Simulation of nanostructures

László Oroszlány

KRFT, ELTE

Trends in Nanotechnology @ BME

DFT

  • Basics
  • Functionals
  • When to use DFT ?
  • When not to use DFT ?
  • state of the art:
    codes and stories

Landauer 

  • Landauer's approach 
  • Green's functions
  • Transmission
  • Generalizations
  • state of the art:
    codes and stories

Outline

This is meant to be an apetizer...

But it might feel like...

Density Functional Theory

Act I

The many body problem

\displaystyle{\hat{H}=\sum_i \frac{\hat{p_i}^2}{2m} +\sum_{ij} \hat{V}_{ee}(r_i-r_j)+\sum_{ij} \hat{V}_{eN}(r_i-R_j) }
H^=∑ipi^22m+∑ijV^ee(ri−rj)+∑ijV^eN(ri−Rj)\displaystyle{\hat{H}=\sum_i \frac{\hat{p_i}^2}{2m} +\sum_{ij} \hat{V}_{ee}(r_i-r_j)+\sum_{ij} \hat{V}_{eN}(r_i-R_j) }
\hat H\psi_n(r_1,r_2,\ldots)=E_n\psi_n(r_1,r_2,\ldots)
H^ψn(r1,r2,…)=Enψn(r1,r2,…)\hat H\psi_n(r_1,r_2,\ldots)=E_n\psi_n(r_1,r_2,\ldots)

Universal

System specific

an AWFULL LOT \(\approx 10^{23}\) of variables..

Hohenberg–Kohn theorems

Theorem 1: The external potential (and hence the total energy), is a unique functional of the electron density.

 

Theorem 2: The ground state density minimizes the total energy functional.

 

\exists! E_{Tot}[n(r)]
∃!ETot[n(r)]\exists! E_{Tot}[n(r)]
\displaystyle n(r)=\int \mathrm{d}r_2 \mathrm{d}r_3\ldots|\psi(r,r_2,r_3,\ldots)|^2
n(r)=∫dr2dr3…∣ψ(r,r2,r3,…)∣2\displaystyle n(r)=\int \mathrm{d}r_2 \mathrm{d}r_3\ldots|\psi(r,r_2,r_3,\ldots)|^2
\left . \displaystyle \delta E_{Tot}[n(r)] \right |_{n_{GS}(r)}=0
δETot[n(r)]∣nGS(r)=0\left . \displaystyle \delta E_{Tot}[n(r)] \right |_{n_{GS}(r)}=0

Phys. Rev. 136, B864  (1964)

Recap on Functionals

a function maps numbers to numbers:

\mathbb{R}\rightarrow \mathbb{R}
R→R\mathbb{R}\rightarrow \mathbb{R}
x\rightarrow x^2-x-2
x→x2−x−2x\rightarrow x^2-x-2

a functional maps functions to numbers:

(\mathbb{R}\rightarrow\mathbb{R})\Rightarrow\mathbb{R}
(R→R)⇒R(\mathbb{R}\rightarrow\mathbb{R})\Rightarrow\mathbb{R}
(x\rightarrow x^2-x-2)\Rightarrow -\frac{9}{2}
(x→x2−x−2)⇒−92(x\rightarrow x^2-x-2)\Rightarrow -\frac{9}{2}
\displaystyle\int_{-1}^2 (x^2-x-2)\mathrm{d}x=-\frac{9}{2}
∫−12(x2−x−2)dx=−92\displaystyle\int_{-1}^2 (x^2-x-2)\mathrm{d}x=-\frac{9}{2}

