean to build the Hamiltonian of the atomi system losely resembling experimental geometry of the CNTmetal ontat (Fig. 2). Namely, we an distinguish in this struture (1) embedded segment of the CNT with arbitrary length L, (2) a hannel whih is in fat unovered CNT portion whih has the length L h, (3) infinite eletrodes omprised of the metal. To build the Hamiltonian of the system (Fig. 2) in the basis of loal atomi orbitals we have simulated three finite systems, two of whih were used to take the influene of the metal on embedded portion of the CNT into aount; Hamiltonian of the third system served to build the Hamiltonian of the CNT itself (both free and embedded portions). This information is available in [5]. What is important in the urrent work, whih view has an effetive Hamiltonian of the system depited in Fig. 3, a after we have deimated the metal (see Fig. 3,b). Then the system an be brought to the view (Fig. 3,) by deimation the part, whih has left after metal deimation. Two parameters (whih are matrix funtions) σ ε and σ t in Fig. 3,b is alulated in [5]. Green funtion of the system depited in Fig. 3, is the same as of the orrespondent portion of the initial system. Transmission oeffiient through the hannel of the CNTFET of the system from Fig. 3, is the same as for those in Fig. 3,a. Parameter σ depen on the length of the ontat, so does T( E ). Funtion T = T( E, L ) is the data from equilibrium alulations whih we use in this work to alulate drain urrent under arbitrary drain and gate potentials. Transmission oeffiient as funtion of energy is depited in Fig. 1 for several ontat lengths. The ontat length ( L ) is expressed in terms of the number of CNT unit ells ( N ). III. CALCULATION OF THE DRAIN CURRENT WITHIN NEGF FORMALISM Our intention is to onstrut an approximate model, whih allows alulating drain-soure urrent I for any drain-soure ( ϕ ) and gate-soure ( gs ) ϕ voltages. From ab-initio Fig. 1. Transmission oeffiient as a funtion of energy for different ontat lengths given in a number of CNT unit ells N for Pd and Al ontats. Fig. 2. Overall system of interest inludes two embedded ontats and free CNT (16,0) segment in between. Transverse setion of the ontats is infinite, whereas the ontat length L is finite and arbitrary. Within Landauer-Buttiker formalism drain urrent an be alulated as follows: 2e + I = T( E, ϕgs) ( fs( E, ϕ) fd ( E, ϕ)) de, h f ( E) = f ( E U ) (1) s( d) F s( d) where ff ( E ) is a Fermi funtion with E F =

2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) Aording to our assumption, drain-soure voltage ϕ affets only Fermi level position in soure and drain eletrodes, namely:

Gate-soure voltage, in turns, alters transmission oeffiient in non-equilibrium state T as follows: T( E, ϕgs) = T( E eϕgs) (equilibrium oeffiient depen also on ontat length, so does non-equilibrium

They must apture the main tren in dependene of drain urrent on eletrode potentials and on ontat length as drain-soure drops indeed mainly at the ontats; we expet, however, underestimation of the

yiel its degradation). or ambipolar), dependene of the ontat resistane on ontat length and material.

Transfer harateristis We have simulated transfer harateristis of the CNTFET [8] with 9 nm hannel represented by one semionduting CNT with unknown hirality and diameter lose to 1.3 nm.

To predit experimental data hereafter we have used CNT with hirality (16,0), whih has a diameter of approximately 1.25 nm and hannel length 9.4 nm.

Pd eletrodes in [8] and other experimental works yield always p-type FET.

