Tunneling conductance of molecular wires

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1 Tunneling conductance of molecular wires Emil Prodan Deptartment of Physics, Yeshiva University, New York Work in collaboration with Roberto Car (Dept. of Chemistry, Princeton U) This work relies heavily on the analytic methods developed during the Nearsightedness project coordinated by Walter Kohn ular chain. Acknowledgments: Partial support for this work was provided by the NSF-MRSEC program through the Princeton Center for Complex Materials (PCCM), grant DMR , and by DOE through grant DE-FG02-05ER E.P acknowledges additional financial support from Yeshiva University.

Exciting developments on the experimental side Measured Conductance (G 0 ) 10-2 10-3 10-4 10-5 10-6 Exp Alkane Exp Alkane Theory Acene Exp Acene Theory PolyPh Exp PolyPh Theory 4 6 8 10 12

2 Exciting developments on the experimental side Measured Conductance (G 0 ) Exp Alkane Exp Alkane Theory Acene Exp Acene Theory PolyPh Exp PolyPh Theory (b) Theory Calculated Tunneling Probability (ev 2 ) Latha Venkataram, first highly reproducible results generated in 2006 N-N Length (Å) N. Tao et al, JACS 2006

3 Transport: a Linear Response Approach within Time Dependent Current-Density Functional Theory Framework: - an equilibrium system that is perturbed by a weak, time oscillating electric field - DC regime obtained by letting the frequency go to zero Eext(!) j(r, ω) = ˆσ KS (r, r ; ω)e eff 1 (r, ω)dr, E eff 1 [j] = E ext + v HXC [j] + E dyn 1 [j]

4 We are after the two-point conductance: g = I φ φ = γ (E ext + E H 1 )dl I = Σ j ds " with j the self-consistent solution of j(r, ω) = ˆσ KS (r, r ; ω)e eff 1 (r, ω)dr, (ω 0)

5 The exact expression of g within the Linear Response Time Dependent Current-Density Functional Theory g I φ = dx dx [(1 ˆσ KS ˆF) 1 ˆσ KS ] zz (x, z; x, z ) F αβ (r, r ) δedyn α (r) δj β (r ) Implications: - the adiabatic vhxc gives no corrections to the bare Kohn-Sham conductance - the dynamical effects renormalize the Kohn-Sham states - an exact F matrix will put the resonances at the correct energies and widen to spectral gap to the correct value

6 The Adiabatic Approximation g I φ = dx dx [(1 ˆσ KS ˆF) 1 ˆσ KS ] zz (x, z; x, z ) Whenever # KS is small, this part can be neglected and g dr σ KS αβ(r, r ) = 1 4π GKS ɛ + F dr σ KS zz (r, z, r, z ), (r, r ) α β G KS (r, r), ɛ F (equivalent with Landauer formula) The problem remains extremely challenging for long molecular chains: The super-cells become extremely large The conductances become extremely small

7 Tunneling Transport - a crude model will be to consider tunneling through a square barrier everybody can then understand the typical tunneling behavior: g = gc e -$N (N = number of monomers)

8 Tunneling Transport in Modern Formulation Science 294, 571 (2001). P. Mavropoulos, N. Papanikolaou, and P. Dederichs, Phys. Rev. Lett. 85, 1088 (2000). J. Tomfohr and O. Sankey, Phys. Rev. B 65, (2002). J. Tomfohr and O. Sankey, J. Chem. Phys. 120, 1542 (2004). G. Fagas, A. Kambili, and M. Elstner, Chem. Phys. Lett. 389, 268 (2004). Im k Re k - the complex band structure of the infinite chain is aligned with the spectral gap of the device $ = 2 Im kf - the link between the tunneling conductance and complex band structure was established empirically - no expression for gc was available

9 Our contribution Start from the following decomposition: V eff H KS = 2 + V 0 + V L + V R V 0 (Nearsightedness setup: periodic potential perturbed by distant perturbations) V L + V R The reason for decomposition is to use new analytic results on periodic systems (E. P., PRB 2006)

10 A lesson from a 1 dimensional problem V(x) Remember: we need σ KS αβ(r, r ) = 1 4π GKS ɛ + F (r, r ) α β G KS (r, r), ɛ F H = d2 dx 2 + V (x) x Green s function: In 1D, there is an alternative expression G E (x, x ) = (H E) 1 = n ψ n (x)ψn(x ) E ɛ n G E (x, x ) = ψ <(x < )ψ > (x > ) W (ψ <, ψ > ) (not very useful) (H E)ψ < = 0, with the boundary condition to the left (H E)ψ > = 0, with the boundary condition to the right

11 The expression for the Green s function is so simple that one can compute g analytically!! g 0 (L) = 4 π Im[R L (k + F )]Im[R R(k + F )]e 2βL 1 e 2βL R L (k + F )R R(k + F ) 2 Unfortunately, the textbooks tell that no such expression for the Green s function exists in higher dimensions! We found the exception, which is the case of periodic potentials!

