steady-state density functional theory for finite-bias

i-DFT for steady-state electron transportFinite-bias Coulomb blockade: requirements for xc potentials

Finite-bias Coulomb blockade for benzeneSummary

Steady-state density functional theory forfinite-bias conductances

Stefan Kurth

1. Universidad del Paıs Vasco UPV/EHU, San Sebastian, Spain2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

3. European Theoretical Spectroscopy Facility (ETSF), www.etsf.eu

S. Kurth Steady-state DFT for finite-bias conductances



In collaboration with:

Gianluca StefanucciUniv. of Rome“Tor Vergata”Rome, Italy




Coulomb Blockade in Transport at Finite BiasExperimental conductances of graphene quantum dot as functionof gate and bias (J. Guttinger et al, PRL 103, 046810 (2009))

Questions:

Can we design a proper DFT framework to calculatefinite-bias conductances?

If yes, what are the essential features for the xc potential(s) todescribe the Coulomb blockade diamonds?




Outline

Steady-state DFT formalism (i-DFT) for electron transport

Choice of variables and 1-1 mapKohn-Sham system: Hxc gate and xc bias potentials

Finite-bias Coulomb blockade: requirements on xc potentials

Single impurity Anderson modelConstant-Interaction Model for multiple levelsApproximate xc potentials for general case

Finite-bias Coulomb blockade for benzene

Summary




Choice of variables and 1-1 mapxc potentials of i-DFT: Hxc gate and xc biasi-DFT self-consistent KS equations

i-DFT for steady-state electron transport

Schematic transport setup with arbitrary molecular region R (withmolecular potential v(r)) and applied bias V , interested only insteady state

Choice of variables for steady-state DFT:

molecular steady-state density n(r) in region R and steady-statecurrent I through R






Theorem:For any finite temperature the map (v(r), V )→ (n(r), I)) isinvertible in a finite, gate-dependent window around bias V = 0.Proof: show that Jacobian

J0 = Det

(δn(r)δv(r′)

∂n(r)∂V

δIδv(r′)

∂I∂V

)V=0

is non-vanishing.note:δI

δv(r′)

∣∣V=0

= 0 since at zero bias a variation of the gate does notinduce a steady current.






Lehmann representation of static density response functionχ0(r, r

′) = δn(r)δv(r′)

∣∣V=0

at finite temperature

χ0(r, r′) =

1

Z

∑i,j

fij(r)fij(r′)

Ω2ij + η2

Ωij(e−βEi − e−βEi)eβµNi

with partition function Z, fij(r) = 〈Ψi|n(r)|Ψj〉 − δijn(r), andΩij = Ei −Ej −→ for Ei ≶ Ej , Ωij ≶ 0 and (e−βEi − e−βEj ) ≷ 0−→ Det[χ0] < 0 for all v(r).

Similarly, using Lehmann representation of G0 = ∂I∂V

∣∣V=0

(D. Bohret al, EPL 73, 246 (2006)) one finds that G0 > 0

−→ J0 = Det[χ0]G0 < 0 for all gates v(r).





xc potentials of i-DFT

let (n, I) be the density and steady-state current of an interactingsystem with gate potential and bias (v, V ). Under assumption ofnon-interacting v-representability there is a unique pair ofpotentials (vs, Vs) which reproduces (n, I) in a non-interactingsystem −→

Hxc gate and xc bias potentials

vHxc[n, I](r) = vs[n, I](r)− v[n, I](r)

Vxc[n, I] = Vs[n, I]− V [n, I]





i-DFT self-consistent KS equations

i-DFT KS equations for density and steady current

n(r) = 2∑α=L,R

∫dω

2πf

(ω + sα

V + Vxc2

)Aα(r, ω)

I = 2∑α=L,R

∫dω

2πf

(ω + sα

V + Vxc2

)sαT (ω)

with KS partial spectral functionAα(r, ω) = 〈r|Gs(ω)Γα(ω)G†s(ω)|r〉, KS transmission functionT (ω) and sR/L = ±1

Note: equivalent to Landauer+DFT formalism if Vxc set to zero





i-DFT expression for zero-bias conductance

linearize i-DFT selfconsistency conditions −→

exact expression for zero-bias conductance

G0 =G0,s

1−G0,s∂Vxc∂I

∣∣I=0

with zero-bias KS conductance G0,s.

Remark: The G0,s of i-DFT is exactly the same as the usualLandauer-DFT zero-bias KS conductance!