3 2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) Aording to our assumption, drain-soure voltage ϕ affets only Fermi level position in soure and drain eletrodes, namely: Us = eϕ 2 and Ud = eϕ 2. Gate-soure voltage, in turns, alters transmission oeffiient in non-equilibrium state T as follows: T( E, ϕgs) = T( E eϕgs) (equilibrium oeffiient depen also on ontat length, so does non-equilibrium one). Aepted assumptions mean that we neglet voltage drop at the hannel of the tube and assume that injetion properties of the ontats do not depen on bias voltage. They must apture the main tren in dependene of drain urrent on eletrode potentials and on ontat length as drain-soure drops indeed mainly at the ontats; we expet, however, underestimation of the subthreshold swing as well (reason: our assumption suppose perfet gate ontrol), and overestimation of the drain urrent (non-zero gate voltage usually not only shifts transmission oeffiient, but also yiel its degradation). or ambipolar), dependene of the ontat resistane on ontat length and material. Due to perfetion of the simulated system geometry and aepted assumptions, our results will overestimate drain urrent in partiular and performane of the CNTFET in general. A. Transfer harateristis We have simulated transfer harateristis of the CNTFET [8] with 9 nm hannel represented by one semionduting CNT with unknown hirality and diameter lose to 1.3 nm. Experimental sample has long Pd ontats (longer than the "effetive" length for Pd eletrode). To predit experimental data hereafter we have used CNT with hirality (16,0), whih has a diameter of approximately 1.25 nm and hannel length 9.4 nm. Comparison of our alulations with the existing experiment (Fig. 4) shows that our model orretly predit polarity of the CNTFET with Pd eletrodes (p-type FET). Pd eletrodes in [8] and other experimental works yield always p-type FET. Overestimation of the alulated urrent magnitude and underestimation of the sub-threshold swing log( dv di ) were expeted (see above); as V inrease so do disrepanies, beause the basis for our alulation is stationary state, and as we go farther, our assumptions beome rougher. gs Fig. 3. Initial system, whih onsist of infinite metal eletrode and a CNT (a) is represented by effetive system whih ontains only arbon atoms and metal self-energies (b), whih, in turns, deimates till unovered CNT portion and self-energy affeting only the neighbouring CNT unit ell () IV. RESULTS AND COMPARISON WITH EXISTING EXPERIMENTAL DATA When ompare experimental and simulated by means of our model eletrial harateristis of the CNTFET, it should be noted that theoretial results are obtained having as the input quantities only oordinates of the orrespondent arbon and metalli atoms; no fitting parameters were aepted neither during DFT alulation nor at NEGF stage of alulations; we have in mind also general problem in DFT with orret predition of the band gaps. In this work we try to apture the main tren, previously inaessible at ab-initio level, namely: preferable polarity of the CNTFET (p-, n-type Fig. 4. Transfer harateristis of CNTFET with hannel of 9 nm and CNT with diameter of 1.3 nm. Eletrode material: Pd. Experimental data were adapted from [8] B. Output harateristis depending on ontat length We have ompared our alulations with the output harateristis of the set of CNT CNTFET [4] with hannel length 40 nm and different ontat lengths: 20, 30, 50, 70 and 100 nm. Output harateristis in both ases were measured/alulated at V whih is V larger that the threshold voltage. Comparison of our results with the experimental data shows that we an predit urrent growth with the growth of the ontat length. Theoretial output harateristis losely resembles Ohm s low due to very simple assumption about the eletrostatis of the devie. Overestimation of the theoretial value of the drain urrent is also expeted. Output harateristis are represented in Fig. 5. Experimental data are taken from Fig.3 of [4]. 263

2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) Fig. 6. Transondutane of the CNTFET [4] with Pd eletrode (theory vs. experiment). Fig. 5.

TFET [4] with different ontat length L. Material of the eletrodes: Pd. C.

It defines both in experiments and theory as: gm = di dvgs for the voltages higher than a threshold voltage.

Condution magnitude differene between theoretial and experimental data is kept to be 1 order as in the ase of the drain urrent, whih is expeted. D.

Experimental data are taken from Fig.4 (b) of [6].

exponentially. Experimentally, ontat resistane almost independent on drain voltage up to the work point [6] (whih is itself an evidene of ohmi ontat).

4 2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) Fig. 6. Transondutane of the CNTFET [4] with Pd eletrode (theory vs. experiment). Fig. 5. Experimental and simulated harateristis of 40-nm hannel CNTFET [4] with different ontat length L. Material of the eletrodes: Pd. C. Transondutane as a funtion of the ontat length We have alulated the transondutane as a funtion of the ontat length for the CNTFET [4] from the previous subsetion (Fig. 6). It defines both in experiments and theory as: gm = di dvgs for the voltages higher than a threshold voltage. We an see that both experimental and theoretial dependenes show that transondutane ten to inrease till some saturated magnitude as ontat length inreases. Condution magnitude differene between theoretial and experimental data is kept to be 1 order as in the ase of the drain urrent, whih is expeted. D. Influene of the Contat length on ontat resistane We have defined a ontat resistane as a derivative R= 12 di dv. A work point for resistane definition was hosen at V = 0.5V, Vgs = 0.3V Fig. 7. Experimental data are taken from Fig.4 (b) of [6]. We an see that both theoretial and experimental urves behaves similarly: there is some effetive ontat length, after whih resistane stops to derease; as we approah zero ontat length resistane grow up exponentially. Experimentally, ontat resistane almost independent on drain voltage up to the work point [6] (whih is itself an evidene of ohmi ontat). As our assumptions are justified better at low voltages, we observe in Fig. 7 quantitative agreement with experimental dependene of R ( L ). E. Comparison of the CNTFETs with Pd and Al ontats In our previous work [5] we have disovered that embedded into metalli eletrode tube portion in ase of Pd eletrode is being metal-doped whereas in ase of Al is fully metallized (no band gap). Our alulations (Fig. 8) show, whih onsequenes it has from the point of view of the ontat resistane, whih has been alulated in the same manner as desribed above. Aluminium ontats are not sensitive to the length of the ontat up to ontat length lose to 1 nm in ontrast to Pd ontats. As expeted, Al ontat has several orders higher resistane than Pd one, as high refletion take plae between drastially modified portion of the tube and the unovered tube portion. Fig. 7. Dependene of the ontat resistane on ontat length for Pd ontat. Experimental data are taken from [6]. In Fig. 9 we finally present transfer harateristis for both Pd and Al ontats. The main differene between CNTFET with Pd ontats and that with Al ontats is a polarity of the devie: Pd ontats yield p-type FET; Al ontats yield ambipolar FET with the signs of n-type polarity. In experimental set-ups Alontated CNTFET usually show more prominent n-type behaviour [9], although, there is no results on Al ontats to ompare diretly with our model system. Fig. 8. Dependene of the ontat resistane on ontat length: omparison between Pd and Al ontats Fig. 9. Calulated transfer harateristis for Al- and Pd-ontated CNTFETs 264

