Using TranSIESTA (II): Integration contour and tbtrans

Transcription

1 Usng TranSIESTA (II): Integraton contour and tbtrans Frederco D. Novaes December 15, 2009

2 Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans

3 Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans

4 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

5 Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans

6 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

7 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

8 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

9 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.

10 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

11 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

12 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

13 The TranSIESTA contour TS.ComplexContour.Emn TS.ComplexContour.NCrcle TS.ComplexContour.NLne TS.ComplexContour.NPoles

14 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),

15 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,...

16 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

17 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)

18 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

19 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,

20 Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng tbtrans

21 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

22 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

23 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.