Thermal density-functional theory: Towards the ab-initio description of thermoelectric transport at the nano scale

Transcription

1 Thermal density-functional theory: Towards the ab-initio description of thermoelectric transport at the nano scale F. G. Eich Max Planck Institute for the Structure and Dynamics of Matter 7th Time-dependent Density-Functional Theory: Prospects and Benasque, Spain, September 22, 2016 F. G. E.

2 References and Acknowledgements Max Di Ventra Giovanni Vignale Recent review arxiv: Research supported by DFG Grant No. EI 1014/1-1, and DOE Grant No. DE-FG02-05ER46203/4 F. G. E.

3 Motivation Experimental advances Transport through nanojunctions Measuring temperatures at the nanoscale Mecklenburg et al., Science, 347 (6222): (2015) F. G. E.

4 Typical nanoscale transport setup Experimental advances Transport through nanojunctions F. G. E.

5 Typical nanoscale transport setup Experimental advances Transport through nanojunctions ( ) ( ) ( I σ σst0 δµ = J σπ T 0 κ + σπ 2 δt T 0 ) F. G. E.

6 Microscopic description Functional construction and Luttinger s ψ-field Including the effect of electron-electron interactions (FGE, M. Di Ventra, G. Vignale, Phys. Rev. Lett. 112, (2014)) DFT efficient tool to incorporate interactions Standard formulation focuses on charge density Our proposal: Inclusion of the energy density as basic variable ( ) ( ) ( I σ σst0 δµ = J σπ T 0 κ + σπ 2 δt T 0 ) F. G. E.

7 Microscopic description Functional construction and Luttinger s ψ-field Including the effect of electron-electron interactions (FGE, M. Di Ventra, G. Vignale, Phys. Rev. Lett. 112, (2014)) DFT efficient tool to incorporate interactions Standard formulation focuses on charge density Our proposal: Inclusion of the energy density as basic variable ( ) ( I σ σst0 = J σπ T 0 κ + σπ 2 ) ( ) δµ + δv δt T 0 F. G. E.

8 Microscopic description Functional construction and Luttinger s ψ-field Luttinger s ψ - field (J. M. Luttinger, Phys. Rev. 135, A1505 (1964)) Mechanical proxy for local temperature variations (B. S. Shastry, Rep. Prog. Phys. 72, (2009)) Low frequencies ω 1/τ 0, long wave length k 1/l 0 Allows to compute conductivities via Kubo formalism ( ) ( I σ σst0 = J σπ T 0 κ + σπ 2 ) ( ) δµ + δv + δψ δt T 0 F. G. E.

9 Microscopic description Functional construction and Luttinger s ψ-field Including the effect of electron-electron interactions (FGE, M. Di Ventra, G. Vignale, Phys. Rev. Lett. 112, (2014)) DFT efficient tool to incorporate interactions Standard formulation focuses on charge density Our proposal: Inclusion of the energy density as basic variable ı tφ i (r) = ψ + ψxc 2 m + (1 + ψ)v + ṽ Hxc + v xc φ i (r) F. G. E.

10 Peculiarities of the KS construction Microscopic description Functional construction v xc(r, t) = δ S xc[n, h s] δn(r, t) ψ xc(r, t) = δ S xc[n, h s] δh s(r, t) How to connect the interacting system and the noninteracting KS system? F. G. E.

11 Peculiarities of the KS construction Microscopic description Functional construction v xc(r, t) = δ S xc[n, h s] δn(r, t) ψ xc(r, t) = δ S xc[n, h s] δh s(r, t) How to connect the interacting system and the noninteracting KS system? Interacting system n(r, t) h(r, t) h eq [n(t)](r) = = n(r, t) KS system h s(r, t) h eq s [n(t)](r) The KS system reproduces the excess energy density!! h(r, t) = h s(r, t) + E Hxc [n(t)](r), E Hxc [n(t)](r) = h eq [n(t)](r) h eq s [n(t)](r) F. G. E.

