Contour Tbtrans

transcript

Using TranSIESTA The integration contour Using tbtrans

Using TranSIESTA (II): Integration contour andtbtrans

Frederico D. Novaes

December 15, 2009

Outline

Using TranSIESTA

The integration contourElectron density from GFWhy go complex ?The non-equilibrium situation

Using tbtrans

Outline

Using TranSIESTA

Using tbtrans

It is simples to use: few (simple) key concepts

• Simple to use (doesn’t mean simple theory). Few concepts :

1. The scattering region setup

2. The electrode calculation (and possibleuse of buffer atoms)

3. The energy contour parameters

∫∞

−∞

G<(E )dE

Outline

Using TranSIESTA

Using tbtrans

The dual basis

• A Dual Set may be defined,

〈φµ|φν〉 = Sµ,ν −→ 〈φµ|φν〉 = δµν

• Easier to obtain expressions,

1 =∑

|φµ〉〈φµ| =∑

|φµ〉〈φµ|

H |ψi〉 = Ei |ψi〉 −→∑

H |φµ〉〈φµ|ψi 〉 = Ei

|φµ〉〈φµ|ψi〉

−→∑

Hν,µcµi = Ei

Sν,µcµi

δνξ =

〈φξ|φµ〉〈φµ|ψi 〉〈ψi |φ

ν〉 =∑

Sξ,µcµi cν∗

Calculus of Complex Variables

• From complex analysis (residue theorem),

f (z)dz = 2πı∑

Resz=zkf (z)

f (z) =

∞∑

j=−∞

cj(z)(z − zk)j −→ Resz=zk

f (z) = c−1(zk)

• A useful relation may then be computed,

f (E )

E+ − E0dE = P

[∫ b

f (E )

E − E0dE

︸︷︷︸

−ıπf (E0)

limδ→0

[∫ E0−δ

f (E )

E − E0dE +

f (E )

E − E0dE

Time Reversal Symmetry

• Considering the time dependent Schroedinger equation, and thereality of H,

ı~∂ψ

∂t= Hψ ⇒ ı~

∂ψ∗

∂(−t)= Hψ∗

Hψ = Eψ ⇒ Hψ∗ = Eψ∗

• Two possibilities,

1. ψ and ψ∗ are LI ⇒ ”doubles” the E degeneracy

2. ψ and ψ∗ are not LI ⇒ ψ = ψ∗ (Real) E ”not” degenerate

Density Matrix in SIESTA

• In practice, in SIESTA, the Kohn-Sham orbitals ψi (r) are expandedin a set of (real) localized basis,

ψi (r) =∑

cµi φµ(r)

• The electron density is then,

ρ(r) =∑

µ,µ′

cµ∗i c

i φµ(r)φµ′ (r)

µ,µ′

ρµ,µ′φµ(r)φµ′ (r)

• The solution consist in finding the Density Matrix (.DM file),

ρµ,µ′ =∑

nicµ∗i c

i =∑

niRe[cµ∗i c

i ′ ]

︸︷︷︸

T .R.S.

Spectral Representation of Gr(E )

• The G r (E ) may be written as,

G rµ,ν(E ) =

cµi cν∗

E+ − Ei

E+S − H)

G(r)(E )

E+Sξ,µ − Hξ,µ

cµi cν∗

E+ − Ei

E+Sξ,µ − EiSξ,µ

cµi cν∗

E+ − Ei

Sξ,µcµi cν∗

i = δνξ

The DM from GFs

• If we integrate,

∫∞

−∞

nFD(E )G rµ,ν(E )dE = ???