Kohn–Sham equations and the scf loop

E_{Tot}[n(r)]=T[n]+E_{ext}[n]+E_{H}[n]+E_{xc}[n]
ETot[n(r)]=T[n]+Eext[n]+EH[n]+Exc[n]E_{Tot}[n(r)]=T[n]+E_{ext}[n]+E_{H}[n]+E_{xc}[n]
\displaystyle n(r)=\sum_{i=0}^{N_{occ}} |\phi_i(r)|^2
n(r)=∑i=0Nocc∣ϕi(r)∣2\displaystyle n(r)=\sum_{i=0}^{N_{occ}} |\phi_i(r)|^2
T[n]=\displaystyle\sum_i \left \langle\phi_i\right|-\frac{\hbar^2\Delta}{2m}\left |\phi_i\right\rangle
T[n]=∑i⟨ϕi∣−ℏ2Δ2m∣ϕi⟩T[n]=\displaystyle\sum_i \left \langle\phi_i\right|-\frac{\hbar^2\Delta}{2m}\left |\phi_i\right\rangle
\displaystyle E_{ext}=\int V_{ext}(r)n(r) \mathrm{d}r
Eext=∫Vext(r)n(r)dr\displaystyle E_{ext}=\int V_{ext}(r)n(r) \mathrm{d}r
\displaystyle E_{H}=\frac{e^2}{2}\int \frac{n(r')n(r)}{|r-r'|} \mathrm{d}r\mathrm{d}r'
EH=e22∫n(r′)n(r)∣r−r′∣drdr′\displaystyle E_{H}=\frac{e^2}{2}\int \frac{n(r')n(r)}{|r-r'|} \mathrm{d}r\mathrm{d}r'
\displaystyle E_{xc}=?=\mathcal{F}[n(r)]
Exc=?=F[n(r)]\displaystyle E_{xc}=?=\mathcal{F}[n(r)]
\displaystyle v_{eff}(r)=\frac{\delta}{\delta n}[E_{ext}+E_{H}+E_{xc}]
veff(r)=δδn[Eext+EH+Exc]\displaystyle v_{eff}(r)=\frac{\delta}{\delta n}[E_{ext}+E_{H}+E_{xc}]
\displaystyle \left (\frac{-\hbar^2\Delta}{2m}+v_{eff}(r)\right) \phi_i(r)=E_i \phi_i(r)
(−ℏ2Δ2m+veff(r))ϕi(r)=Eiϕi(r)\displaystyle \left (\frac{-\hbar^2\Delta}{2m}+v_{eff}(r)\right) \phi_i(r)=E_i \phi_i(r)

Phys. Rev. 140, A1133  (1965)

