A Density Functional Theory Study of Schottky Barriers at Metal-nanotube Contacts

1. Symposium on Quantum Mechanical Models of Materials A Density Functional Theory Study of Schottky Barriers at Metal-nanotube Contacts School of Electrical and Computer Engineering Tuo-Hung Hou

2. Outlines • Introduction • What’s carbon nanotube (CNT)? • What’s Schottky barrier ? • What’s DFT? • DFT simulation on (8,0) carbon nanotube • DFT simulation on CNT/Pd and CNT/Au contacts • Summary

3. Carbon Nanotube Discovered by Dr. Sumio Iijima in NEC in 1991 • Extraordinary properties: • Self-assemble nanostructure. (diameter 1 nm / aspect ratio as high as 107) • 1-D carrier transport. Reduced scattering. Very high mobility (100x faster than silicon) • Sustain current density as high as 109 A/cm2. (100x higher than copper) • Stiffer and stronger than steel. Proc. IEEE, vol.91, p.1772, 2003

4. Carbon Nanotube metallic Graphene: a layer of graphite semiconducting Chirality vector C= na1 + ma2 Graphere: 2DEG (3,3) CNT: 1DEG (10,10) CNT (4,2) CNT: 1DEG (20, 0) CNT Proc. IEEE, vol.91, p.1772, 2003

5. M M M S M S Ec Ec BN e EF EF EF BP e Ev Ev Work Function & Schottky Barrier Work Function: Minimal energy necessary to extract an electron from the metal W = Ve - EF Schottky Barrier: The barrier formed at the metal- semiconductor interface BP = EF – Ev ; BN = Ec - EF vacuum e

6. Carbon Nanotube Field Effect Transistor Bottleneck step of charge transport S D CNT Back Gate The Schottky barrier at metal/ CNT interface determines the performance of CNTFET. Proc. IEEE, vol.91, p.1772, 2003

7. M S Evac Evac Ec No charge transfer EF Ev M S Evac Evac charge transfer Ec EF Ev - + M S Evac Evac charge transfer & finite separation Ec EF Ev - + A Closer Look of Schottky Barrier M S Evac Evac W Ec EF Ev

8. Ab-Initio Calculation of Schottky Barrier DFT takes into account the interfacial interactions (dipole formation, geometry relaxation etc. ) in first-principle. Thus, it is very accurate in calculating Schottky barriers heights. Schottky barrier height BP w/o interfacial interaction = [EF]metal – [ EV]CNT w/i interfacial interaction = [EF – <V>]metal – [ EV -<V>]CNT + [<V>metal - <V>CNT]IF _ + EF EF EV EV [EF – <V>]metal [EV – <V>]CNT [<V>metal - <V>CNT]IF J. Vc. Sci. Tech. A, vol.11, p.848, 1993

9. Brief Review of DFT Density Functional Theory (DFT): Describing electrons in a many-body system using the density instead of the many-body wave function.It dramatically reduces the dimension of freedom from 3N for N electrons to just 3. 1st Hohenberg-Kohn Theorem: A one-to-one mapping exists between the ground-state charge density and the ground-state many-body wavefunction. 2nd Hohenberg-Kohn Theorem: There is a variational principle so that So we can continuously refine the charge density to find the ground-state energy.

10. Brief Review of DFT Kohn-Sham Equation & Energy Minimization: Plane Wave Basis LDA Pseudopotential • Errors: • LDA • Pseudopotential approximation • Energy cutoff of plane wave basis • K-point selection for BZ sampling • Finite unit cell size

11. Y X Carbon Nanotube Unit Cell Carbon nanotube 3-D coordinates:generated by the wrapping program. Unit cell:A hexagonal close-pack lattice with larger enough separation between tubes. Periodic in the z direction. a (8, 0) CNT Cross section the hexagonal unit cell

12. Z0 = 4.26 Å a = 22 Bohr Below 1 meV Convergence Plane wave energy cutoff Unit cell size ** Energy difference between a unrelaxed and a relaxed structure. relaxed structure Ecutoff = 40 Ryd Energy cutoff 40 Ryd, unit cell distance 22 Bohr and 11 special k-points along the z direction are found to give good convergence.