Single impurity Anderson model (SIAM)Constant Interaction Model (CIM)Parametrization of CIM xc potentials

Single impurity Anderson model

construct xc potentials for single impurity Anderson model: singlesite (energy v) with a charging energy U for double occupancycoupled (coupling strength γ) to two leads (wide band limit, WBL)





Single impurity Anderson model (SIAM)

density and current from spectral function (in WBL)

N =

∫dω

2π[fβ(ω − V/2) + fβ(ω + V/2)]A(ω)

I =γ

2

∫dω

2π[fβ(ω − V/2)− fβ(ω + V/2)]A(ω)

Model (interacting) spectral function of site connected to leads

A(ω) =N

2Lγ(ω − v − U) +

(1− N

2

)Lγ(ω − v)

with Lorentzian Lγ(ω) = γ

ω2+( γ2 )2

reasonable in Coulomb blockade regime, i.e., for temperaturesabove the Kondo temperature TK





SIAM: reverse engineering for xc potentials

Remark:model spectral function gives exactly the same N and I as the rateequation approach (C.W.J. Beenakker, PRB 44, 1646 (1991))

Reverse engineering for xc potentials:for given (N, I): numerically find the potentials (v, V ) which yieldthis (N, I) first for the interacting case, then find non-interactingpotentials (vs, Vs) which also yield same densities −→ construct(H)xc potentials as(vHxc[N, I], Vxc[N, I]

)=

(vs[N, I]− v[N, I], Vs[N, I]−V [N, I]

)





SIAM: Hxc gate and xc bias potentials

SIAM (H)xc potentials from reverse engineering

Hxc gate

0

0.5

1

1.5

2

-0.4

-0.2

0

0.2

0.4

vHxc

N

I/γ

vHxc

0

0.2

0.4

0.6

0.8

1

xc bias

0

0.5

1

1.5

2

-0.4

-0.2

0

0.2

0.4

Vxc

N

I/γ

Vxc

-1

-0.5

0

0.5

1

Domain: |I| ≤ γ2N for N ∈ [0, 1] and |I| ≤ γ

2 (2−N) for N ∈ [1, 2]






Hxc gate (xc bias) hassmeared steps of heightU/2 (U)

DFT xc discontinuity atN = 1 in vHxc[N, I = 0]bifurcates as currentstarts flowing

xc bias has opposite signof current, i.e., xc biascounteracts external bias






simple parametrization (low temperature regime)

parametrization of xc potentials for SIAM

vHxc[N, I] =U

4

∑s=±

[1 +

2

πatan

(N + s

γ I − 1

W

)]

Vxc[N, I] = −U∑s=±

s

πatan

(N + s

γ I − 1

W

)

parameter W = 0.16γ/U to obtain best fit to reverse-engineeredxc potentials





Constant Interaction Model (CIM)

consider CIM with M degenerate single-particle (sp) levels(SIAM −→M = 1); same coupling γ to leads of all sp levels

all sp levels equivalent → Hxc gate and xc bias depend only ontotal number of electrons N and I





CIM with M=3: Hxc gate and xc bias potentials

Hxc potentials by reverse engineering of (N, I) as function of(v, V ) obtained from Beenakker’s rate equations (RE)

again smeared steps ofheight U/2 (U) for Hxcgate (xc bias)

edges at piecewise linear

functions of ∆(±)K (N, I)

vertices in (N, I) planecorrespond to plateauvalues of (N, I) ingate-bias scan of RE; canbe obtained analytically





parametrization of degenerate CIM xc potentials

simple parametrization (low temperature regime)

parametrization of xc potentials for CIM with M degenerate levels

v(M)Hxc [N, I] =

U

4

2M−1∑K=1

∑s=±

[1 +

2

πatan

(∆

(s)K (N, I)

W

)]

V (M)xc [N, I] = −U

2M−1∑K=1

∑s=±

s

πatan

(∆

(s)K (N, I)

W

)

∆(±)K (N, I) piecewise linear function of N and I which vanishes at

step edges passing through (N = K, I = 0) with positive (s = +)and negative (s = −) slopes





parametrization for general single-particle level structure

parametrization of CIM xc potentials for arbitrary sp level structure

vHxc[n, I] =D∑p=1

v(Mp)Hxc

[N −Np, I

]+U

4

D−1∑p=1

∑s=±

[1 +

2

πatan

(N + 2s

γ I −Np+1

W

)]

and similarly for xc bias Vxc[n, I]

D: number of different sp energies (not counting degeneracies)Mp: degeneracy of p-th sp levelNp: maximal number of particles in first p− 1 degenerate levels





model benzene as a 6-level CIM with ε1 = −ε6 = −5.08 eV andε2 = ε3 = −ε4 = −ε5 = −2.54 eV; U = 0.5 eVsolve i-DFT equations for (N, I) with model xc potentials,calculate differential conductance map dI

dV as function of (v, V )





LB-DFT, i.e. neglecting xc bias, gives spurious Kondoplateaus at finite bias; low-bias CB not correctly describedi-DFT in simple approx. gets all low-bias features of the REsize of RE matrix scales as 4Nlev ; simple i-DFT doesn’t scaleat all (2 coupled nonlinear eqs. for N and I)




Summary

i-DFT: DFT formulation for steady-state transport at arbitrary bias

i-DFT based on 1-1 map (for small bias) between (n(r), I)and (v(r), V )

i-DFT KS formulation with both Hxc gate and xc bias

xc potentials from models to describe Coulomb blockade: stepstructures in xc potentials; xc derivative discontinuity ofequilibrium DFT bifurcates as current starts flowing

simple model xc potentials with correct step structure aresufficient to describe finite-bias CB correctly for not too largebias

Reference:

Stefanucci, Kurth, arXiv:1505.07354


steady-state density functional theory for finite-bias

Documents