12 Microscopic description Functional construction Adiabatic local density approximation ı tφ i (r) = [ ψ + ψ xc ALDA m ] + (1 + ψ) (v + v Hxc ) + v xc ALDA φ i (r) v xc ALDA (r, t) = 1 s xc(n(r, t), h s(r, t)) β n(r, t) ψ xc ALDA (r, t) = 1 s xc(n(r, t), h s(r, t)) β h s(r, t) F. G. E.

13 Microscopic description Functional construction Adiabatic local density approximation ı tφ i (r) = [ ψ + ψ xc ALDA m ] + (1 + ψ) (v + v Hxc ) + v xc ALDA φ i (r) v xc ALDA (r, t) = 1 s xc(n(r, t), h s(r, t)) β n(r, t) ψ xc ALDA (r, t) = 1 s xc(n(r, t), h s(r, t)) β h s(r, t) s xc(n(r, t), h s(r, t)) = s unif (n, h s + ɛ Hxc (n)) s unif s (n, h s)... where s unif (n, h), ɛ unif xc (n) is taken from the uniform electron gas (PRL 110, , 2013, PRL 112, ) F. G. E.

14 Dynamical correction Motivation Microscopic description Functional construction Inverting the linear response relation ( v xc dyn dyn ψ xc ) = ( L 1 L 1 s ) ( ) j j q F. G. E.

15 Dynamical correction Motivation Microscopic description Functional construction Inverting the linear response relation ( v xc dyn dyn ψ xc ) = [( ρ + β 0Π 2 β 0 Π κ κ β 0 Π κ β 0 κ ) ( ρ s + β 0Π 2 s κ s β 0 Π s κ s β 0 Π s κ s β 0 κ s )] ( ) j j q F. G. E.

16 Dynamical correction Motivation Microscopic description Functional construction Inverting the linear response relation ( v xc dyn dyn ψ xc ) = ( 1 n ηxc 1 n + β 0Π 2 κ β 0 Π κ β 0 Π κ β 0 κ ) ( ) j j q F. G. E.

17 Dynamical correction Motivation Microscopic description Functional construction Inverting the linear response relation ( v xc dyn dyn ψ xc ) = ( 1 n ηxc 1 n + β 0Π 2 κ β 0 Π κ β 0 Π κ β 0 κ ) ( ) j j q Vignale-Kohn type relaxation W xc = d 3 r j η xc 2 + β 0 n κ j q + Πj 2 F. G. E.

18 F. G. E.

19 Steady state heat current through single site impurity LB+ Q β V TM U/t 1.0 Q β V TM 0.5 LB U/t F. G. E.

20 Time-dependent density Motivation 0.01 δn(t) 0.00 I R (t) I L (t) t/τ F. G. E.

21 Time-dependent density Motivation 0.01 δn(t) 0.00 I R (t) I L (t) t/τ F. G. E.

22 Time-dependent density Motivation 0.01 δn(t) 0.00 I R (t) I L (t) t/τ F. G. E.

23 Time-dependent density Motivation 0.01 δn(t) 0.00 I R (t) I L (t) t/τ F. G. E.

24 Temperature induced charge and energy wave in atomic chain (100 sites) δn i (t) 45.0τ 40.0τ 35.0τ 30.0τ 25.0τ 20.0τ 15.0τ 10.0τ 5.0τ δh i (t) 45.0τ 40.0τ 35.0τ 30.0τ 25.0τ 20.0τ 15.0τ 10.0τ 5.0τ F. G. E.

25 Defining a local temperature via zero-current conditions I α = 1 J α = 1 1 2π α 1 2π α dɛ T αα (ɛ) (f α f α ) dɛ ɛt αα (ɛ) (f α f α ) F. G. E.

26 Defining a local temperature via zero-current conditions n 0 = α h 0 = α 1 2π 1 2π dɛ D α (ɛ)f α = 1 2π dɛ ɛd α (ɛ)f α = 1 2π dɛ D(ɛ)f probe dɛ ɛd(ɛ)f probe F. G. E.

27 for single site impurity ψprobe 0 LB 2 TM ψprobe ψ TM ψ LB F. G. E.