13. Min. 4.221 Å Geometry Optimization of CNT (8, 0) CNT Ecutoff = 40 Ryd a = 22 Bohr Y Z Z0 = 4.221 Å Geometry optimization in the XY plane was carried out for each Z0 to find the most stable structure. (Force < 20 meV/ Å in X,Y,Z directions) Unrelaxed structure from the graphene sheet: Z0 = 4.26 Å , D = 6.26 Å Y X D = 6.29 Å

14. Band Structure of CNT (8, 0) CNT EG = 0.6 eV Γ X (8,0) CNT is semiconducting with EG 0.6eV, agreed with the value reported by Blase etc. ( 0.62 eV by LDA, Phys. Rev. Lett. 72, 1878 (1994) )

15. Work Function of CNT Potential Charge Density A A’ A A’ Y High density low potential X The potential and charge density are averaged over the z direction. VASP, Shan and Cho, Phys. Rev. Lett. 94, 236602 (2005) Total potential V = VIon + VH-F + Vxc A A’ Important!! Unit is Ry not eV

16. Y Y X Z Metal/Nanotube Contact Unit Cell Cross section the tetragonal unit cell b a Y X The initial structure has a relaxed (8,0) CNT on top of (100) surface of a two or three atomic-layer metal slab. The lattice parameters of metals are first calculated from the bulk (Pd 3.88Å , Au 4.05Å). The tensile strain is applied on metal at the z direction to match the lattice constant of CNT. The strains at x y directions are calculated by the Poisson ratio.

17.  z Geometry Optimization of Contact Rotational Angel Translational Distance d = 2.0Å z = 0 Å d = 2.0Å  = 0o Major degrees of freedom are first sampled before full ab-initio optimization to avoid trapping at local minima.

18. d Binding Energy Interfacial Distance z = 0 Å  = 0o EBinding = ECNT + EMetal – ECNT/Metal The equilibrium interfacial distance between CNT and Pd is smaller than CNT and Au with stronger binding energy.

19. Full ab-initio Optimization CNT/Pd Energy = -1502.0944 rdy Step1 fixed Y Y X Z Energy = -1502.1555 rdy All force < 0.02 eV/Å Step25

20. Potential Energy A CNT / Pd No physical tunneling barrier existed between CNT and either Pd or Au. Although Au with larger interfacial distance does show an additional bump. The carrier transportation across the interface is then determined by the band lineup, i.e. Schottky barrier. A’ A CNT / Au A’

21. Charge Density CNT / Pd CNT / Au C C Pd Au Less charge density between C / Au due to the additional bump in the potential profile.

22. + + + _ _ _ Charge Transfer + + _ _ _ Charge difference = [e] CNT/Metal – [e] CNT – [e] Metal Electron transfers from CNT to Pd and Au. More dipole formation at CNT/Pd is due to its proximity, but the dipole moment is not necessarily larger (p=qd).

23. Schottky Barrier BP = [EF]metal – [ EV]CNT w/o interaction = [EF – <V>]metal – [ EV -<V>]CNT + [<V>metal - <V>CNT]IF w/i interaction Pd WF = -4.79 eV; CNT EV= -5.01 eV; w/o interaction Ep = 0.22 eV w/i interaction Ep = 0.185 eV Au WF = -5.27 eV; CNT EV= -5.01 eV; w/o interaction Ep = -0.26 eV w/i interaction Ep = 0.39 eV

24. Summary • Review the theory of the ab initio Schottky barrier calculation based on DFT. The method is applicable for many interfacial problems in nanoscale. • Detailed DFT calculations on (8,0) carbon nanotube are performed, including the geometry optimization, band structure, and work function. • Geometry optimization at CNT/metal contacts are carefully examined though a 2-step process. The major degrees of freedom are first optimized, and followed by the ab initio relaxation. • CNT is more closely bounded to Pd than Au with larger binding energy and shorter interfacial distance. • Although the number of dipoles is larger in CNT/Pd, the total dipole moment, which is responsible for the potential shift across the interface is larger in CNT/Au, which therefore shows larger Schottky barrier. • Very good quantitative agreement in this study in comparison with previous works and experimental results.