Functionals of exchange and correlation

  • S: The Slater exchange, ρ4/3 with theoretical coefficient of 2/3, also referred to as Local Spin Density exchange [Hohenberg64, Kohn65, Slater74]. Keyword if used alone: HFS.
  • XA: The XAlpha exchange, ρ4/3 with the empirical coefficient of 0.7, usually employed as a standalone exchange functional, without a correlation functional [Hohenberg64, Kohn65, Slater74]. Keyword if used alone: XAlpha.
  • B: Becke’s 1988 functional, which includes the Slater exchange along with corrections involving the gradient of the density [Becke88b]. Keyword if used alone: HFB.
  • PW91: The exchange component of Perdew and Wang’s 1991 functional [Perdew91, Perdew92, Perdew93a, Perdew96, Burke98].
  • mPW: The Perdew-Wang 1991 exchange functional as modified by Adamo and Barone [Adamo98].
  • G96: The 1996 exchange functional of Gill [Gill96, Adamo98a].
  • PBE: The 1996 functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97].
  • O: Handy’s OPTX modification of Becke’s exchange functional [Handy01, Hoe01].
  • TPSS: The exchange functional of Tao, Perdew, Staroverov, and Scuseria [Tao03].
  • RevTPSS: The revised TPSS exchange functional of Perdew et. al. [Perdew09, Perdew11].
  • BRx: The 1989 exchange functional of Becke [Becke89a].
  • PKZB: The exchange part of the Perdew, Kurth, Zupan and Blaha functional [Perdew99].
  • wPBEh: The exchange part of screened Coulomb potential-based final of Heyd, Scuseria and Ernzerhof (also known as HSE) [Heyd03, Izmaylov06, Henderson09].
  • PBEh: 1998 revision of PBE [Ernzerhof98].
  • VWN: Vosko, Wilk, and Nusair 1980 correlation functional(III) fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation [Vosko80] (functional III in this article).
  • VWN5: Functional V from reference [Vosko80] which fits the Ceperly-Alder solution to the uniform electron gas (this is the functional recommended in [Vosko80]).
  • LYP: The correlation functional of Lee, Yang, and Parr, which includes both local and non-local terms [Lee88, Miehlich89].
  • PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981) [Perdew81].
  • P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local correlation functional [Perdew86].
  • PW91 (Perdew/Wang 91): Perdew and Wang’s 1991 gradient-corrected correlation functional [Perdew91, Perdew92, Perdew93a, Perdew96, Burke98].
  • B95 (Becke 95): Becke’s τ-dependent gradient-corrected correlation functional (defined as part of his one parameter hybrid functional [Becke96]).
  • PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97].
  • TPSS: The τ-dependent gradient-corrected functional of Tao, Perdew, Staroverov, and Scuseria [Tao03].
  • RevTPSS: The revised TPSS correlation functional of Perdew et. al. [Perdew09, Perdew11].
  • KCIS: The Krieger-Chen-Iafrate-Savin correlation functional [Rey98, Krieger99, Krieger01, Toulouse02].
  • BRC: Becke-Roussel correlation functional [Becke89a].
  • PKZB: The correlation part of the Perdew, Kurth, Zupan and Blaha functional [Perdew99].
  • VP86: VWN5 local and P86 non-local correlation functional.
  • V5LYP: VWN5 local and LYP non-local correlation functional.
  • VSXC: van Voorhis and Scuseria’s τ-dependent gradient-corrected correlation functional [VanVoorhis98].
  • HCTH/*: Handy’s family of functionals including gradient-corrected correlation [Hamprecht98, Boese00, Boese01]. HCTH refers to HCTH/407, HCTH93 to HCTH/93, HCTH147 to HCTH/147, and HCTH407 to HCTH/407. Note that the related HCTH/120 functional is not implemented.
  • tHCTH: The τ-dependent member of the HCTH family [Boese02]. See also tHCTHhyb below.
  • B97D: Grimme’s functional including dispersion [Grimme06]. B97D3 requests the same but with Grimme’s D3BJ dispersion [Grimme11].
  • M06L [Zhao06a], SOGGA11 [Peverati11], M11L [Peverati12], MN12L [Peverati12c] N12 [Peverati12b] and MN15L [Yu16a] request these pure functionals from the Truhlar group.
\displaystyle E^{LDA}_{xc}[n]=\int \epsilon_{LDA}(n)n(r) \mathrm{d}r
ExcLDA[n]=∫ϵLDA(n)n(r)dr\displaystyle E^{LDA}_{xc}[n]=\int \epsilon_{LDA}(n)n(r) \mathrm{d}r
\displaystyle E^{GGA}_{xc}[n]=\int \epsilon_{GGA}(n,|\nabla n|)n(r) \mathrm{d}r
ExcGGA[n]=∫ϵGGA(n,∣∇n∣)n(r)dr\displaystyle E^{GGA}_{xc}[n]=\int \epsilon_{GGA}(n,|\nabla n|)n(r) \mathrm{d}r
\displaystyle E^{LSDA}_{xc}[n_\uparrow,n_\downarrow]=\int \epsilon_{LSDA}(n_\uparrow,n_\downarrow)n(r) \mathrm{d}r
ExcLSDA[n↑,n↓]=∫ϵLSDA(n↑,n↓)n(r)dr\displaystyle E^{LSDA}_{xc}[n_\uparrow,n_\downarrow]=\int \epsilon_{LSDA}(n_\uparrow,n_\downarrow)n(r) \mathrm{d}r

The choice of basis

\displaystyle\phi_i(r)=\sum_pa_{ip}\chi_p(r)
ϕi(r)=∑paipχp(r)\displaystyle\phi_i(r)=\sum_pa_{ip}\chi_p(r)
PRO CON
Plane waves
 		 -simple to implement
-one convergence parameter		 - resource hungry
- big basis set needed
Atomic like orbitals - small basis
- sparse matrix methods
- scales to BIG systems		 -harder to implement
-many things influence
 convergence
KKR method
baseless method..
relies on scattering approach and Green's functons		 -relativistic effects are easy
-standard for magnetic systems
-most accurate for some  problems		 -harder to implement
-smaller community
-some features not yet  implemented..
\chi_p(r)=\mathrm{e}^{\mathrm{i}pr}
χp(r)=eipr\chi_p(r)=\mathrm{e}^{\mathrm{i}pr}
\chi_{nlm}(r)=R_{nl}(\rho)Y_{l}^m(\hat{r})
χnlm(r)=Rnl(ρ)Ylm(r^)\chi_{nlm}(r)=R_{nl}(\rho)Y_{l}^m(\hat{r})

