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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 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 Carbon Nanotube Discovered by Dr. Sumio Iijima in NEC in 1991 Extraordinary properties: 1. Self-assemble nanostructure. (diameter 1 nm / aspect ratio as high as 107) 2. 1-D carrier transport. Reduced scattering. Very high mobility (100x faster than silicon) 3. Sustain current density as high as 109 A/cm2. (100x higher than copper) 4. Stiffer and stronger than steel. Proc. IEEE, vol.91, p.1772, 2003 Carbon Nanotube Graphene: a layer of graphite Chirality vector C= na1 + ma2 metallic semiconducting Graphere: 2DEG (3,3) CNT: 1DEG (10,10) CNT (4,2) CNT: 1DEG (20, 0) CNT Proc. IEEE, vol.91, p.1772, 2003 Work Function & Schottky Barrier Work Function: Schottky Barrier: Minimal energy necessary to extract an electron from the metal The barrier formed at the metalsemiconductor interface W = Ve - EF BP = EF – Ev ; BN = Ec - EF vacuum M M M S M Ec e e EF e BP S BN Ec EF EF Ev Ev Carbon Nanotube Field Effect Transistor S D CNT Bottleneck step of charge transport Back Gate The Schottky barrier at metal/ CNT interface determines the performance of CNTFET. Proc. IEEE, vol.91, p.1772, 2003 A Closer Look of Schottky Barrier M S Evac Evac Ec No charge transfer EF M S Evac Evac W Ec EF Ev Ev M S Evac Evac charge transfer Ec EF Ev -+ M charge transfer & finite separation S Evac Evac Ec EF - + Ev 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 [EF – <V>]metal EV [EV – <V>]CNT _ + EV [<V>metal - <V>CNT]IF J. Vc. Sci. Tech. A, vol.11, p.848, 1993 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 E[ (r )] T [ (r )] Ee e [ (r )] vext (r ) (r )dr E0 So we can continuously refine the charge density to find the ground-state energy. Brief Review of DFT E[ (r )] T [ (r )] Ee e [ (r )] vext (r ) (r )dr E0 Kohn-Sham Equation & Energy Minimization: 1 1 ( r ) ( r ) E[ (r )] i* (r ) 2 i (r )dr 1 2 dr1dr2 Exc[ (r )] vext (r ) (r )dr E0 2 2 r1 r2 i 1 n Plane Wave Basis Errors: 1. LDA 2. Pseudopotential approximation 3. Energy cutoff of plane wave basis 4. K-point selection for BZ sampling 5. Finite unit cell size LDA Pseudopotential 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. Y X a (8, 0) CNT Cross section the hexagonal unit cell Convergence Plane wave energy cutoff Unit cell size relaxed structure -365.186 Z0 = 4.26 Å -4 8.0x10 Total Energy [ Ryd ] Energy Convergence [ Ryd ] ** Energy difference between a unrelaxed and a relaxed structure. a = 22 Bohr -4 6.0x10 -4 4.0x10 Below 1 meV -4 2.0x10 20 25 30 35 Energy Cutoff [Ryd] 40 45 Ecutoff = 40 Ryd a = 27 Bohr -365.188 -365.190 a = 22 Bohr -365.192 -365.194 -365.196 4.18 4.20 4.22 4.24 4.26 Z0 [Å] Energy cutoff 40 Ryd, unit cell distance 22 Bohr and 11 special k-points along the z direction are found to give good convergence. 4.28 Geometry Optimization of CNT Y Z Z0 = 4.221 Å Total Energy [ Ryd ] (8, 0) CNT -365.178 -365.180 -365.182 Ecutoff = 40 Ryd a = 22 Bohr -365.184 -365.186 -365.188 Min. 4.221 Å -365.190 -365.192 -365.194 -365.196 4.18 4.20 4.22 4.24 4.26 4.28 4.30 Z0 [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) Y X D = 6.29 Å Unrelaxed structure from the graphene sheet: Z0 = 4.26 Å , D = 6.26 Å Band Structure of CNT (8, 0) CNT EG = 0.6 eV Γ X (8,0) CNT is semiconducting with EG 0.6 eV, agreed with the value reported by Blase etc. ( 0.62 eV by LDA, Phys. Rev. Lett. 72, 1878 (1994) ) Work Function of CNT Potential Charge Density A’ A A A’ Y High density low potential 1x10 1 1x10 -1 1x10 -3 1x10 -5 1x10 -7 -3 Total Potential [ eV ] 4 2 0 -2 -4 -6 WF 4.7eV EF -8 -10 -10 A Charge Density [ Bohr ] X The potential and charge density are averaged over the z direction. C C -5 0 X [Å] 5 10 A’ VASP, Shan and Cho, Phys. Rev. Lett. 94, 236602 (2005) Total potential V = VIon + VH-F + Vxc Important!! Unit is Ry not eV Metal/Nanotube Contact Unit Cell Cross section the tetragonal unit cell b Y a Y X Z 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. Geometry Optimization of Contact Rotational Angel Translational Distance -1122.88 d = 2.0Å -1122.88 Total Energy [ Ryd ] Total Energy [ Ryd ] -1122.87 z=0Å -1122.89 -1122.90 -1122.91 -1122.92 -1122.93 -1122.94 0 5 10 15 20 [ 25 o ] 30 35 40 45 -1122.89 -1122.90 -1122.91 -1122.92 z -1122.93 -1122.94 -1122.95 d = 2.0Å -1122.96 = 0o -1122.97 -0.5 0.0 0.5 1.0 1.5 2.0 Z [Å] Major degrees of freedom are first sampled before full ab-initio optimization to avoid trapping at local minima. 2.5 Binding Energy Interfacial Distance Total Energy [ Ryd ] -1122.60 -1033.00 -1122.65 z=0Å -1033.05 -1122.70 = 0o -1033.10 -1122.75 Au Pd -1033.15 -1122.80 -1033.20 -1122.85 -1033.25 -1122.90 -1033.30 d Pd Au (8,0) CNT -365.1943 -365.1943 Metal slab -757.5965 -667.9404 CNT/Metal -1122.9598 -1033.1867 Binding Energy (meV) 12.4285 3.8194 EBinding = ECNT + EMetal – ECNT/Metal -1033.35 -1122.95 -1123.00 Total Energy (Ryd) 1.6 1.8 2.0 2.2 2.4 2.6 2.8 -1033.40 3.0 d [Å] The equilibrium interfacial distance between CNT and Pd is smaller than CNT and Au with stronger binding energy. Full ab-initio Optimization CNT/Pd Energy = -1502.0944 rdy Step1 fixed Y Y X Z Energy = -1502.1555 rdy Step25 All force < 0.02 eV/Å Potential Energy A CNT / Pd Total Potential [ eV ] 10 A’ A EF -10 C Pd Pd Pd C -20 -10 -5 0 5 10 Y [Å] 10 Total Potential [ eV ] CNT / Au A’ 0 0 EF -10 Au Au Au C -20 -10 -5 C 0 Y [Å] 5 10 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. Charge Density CNT / Pd C Pd CNT / Au C Au Less charge density between C / Au due to the additional bump in the potential profile. 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). Schottky Barrier CNT/Pd 10 Pd -10 -20 -40 -50 5.2 5.0 4.8 -60 -70 Pd WF = -4.79 eV; CNT EV= -5.01 eV; 4.6 C Pd Pd Pd C -5 Total Potential [ eV ] -30 -64.2 -64.4 w/o interaction Ep = 0.22 eV -64.6 -64.8 0 5 10 Y [Å] 10 CNT/Au 0 CNT Au WF = -5.27 eV; CNT EV= -5.01 eV; -20 w/o interaction Ep = -0.26 eV -3.2 -3.4 -3.8 -60 -70 C Au Au Au C -80 -10 w/i interaction Ep = 0.39 eV -3.6 -5 0 Y [Å] Total Potential [ eV ] Total Potential [ eV ] -30 -50 w/i interaction Ep = 0.185 eV Au -10 -40 BP = [EF]metal – [ EV]CNT w/o interaction = [EF – <V>]metal – [ EV -<V>]CNT + [<V>metal - <V>CNT]IF w/i interaction 5.4 -80 -10 Total Potential [ eV ] CNT 0 Total Potential [ eV ] Total Potential [ eV ] 20 -64.6 -64.8 -65.0 -65.2 5 10 Summary 1. Review the theory of the ab initio Schottky barrier calculation based on DFT. The method is applicable for many interfacial problems in nanoscale. 2. Detailed DFT calculations on (8,0) carbon nanotube are performed, including the geometry optimization, band structure, and work function. 3. 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. 4. CNT is more closely bounded to Pd than Au with larger binding energy and shorter interfacial distance. 5. 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. 6. Very good quantitative agreement in this study in comparison with previous works and experimental results.