5 2015 IEEE 35th International Conferene on Eletronis and Nanotehnology (ELNANO) CONCLUSION In onlusion, we have developed DFT-NEGF based model to predit stati eletrial harateristis of the CNTFET with Al and Pd ontats. The model is shown to be able to predit orret polarity of the CNTFET with Pd and Al eletrodes. This follows from the omparison of with experimental transfer harateristis. Our model predit experimental tren in the dependenes of transondutane and ontat resistane for Pd-ontated FET. Namely, ontat resistane in both our simulation and experimental data drops as ontat length inrease up to some effetive length. We have shown also that Al-ontated CNTFETs are not sensible to ontat length in ontrast to Pd-ontated ones. This finding should be yet onfirmed experimentally. Finally, we demonstrate higher ontat resistane for Al ontat ompared to Pd ontats, whih an be explained by stronger interation with Carbon; preferable polarity of the Pd- and Al-ontated CNTFETs is opposite whih is in qualitative agreement with existing experiments. ACKNOWLEDGMENT We aknowledge the Center for Information Servies and High Performane Computing (ZIH) at TU Dresden for omputational resoures. This work is partly supported by the German Researh Foundation (DFG) within the Cluster of Exellene "Center for Advaning Eletronis Dresden". REFERENCES [1] T. Drkop, S. A. Getty, E. Cobas, and M. S. Fuhrer, Extraordinary mobility in semionduting arbon nanotubes, Nano Letters, vol. 4, no. 1, pp , [Online]. Available: /nl034841q [2] N. Neme, D. Tomanek, and G. Cuniberti, Contat dependene of arrier injetion in arbon nanotubes: An ab initio study, Phys. Rev. Lett., vol. 96, p , Feb [Online]. Available: [3] N. Neme, D. Tom anek, and G. Cuniberti, Modeling extended ontats for nanotube and graphene devies, Phys. Rev. B, vol. 77, p , Mar [Online]. Available: /PhysRevB [4] A. D. Franklin and Z. Chen, Length saling of arbon nanotube transistors, Nat Nano, vol. 5, no. 12, pp , De [Online]. Available: [5] A. Fediai, D. Ryndyk, and G. Cuniberti, Eletron transport in extended nt-metal ontats: rigorous ab-initio based green funtion method, (submitted to Phys. Rev. B). [6] A. D. Franklin, D. B. Farmer, and W. Haensh, Defining and overoming the ontat resistane hallenge in saled arbon nanotube transistors, ACS Nano, vol. 8, no. 7, pp , 2014, pmid: [Online]. Available: [7] J. VandeVondele, M. Krak, F. Mohamed, M. Parrinello, T. Chassaing, and J. Hutter, Quikstep: Fast and aurate density funtional alulations using a mixed gaussian and plane waves approah, Computer Physis Communiations, vol. 167, no. 2, pp , [Online]. Available: /artile/pii/s [8] A. D. Franklin, M. Luisier, S.-J. Han, G. Tulevski, C. M. Breslin, L. Gigna, M. S. Luntrom, and W. Haensh, Sub-10 nm arbon nanotube transistor, Nano Letters, vol. 12, no. 2, pp , 2012, pmid: [Online]. Available: / /nl203701g [9] A. Javey, Q. Wang, W. Kim, and H. Dai, Advanements in omplementary arbon nanotube field-effet transistors, in Eletron Devies Meeting, IEDM 03 Tehnial Digest. IEEE International, De 2003, pp

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