Usng TranSIESTA (II): Integraton contour and tbtrans Frederco D. Novaes December 15, 2009
Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans
Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans
It s smples to use: few (smple) key concepts Smple to use (doesn t mean smple theory). Few concepts : 1. The scatterng regon setup 2. The electrode calculaton (and possble use of buffer atoms) 3. The energy contour parameters G < (E)dE
Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans
A Dual Set may be defned, The dual bass Easer to obtan expressons, φ µ φ ν = S µ,ν φ µ φ ν = δ µ ν ˆ1 = µ φ µ φ µ = µ φ µ φ µ Ĥ ψ = E ψ µ µ Ĥ φ µ φ µ ψ = E φ µ φ µ ψ H ν,µ c µ = E S ν,µ c µ µ µ δξ ν = φ ξ φ µ φ µ ψ ψ φ ν = µ, µ, S ξ,µ c µ c ν
b a lm δ 0 Calculus of Complex Varables From complex analyss (resdue theorem), f (z)dz = 2πı Res z=zk f (z) C k f (z) = j= c j (z)(z z k ) j Res z=zk f (z) = c 1 (z k ) A useful relaton may then be computed, [ ] f (E) b f (E) E + de = P de ıπf (E 0 ) E 0 a E E 0 }{{} [ E0 δ a ] f (E) b f (E) de + de E E 0 E 0+δ E E 0
Tme Reversal Symmetry Consderng the tme dependent Schroednger equaton, and the realty of H, ı ψ t = Hψ ı ψ ( t) = Hψ Hψ = Eψ Hψ = Eψ Two possbltes, 1. ψ and ψ are LI doubles the E degeneracy 2. ψ and ψ are not LI ψ = ψ (Real) E not degenerate
Densty Matrx n SIESTA In practce, n SIESTA, the Kohn-Sham orbtals ψ (r) are expanded n a set of (real) localzed bass, ψ (r) = µ c µ φ µ (r) The electron densty s then, ρ(r) = n ( µ,µ c µ c µ φ µ (r)φ µ (r) ) = µ,µ ρ µ,µ φ µ (r)φ µ (r) The soluton consst n fndng the Densty Matrx (.DM fle), ρ µ,µ = n c µ c µ = n Re[c µ c µ ] } {{ } T.R.S.
Spectral Representaton of G r (E) The G r (E) may be wrtten as, G r µ,ν(e) = c µ c ν E + E ( ) ( ) E + S H G (r) (E) = µ ξ,ν = µ (E + S ξ,µ E S ξ,µ ) ( = µ (E + S ξ,µ H ξ,µ ) ( S ξ,µ c µ c ν = δ ν ξ ) c µ c ν E + E ) c µ c ν E + E
The DM from GFs If we ntegrate, n FD (E)G r µ,ν(e)de =??? [ c µ c ν ] n FD (E) E + de = E Im c µ c ν [ ] n FD (E)Gµ,ν r (E)dE = π [ ] ρ = 1 π Im n FD (E)G r (E)dE n FD (E) E + E de }{{} P[ ] ıπn FD (E ) c µ c ν n
Equlbrum DM Two ways of computng the Densty Matrx, 1. From the Kohn-Sham orbtals, ρ = X n c c 2. From the Retarded Green s Functon " Z # ρ = 1 π Im n FD (E)G r (E)dE Wth GFs a Self Consstent procedure can be used n the same way as the standard Kohn-Sham orbtals
The TranSIESTA contour TS.ComplexContour.Emn TS.ComplexContour.NCrcle TS.ComplexContour.NLne TS.ComplexContour.NPoles
Smooth n the complex plane G r (E) s smoother n the complex plane. Smaller number of ponts to get accurate results. As an example, the spectral functon (DOS),
Thngs we know... The G r (E) s smoother for E = E r + ıe c = Z, G r µ,ν (Z) = c µ c ν Z E G r (Z) s analytc for Im[Z] > 0. n FD (E) has poles at known places and known resdues, n FD (Z) = ( e Z E f k B T }{{} 1 ) 1 +1 Z j = E f + ık B T(2j + 1)π, j = 0, ±1, ±2,...
Contour Integraton: Equlbrum The ntegral may be obtaned n a contour ntegraton, N p n FD (E)G r (E)dE = n FD (Z)G r (Z)dZ 2πık B T G r (Z j ) C j=1
Default values n TS TS.ComplexContour.Emn = -3.0 Ry TS.ComplexContour.NCrcle = 24 TS.ComplexContour.NLne = 6 TS.ComplexContour.NPoles = 6 DANGER : Start the contour bellow the lowest egenvalue of the system! For that a good practce s to always do frst a SIESTA calculaton and check the egenvalues (.EIG fle)
From NEGF In the non-equlbrum case, the charge densty s gven by, ρ CC = 1 ( G r 2π CC(E) ( ffd(e)γ E E (E) + ffd(e)γ D D (E) ) ) GCC(E) a de Ths ntegrand s however non analytc: presence of retarded and advanced. The ntegraton could be done at the real axs, but... too expensve. The soluton s make a transformaton, and get, ρ CC = ρ eq CC + ρneq CC ρ eq CC = 1 π Im[ ffd(e)g E CC(E)dE] r ρ neq CC = 1 GCC r 2π (E)Γ D(E)GCC a (E)( ffd D (E) f FD E (E)) de
Fnal remarks on contours The ntegraton on the bas range can be more demandng. Ths s controled by the flag: TS.basContour.NumPonts If you look at the.contour fle (wth bas), you ll see somethng lke ths,
Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans
What s tbtrans? The current I s obtaned by the relaton, I = e (f E h FD(E) ffd(e) D ) Tr[Γ E (E)G r (E)Γ D (E)G a (E)] de }{{} T(E) These matrces depend only on the Hamltonan of the Scatterng setup that was stored n a TranSIESTA calculaton. = Transport propertes are obtaned wth a post prcessng code: tbtrans
How to use t I = e h (f E FD (E) f FD D (E)) Tr[Γ E (E)G r (E)Γ D (E)G a (E)] de }{{} T(E) TranSIESTA stores the Hamltonan (and Overlap) n fles.tshs tbtrans wll need the electrode s.tshs fle(s), and the scatterng regon TSHS. The energy nterval s defned by TS.TBT.Emn, TS.TBT.Emax To calculate the current be sure to defne the energy nterval bg enough The number of ponts (mesh) n ths nterval s defned by TS.TBT.NPonts For the mesh, also, be sure to have a suffcently dense mesh
Remark on k-ponts samplng Warnng: Even f the real-space Hamltonan s suffcently converged for a gven k-pont samplng, the transmsson functon mght not be for the same samplng.