12 Green s function from the Riemann surface of the bands Globally defined %&, P& on a Riemann surface describe the whole band structure (&=e ikb ). %& If we evaluate %& along the circle we obtain the usual energy bands The Riemann surface of the bands was discovered by Walter Kohn in 1959.

13 G 0 ɛ(r, r ) = n λ =1 ψ n,1/λ (r)ψ n,λ (r ) ɛ ɛ n,λ dλ 2πiλ (eigenfunction expansion) G 0 ɛ(r, r ) = Γ ψ 1/λ (r < )ψ λ (r >) ɛ ɛ(λ) dλ 2πiλ But the contour " can be deformed to a point: G 0 ɛ(r, r ) = ψ 1/λ(r < )ψ λ (r >) λ λ ɛ(λ)! = the point on the Riemann surface so that: ɛ = ɛ(λ) The existence of the Riemann structure gives the simple expression for the Green s function!

14 Molecular wires Existence and characterization of the Riemann surface for molecular wires was given in E. Prodan, PRB 2006 G 0 ɛ(r, r ) = Γ ψ 1/λ (r < )ψ λ (r >) ɛ ɛ(λ) dλ 2πiλ But the contour " can be deformed to a point: G 0 ɛ(r, r ) = j ψ 1/λj (r < )ψ λj (r >) λ j λ ɛ(λ j )! j are all! on the Riemann surface so that: ɛ = ɛ(λ j )

15 Tunneling Conductance g 0 (L) = 1 π i,j Θ ij L Θ ij R e i(k i+k j )L. k ɛ ki k ɛ kj E. Prodan & R. Car, PRB 2007 #(k i ) = #F Localized at the contacts

16 Amine Linked Alkyl Chains (E. P. and R. Car, Nano Lett 2008)

17 Our Theoretical Predictions Experiment, Venkataram et al, NanoLett 2006 Theory Tao et all, JACS 2006 Transmission Im k!-!f (ev) Re k Fermi level shifts lead to small changes in conductance: band alignment not so important in alkyl chains

18 New Insight into the Tunneling Transport of the devices g = 1 π Θ L Θ R ( k ɛ k ) 2 e2ikl. Θ L = 2πi dr dr V L (r)ψ k0 (r)ρ ɛf +eφ(r, r ; T ) V L (r )ψ k0 (r ), Θ R = 2πi dr dr V R (r)ψ k0 (r)ρ ɛf +eφ(r, r ; T ) V R (r )ψ k0 (r ),

19 The self-consistent potential and!v Veff (isosurface and xy average) 'V (isosurface and xy average)

20 The evanescent Block functions

21 Why contact conductance is a contact conductance (a) Ψ L/R (r) = ψ k (r) V L/R (r) (b) (c)!"!# (d)

22 Benzene chains

23 Theory vs Experiment Theory Experiment Venkataraman et al, Nature 2006 Transmission Conductance (G0) ln g!-!f (ev) Number of monomers

24 New Insight into the Tunneling Transport of the devices (a) (b) (c) (d) (e)

25 The evanescent Block functions (a) (b)

26 Again,Why contact conductance is a contact conductance Ψ L/R (r) = ψ k (r) V L/R (r)!"!#

27 Conclusions - a newly formulated tunneling transport theory give a rigorous way to compute beta and the contact conductance - the analytic expression of the contact conductance give insight into the transport characteristics of the devices - we hope that the formalism will become a useful tool for device design

28 Further directions { Tunneling Magneto-Resistance Generalization to the spin polarized case: Spin Transport g(l) = 1 π Θ 2 maj + Θ 2 min ( k ɛ k0 ) 2 e βl g(l) = 1 π 2Θ maj Θ min ( k ɛ k0 ) 2 e βl G P G A G P + G A = [ Θmaj Θ min Θ maj + Θ min ] 2

The analytic structure of Bloch functions for linear molecular chains

The analytic structure of Bloch functions for linear molecular chains E. Prodan PRISM, Princeton University, Princeton, NJ 08544 This paper deals with Hamiltonians of the form H = 2 + v(r), with v(r) periodic