contour tbtrans
TRANSCRIPT
Using TranSIESTA The integration contour Using tbtrans
Using TranSIESTA (II): Integration contour andtbtrans
Frederico D. Novaes
December 15, 2009
Using TranSIESTA The integration contour Using tbtrans
Outline
Using TranSIESTA
The integration contourElectron density from GFWhy go complex ?The non-equilibrium situation
Using tbtrans
Using TranSIESTA The integration contour Using tbtrans
Outline
Using TranSIESTA
The integration contourElectron density from GFWhy go complex ?The non-equilibrium situation
Using tbtrans
Using TranSIESTA The integration contour 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
Using TranSIESTA The integration contour Using tbtrans
Outline
Using TranSIESTA
The integration contourElectron density from GFWhy go complex ?The non-equilibrium situation
Using tbtrans
Using TranSIESTA The integration contour Using tbtrans
The dual basis
• A Dual Set may be defined,
〈φµ|φν〉 = Sµ,ν −→ 〈φµ|φν〉 = δµν
• Easier to obtain expressions,
1 =∑
µ
|φµ〉〈φµ| =∑
µ
|φµ〉〈φµ|
H |ψi〉 = Ei |ψi〉 −→∑
µ
H |φµ〉〈φµ|ψi 〉 = Ei
∑
µ
|φµ〉〈φµ|ψi〉
−→∑
µ
Hν,µcµi = Ei
∑
µ
Sν,µcµi
δνξ =
∑
µ,i
〈φξ|φµ〉〈φµ|ψi 〉〈ψi |φ
ν〉 =∑
µ,i
Sξ,µcµi cν∗
i
Using TranSIESTA The integration contour Using tbtrans
Calculus of Complex Variables
• From complex analysis (residue theorem),
∫
C
f (z)dz = 2πı∑
k
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,
∫ b
a
f (E )
E+ − E0dE = P
[∫ b
a
f (E )
E − E0dE
]
︸ ︷︷ ︸
−ıπf (E0)
limδ→0
[∫ E0−δ
a
f (E )
E − E0dE +
∫ b
E0+δ
f (E )
E − E0dE
]
Using TranSIESTA The integration contour Using tbtrans
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
Using TranSIESTA The integration contour Using tbtrans
Density Matrix in SIESTA
• In practice, in SIESTA, the Kohn-Sham orbitals ψi (r) are expandedin a set of (real) localized basis,
ψi (r) =∑
µ
cµi φµ(r)
• The electron density is then,
ρ(r) =∑
i
ni
(∑
µ,µ′
cµ∗i c
µ′
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′ =∑
i
nicµ∗i c
µ′
i =∑
i
niRe[cµ∗i c
µ′
i ′ ]
︸ ︷︷ ︸
T .R.S.
Using TranSIESTA The integration contour Using tbtrans
Spectral Representation of Gr(E )
• The G r (E ) may be written as,
G rµ,ν(E ) =
∑
i
cµi cν∗
i
E+ − Ei
((
E+S − H)
G(r)(E )
)
ξ,ν
=∑
µ
(
E+Sξ,µ − Hξ,µ
)(∑
i
cµi cν∗
i
E+ − Ei
)
=∑
µ
(
E+Sξ,µ − EiSξ,µ
)(∑
i
cµi cν∗
i
E+ − Ei
)
=∑
µ
∑
i
Sξ,µcµi cν∗
i = δνξ
Using TranSIESTA The integration contour Using tbtrans
The DM from GFs
• If we integrate,
∫∞
−∞
nFD(E )G rµ,ν(E )dE = ???
∫∞
−∞
nFD(E )[∑
i
cµi cν∗
i
E+ − Ei
]
dE =∑
i
cµi cν∗
i
∫∞
−∞
nFD(E )
E+ − Ei
dE
︸ ︷︷ ︸
P[ ]−ıπnFD(Ei )
⇛ Im
[∫
∞
−∞
nFD(E )G rµ,ν(E )dE
]
= −π∑
i
cµi cν∗
i ni
⇛ ρ = −1
πIm
[∫
∞
−∞
nFD(E )Gr (E )dE
]
Using TranSIESTA The integration contour Using tbtrans
Equilibrium DM
• Two ways of computing the Density Matrix,
1. From the Kohn-Sham orbitals,
ρ =X
i
nic∗
i ci
2. From the Retarded Green’s Function
ρ = −1
πIm
"
Z
∞
−∞
nFD(E)Gr(E)dE
#
• With GFs a Self Consistent procedure can be used in the same wayas the “standard” Kohn-Sham orbitals
Using TranSIESTA The integration contour Using tbtrans
The TranSIESTA contour
• TS.ComplexContour.Emin
• TS.ComplexContour.NCircle
• TS.ComplexContour.NLine
• TS.ComplexContour.NPoles
Using TranSIESTA The integration contour Using tbtrans
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),
Using TranSIESTA The integration contour Using tbtrans
Things we know ...
• The G r (E ) is smoother for E = Er + ıEc = Z ,
G rµ,ν(Z ) =
∑
i
cµi cν∗
i
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, . . .
Using TranSIESTA The integration contour Using tbtrans
Contour Integration: Equilibrium• The integral may be obtained in a contour integration,
∫
nFD(E )G r (E )dE =
∫
C
nFD(Z )G r (Z )dZ − 2πıkBT
Np∑
j=1
G r (Zj)
Using TranSIESTA The integration contour Using tbtrans
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)
Using TranSIESTA The integration contour Using tbtrans
From NEGF
• In the non-equilibrium case, the charge density is given by,
ρCC =1
2π
∫ (
G rCC (E )
(f EFD(E )ΓE (E ) + f D
FD(E )ΓD(E ))G a
CC (E ))
dE
• 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
neqCC
ρeqCC = −
1
πIm[
∫
f EFD(E )G r
CC (E )dE ]
ρneqCC =
1
2π
∫
G rCC (E )ΓD(E )G a
CC (E )(f DFD(E ) − f E
FD(E ))dE
Using TranSIESTA The integration contour Using tbtrans
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,
Using TranSIESTA The integration contour Using tbtrans
Outline
Using TranSIESTA
The integration contourElectron density from GFWhy go complex ?The non-equilibrium situation
Using tbtrans
Using TranSIESTA The integration contour Using tbtrans
What is tbtrans ?
• The current I is obtained by the relation,
I =e
h
∫(f EFD(E ) − f D
FD(E ))Tr [ΓE (E )G r (E )ΓD(E )G a(E )]︸ ︷︷ ︸
T (E)
dE
• 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
Using TranSIESTA The integration contour Using tbtrans
How to use it
I =e
h
∫(f EFD(E ) − f D
FD(E ))Tr [ΓE (E )G r (E )ΓD(E )G a(E )]︸ ︷︷ ︸
T (E)
dE
• 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
Using TranSIESTA The integration contour Using tbtrans
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.