Pseudopotentials

Motivation:

  • Reduction of basis set size
  • Reduction of number of electrons
  • Inclusion of relativistic and other effects

Approximations:

  • One-electron picture.
  • The small-core approximation 

 

What is DFT good at?

  • structural properties



     
  • electronic structure of simple systems

Failures of DFT

  • GAPs
     

 

 

 

  • Strong(ish) interactions

LDA+U:

J. Phys.: Cond. Matt. 9,767 (1997)

DMFT:

Phys. Rev. Lett. 62, 324 (1989)

QMC:

Rev. Mod. Phys. 73, 33 (2001)

GW:

Phys. Rev. 139, A796 (1965)

BSE:

Phys. Rev. Lett. 75, 818 (1995)

State of the art-VASP

6000 atom

3000 proc

120s/scf

Massively  

parallel

Phonon spectrum

J. Chem. Phys. 143, 064710 (2015)

Excitations with

GW method

Phys. Rev. B 75, 235102 (2007)

https://www.vasp.at/

State of the art-Siesta

J. Phys.: Cond. Matt. 27, 054004 (2015)

J. Phys.: Cond. Matt. 26, 305503 (2014)

6000 atom

2000 proc

10 sec/scf

SIESTA-PEXSI parallelization

New J.  Phys., 16, 093029 (2014)

Phys. Rev. B 65, 165401 (2002)

Local quantities (e.g. STM)

Ideal as an input for transport calculations

https://launchpad.net/siesta

State of the art-KKR

KKRNano @ http://www.judft.de/   massive parallelization

PRB 94, 104511 (2016)

abinitio superconductivity

PRB 89, 224401 (2014)

finite temperature magnetism

PRB 82, 024411 (2010)

ARPES+DMFT

Budapest

München

Jülich

Landauer's Approach

Act II

Conductance quantum

V_L
VLV_L
V_R
VRV_R
\sigma=\frac{e^2}{h}
σ=e2h\sigma=\frac{e^2}{h}
I
II
I=\sigma\delta V
I=σδVI=\sigma\delta V
\approx 3.874\times 10^{-5} \Omega^{-1}
≈3.874×10−5Ω−1\approx 3.874\times 10^{-5} \Omega^{-1}

IBM J. Res. Dev. 1, 223 (1957 )

Conductance of a sample

V_L
VLV_L
V_R
VRV_R
\sigma=\frac{e^2}{h}T
σ=e2hT\sigma=\frac{e^2}{h}T
T
TT

IBM J. Res. Dev. 1, 223 (1957 )

Green's functions

\hat{H}_0 = \begin{pmatrix} \hat{H}_{s} & 0\\ 0 & \hat{H}_{leads} \end{pmatrix}
H^0=(H^s00H^leads) \hat{H}_0 = \begin{pmatrix} \hat{H}_{s} & 0\\ 0 & \hat{H}_{leads} \end{pmatrix}
\hat{V}=\left(\begin{array}{cc} 0 & \hat{V}_{SL}\\ \hat{V}_{SL}^{\dagger} & 0 \end{array}\right)
V^=(0V^SLV^SL†0)\hat{V}=\left(\begin{array}{cc} 0 & \hat{V}_{SL}\\ \hat{V}_{SL}^{\dagger} & 0 \end{array}\right)
\hat{G}_0=(E-\hat{H}_0)^{-1}
G^0=(E−H^0)−1\hat{G}_0=(E-\hat{H}_0)^{-1}
\hat{G}=(\hat{G}_0^{-1}-\hat{V})^{-1}
G^=(G^0−1−V^)−1\hat{G}=(\hat{G}_0^{-1}-\hat{V})^{-1}

The structure of \( V\) is such that in order to find matrix elements of \(G\) in the neighborhood of the scatterer one only need to invert a FINITE matrix!