∫∞

−∞

nFD(E )[∑

cµi cν∗

E+ − Ei

dE =∑

cµi cν∗

∫∞

−∞

nFD(E )

E+ − Ei

︸︷︷︸

P[ ]−ıπnFD(Ei )

⇛ Im

−∞

nFD(E )G rµ,ν(E )dE

= −π∑

cµi cν∗

⇛ ρ = −1

−∞

nFD(E )Gr (E )dE

Equilibrium DM

• Two ways of computing the Density Matrix,

1. From the Kohn-Sham orbitals,

nic∗

2. From the Retarded Green’s Function

ρ = −1

−∞

nFD(E)Gr(E)dE

• With GFs a Self Consistent procedure can be used in the same wayas the “standard” Kohn-Sham orbitals

The TranSIESTA contour

• TS.ComplexContour.Emin

• TS.ComplexContour.NCircle

• TS.ComplexContour.NLine

• TS.ComplexContour.NPoles

Smooth in the complex plane

• G r (E ) is smoother in the complex plane.

• Smaller number of points to get accurate results.

• As an example, the spectral function (DOS),

Things we know ...

• The G r (E ) is smoother for E = Er + ıEc = Z ,

G rµ,ν(Z ) =

cµi cν∗

Z − Ei

• G r (Z ) is analytic for Im[Z ] > 0.

• nFD(E ) has poles at known places and known residues,

nFD(Z ) =(

eZ−EfkB T

︸︷︷︸

→−1

+1)−1

Zj = Ef + ıkBT (2j + 1)π, j = 0,±1,±2, . . .

Contour Integration: Equilibrium• The integral may be obtained in a contour integration,

nFD(E )G r (E )dE =

nFD(Z )G r (Z )dZ − 2πıkBT

G r (Zj)

Default values in TS

• TS.ComplexContour.Emin = -3.0 Ry

• TS.ComplexContour.NCircle = 24

• TS.ComplexContour.NLine = 6

• TS.ComplexContour.NPoles = 6

• DANGER : Start the contour bellow the lowest eigenvalue of thesystem !

• For that a good practice is to always do first a SIESTAcalculation and check the eigenvalues (.EIG file)

From NEGF

• In the non-equilibrium case, the charge density is given by,

ρCC =1

G rCC (E )

(f EFD(E )ΓE (E ) + f D

FD(E )ΓD(E ))G a

CC (E ))

• This integrand is however non analytic: presence of retarded andadvanced.

• The integration could be done at the real axis, but ... too expensive.

• The solution is make a transformation, and get,

ρCC = ρeqCC + ρ

ρeqCC = −

f EFD(E )G r

CC (E )dE ]

ρneqCC =

G rCC (E )ΓD(E )G a

CC (E )(f DFD(E ) − f E

FD(E ))dE

Final remarks on contours

• The integration on the bias range can be more demanding. This iscontroled by the flag: TS.biasContour.NumPoints

• If you look at the .CONTOUR file (with bias), you’ll see somethinglike this,

Outline

Using TranSIESTA

Using tbtrans

What is tbtrans ?

• The current I is obtained by the relation,

∫(f EFD(E ) − f D

FD(E ))Tr [ΓE (E )G r (E )ΓD(E )G a(E )]︸︷︷︸

• These matrices depend only on the Hamiltonian of the Scatteringsetup that was stored in a TranSIESTA calculation.

=⇒ Transport properties are obtained with a post prcessing code:tbtrans

How to use it

∫(f EFD(E ) − f D

FD(E ))Tr [ΓE (E )G r (E )ΓD(E )G a(E )]︸︷︷︸

• TranSIESTA stores the Hamiltonian (and Overlap) in files .TSHS

• tbtrans will need the electrode’s .TSHS file(s), and the scatteringregion TSHS.

• The energy interval is defined by TS.TBT.Emin, TS.TBT.Emax

• To calculate the current be sure to define the energy interval bigenough

• The number of points (mesh) in this interval is defined byTS.TBT.NPoints

• For the mesh, also, be sure to have a sufficiently dense mesh

Remark on k-points sampling• Warning: Even if the real-space Hamiltonian is sufficiently converged

for a given k-point sampling, the transmission function might not befor the same sampling.

Contour Tbtrans

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