28 for atomic chain (100 sites) (I) δt i /T 0 for T 0 = 0.025V δt i /T 0 for T 0 = 0.25V F. G. E.

29 for atomic chain (100 sites) (II) δt i /T 0 for T 0 = 0.025V δt i /T 0 for T 0 = 0.25V 0.3 F. G. E.

30 Summary Motivation Conclusion : ab-initio description of thermoelectric transport Time-dependent Thermoelectrics at the nano scale F. G. E.

31 Summary Motivation Conclusion : ab-initio description of thermoelectric transport Time-dependent Thermoelectrics at the nano scale Outlook and open questions Development of functionals Implementation of thermal KS equations When does the electron-electron interaction become important for energy transport (Hydrodynamic regime)? What is this ψ potential? Can it be microscopically justified? F. G. E.

32 Thank you for you attention! F. G. E.

33 The action functional Motivation complex time plane t 0 T t 0 i hβ Keldysh contour C A[ṽ, ψ] ı lntr {T τ e ı C[Ĥ+ d 3 r {ψ(r,τ)ĥ(r)+ṽ(r,τ)ˆn(r)}]} F. G. E.

34 The action functional Motivation complex time plane t 0 T t 0 i hβ Keldysh contour C A[ṽ eq, ψ eq ] = ı lntr {e β[ĥ µ ˆN+ d 3 r {ψ(r)ĥ(r)+ṽ(r)ˆn(r)}]} F. G. E.

35 The action functional Motivation complex time plane t 0 T t 0 i hβ Keldysh contour C ıβω[ṽ eq, ψ eq ] = ı lntr {e β[ĥ µ ˆN+ d 3 r {ψ(r)ĥ(r)+ṽ(r)ˆn(r)}]} F. G. E.

36 The action functional Motivation complex time plane t 0 T t 0 i hβ Keldysh contour C A[ṽ, ψ] ı lntr {T τ e ı C[Ĥ+ d 3 r {ψ(r,τ)ĥ(r)+ṽ(r,τ)ˆn(r)}]} h(r, τ) = δa[ṽ, ψ] δψ(r, τ) n(r, τ) = δa[ṽ, ψ] δṽ(r, τ) F. G. E.

37 The action functional Motivation complex time plane t 0 T t 0 i hβ Keldysh contour C A[ṽ, ψ] ı lntr {T τ e ı C[Ĥ+ d 3 r {ψ(r,τ)ĥ(r)+ṽ(r,τ)ˆn(r)}]} h(r, τ) = δa[ṽ, ψ] δψ(r, τ) n(r, τ) = δa[ṽ, ψ] δṽ(r, τ) Legendre transformation yields A[n, h] δa[n, h] ψ(r, τ) = δh(r, τ) δa[n, h] ṽ(r, τ) = δn(r, τ) F. G. E.

38 Decomposition of the action functional A adia [n, h] = C = C Adiabatic approximation F [n(τ), h(τ)] = C d 3 r h(r, τ) 1 S adia [n, h] β d 3 r h(r, τ) 1 S[n(τ), h(τ)] β C F. G. E.

39 Decomposition of the action functional A adia [n, h] = C = C Adiabatic approximation F [n(τ), h(τ)] = C d 3 r h(r, τ) 1 S adia [n, h] β d 3 r h(r, τ) 1 S[n(τ), h(τ)] β C Dynamical contribution (everything else!) A[n, h] = d 3 r h(r, τ) 1 ( Sadia [n, h] + S dyn [n, h]) C β F. G. E.

40 Decomposition of the action functional A adia [n, h] = C = C Adiabatic approximation F [n(τ), h(τ)] = C d 3 r h(r, τ) 1 S adia [n, h] β d 3 r h(r, τ) 1 S[n(τ), h(τ)] β C Dynamical contribution (everything else!) A[n, h] = d 3 r h(r, τ) 1 ( Sadia [n, h] + S dyn [n, h]) C β Exchange-correlation contribution S xc[n, h s] = S [ n, h s + ɛ Hxc [n] ] S s [ n, hs ] F. G. E.