Dyson's equation

\hat{H}=\hat{H}_0+\hat{V}
H^=H^0+V^\hat{H}=\hat{H}_0+\hat{V}

How to get \(T\)?

\hat{G}\text{ from :}
G^ from :\hat{G}\text{ from :}
\hat{H}\text{ from :}
H^ from :\hat{H}\text{ from :}
  • from TB
  • fitting to DFT
  • directly from DFT
T=\mathrm{Tr}[t^\dagger t]
T=Tr[t†t]T=\mathrm{Tr}[t^\dagger t]
  1. take \(\hat{H}\)
  2. get \(\hat{G}_0\)
  3. use Dyson's
S_{pq}=\mathrm{i}\sqrt{v_pv_q}\left \langle \phi_p \right |G\left | \phi_q \right \rangle-\delta_{pq}
Spq=ivpvq⟨ϕp∣G∣ϕq⟩−δpqS_{pq}=\mathrm{i}\sqrt{v_pv_q}\left \langle \phi_p \right |G\left | \phi_q \right \rangle-\delta_{pq}

the scattering matrix:

S=\begin{pmatrix} r&t'\\ t&r' \end{pmatrix}
S=(rt′tr′)S=\begin{pmatrix} r&t'\\ t&r' \end{pmatrix}

two terminal case:

Fisher-Lee relation: Phys. Rev. B 23, 6851(R) (1981)

Extension to non-equilibrium

Steady state current due to a finite bias.

\displaystyle I=\int_{\mu_R}^{\mu_L}T(E)\mathrm{d}E
I=∫μRμLT(E)dE\displaystyle I=\int_{\mu_R}^{\mu_L}T(E)\mathrm{d}E

Non-equilibrium population of scatterer needs to be taken in to account! \(\Rightarrow\) Keldysh formalism
H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors

Example

Negative differential resistance

in molecular junctions:

Nanotechnology 19, 455203 (2008)

Extension to include Interactions

New Journal of Physics 16 093029 (2014)

Extension to include Superconductivity

 Phys. Rev. B 93, 224510 (2016)

\displaystyle J=\frac{1}{\pi}\mathrm{Im}\int_\Gamma f(z)\mathrm{Tr}[\hat{\mathcal{G}}_{BdG}(z)\hat{J}_{BdG}]
J=1πIm∫Γf(z)Tr[G^BdG(z)J^BdG]\displaystyle J=\frac{1}{\pi}\mathrm{Im}\int_\Gamma f(z)\mathrm{Tr}[\hat{\mathcal{G}}_{BdG}(z)\hat{J}_{BdG}]
\mathcal{\hat H}_{BdG}=\begin{pmatrix} \hat{H} & \hat{\Delta} \\ \hat{\Delta}^\dagger & -\hat{H}^* \end{pmatrix}
H^BdG=(H^Δ^Δ^†−H^∗)\mathcal{\hat H}_{BdG}=\begin{pmatrix} \hat{H} & \hat{\Delta} \\ \hat{\Delta}^\dagger & -\hat{H}^* \end{pmatrix}

https://kwant-project.org/

written in python, ideal for quick and dirty calculations

2x100x400 site disordered topological insulator

quantum Hall effect

flying qbit

Equus

http://eqt.elte.hu/equus/home

conductivity of a 4\(\mu m\) wide graphene ribbon

graphene ribbon antidot in \(\mathbf{B}\) field

1\(\mu m \times \) 1\(\mu m\)

BiTeI/graphene/BiTeI sandwitch

graphene Josephson junction

Gollum

Multi terminal calculations

Statistical analysis of environmental effects

http://www.physics.lancs.ac.uk/gollum/

Magneto transport

thermoelectric properties

A Multi scale story: Indium MCBJ

Makk et al.,  Phys. Rev. Lett. 107 276801 (2011)

Take home message

  • Chemically specific simulation of simple quantities with a cople of thousand atoms!
  • Simple quantities of simple models with milions of degrees of freedom!
  • Quantites that require an energy integral as a rule of thumb require one order of magnitude more resources!
  • Full many body treatment of large scales is still a challenge!
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