Spectral analysis, perturbation theory, and applications to molecular ...

Faculty of Science Institute for Theoretical Physics, Condensed Matter Theory 

The master equation for nanoscopic transport: 

Spectral analysis, perturbation theory, and 

applications to molecular devices 

Carsten Timm 

source 

drain 

gate 

Spin-Orbit and Interaction Effects in Nano-Electronics 

04.02.-06.02.2013

Overview 

• The master equation 

• Perturbative expansion of Pauli master equations 

• Spectrum of the transition-rate matrix 

• Realistic modeling for N@C 60 

• Summary

Overview 





• Summary

Quantum master equation(s) 

source 

drain 

gate 

metal, semiconductor… 

Small system (“dot”) coupled to large reservoirs, out of equilibrium 

local dot observable 

current

only depends on the dot: 

with reduced density operator (in “small” dot Fock space) 

basis of lead states only 

Only useful if we can find an equation of motion for 

Master Equation 

Derivation starts from the exact von Neumann equation 

with the Liouvillian superoperator

Projection onto relevant states: projection superoperators 

with 

relevant part 

Example (a): 

irrelevant part 

leads in (separate) equilibrium; 

actual lead state ignored; 

same information as 

Example (b): 

master equations 

for and ? 

diagonal projection: consider only 

diagonal components of 

(probabilities ) 

→ Pauli master equation 

cf. R. Zwanzig, J. Chem. Phys. 33, 

1338 (1960)

More specifically: 

Hamiltonian 

Liouvillian 

source 

drain 

• includes local interactions 

(Coulomb, electron-vibron, electron-spin) 

gate 

• hybridization between dot state and lead state in lead : 

: bilinear hopping term

1. Nakajima-Zwanzig master equation 

S. Nakajima, Prog. Theor. Phys. 20, 948 (1958); 

R. Zwanzig, J. Chem. Phys. 33, 1338 (1960); Physica 30, 1109 (1964) 

allow time-dependent tunneling, specifically switching 

with rate 

von Neumann equation 

and 

solve second equation for , assume , insert into first equation… 

see, e.g., Breuer and Petruccione, The Theory of Open Quantum Systems 

(Oxford)

Nakajima-Zwanzig master equation 

bare time evolution 

exact equation of motion for the relevant part 

alone 

note the integro-differential form 

memory kernel

• stationary state: 

• non-stationary state but “poor memory”: 

decays rapidly with 

“Nakajima-Zwanzig-Markov” 

(NZM) generator 

Perturbation theory to arbitrary order: equivalent to real-time diagrammatics 

of Schoeller, Schön, König [PRB 50, 18436 (1994); EPL 31, 31 (1995); PRL 

76, 1715 (1996)], see C.T., PRB 77, 195416 (2008)

2. Time-convolutionless (TCL) master equation 

Tokuyama and Mori, Prog. Theor. Phys. 55, 411 (1976); Hashitsume et al., J. 

Stat. Phys. 17, 155 (1977); see also Breuer and Petruccione, The Theory of 

Open Quantum Systems (Oxford) 

beyond Nakajima-Zwanzig: 

make equation local in time by propagating backward (exactly) 

requires additional 

perturbative expansion

TCL master equation 

with 

is of the form

Overview 





• Summary

Perturbative expansion of Pauli master equations 

Nakajima-Zwanzig-Markov (NZM) and time-convolutionless (TCL) equations 

can be derived for diagonally projected density operator 

→ Pauli master equations, C.T., PRB 83, 115416 (2011) 

• TCL generator 

can be expressed in terms of T-matrix generator 

• TCL generator regular for to all orders, divergences cancel 

[dito for NZM generator, for 4 th order shown by S. Koller et al., Phys. 

Rev. B 82, 235307 (2010)] 

• explicit construction of all expansion terms of TCL generator 

(also of NZM generator), first two terms: 

TCL correction 

correction is relevant close to Coulomb-blockade threshold

Perturbative expansion of Pauli master equations 

Nakajima-Zwanzig-Markov (NZM) and time-convolutionless (TCL) equations 

can be derived for diagonally projected density operator 

→ Pauli master equations, C.T., PRB 83, 115416 (2011) 

• TCL and NZM master equations agree for stationary states (as they should) 

• … but only if full perturbation series is summed up, not at finite order > 2 

To do: 

• physical relevance for realistic model 

• TCL master equation without diagonal projection

Overview 





• Summary

Spectrum of the transition-rate matrix 

master equation can be written in superoperator form as 

eigenvalues of A describe dynamics of open system and are interesting: 

A. Donabidowicz-Kolkowska, and C.T., New J. Phys. 14, 103050 (2012) 

Ansatz 

eigenvalue equation for eigenvalues ® 

• solution decays with rate 

• … and oscillates with frequency 

• one eigenvalue is 

→ eigen“vector” ³ 0 is stationary state


Model: molecule with electrons coupled to vibrational mode, coupled to leads 

for large electron-vibron coupling find Franck-Condon blockade since lowenergy 

vibrational eigenstates for different charge overlap weakly 

Koch, von Oppen, PRL 94, 206804 (2005): 

dI/dV very small 

at low bias


does the eigenvalue spectrum of A show characteristic differences between 

• transmitting dot 

• Coulomb blockade 

• Franck-Condon blockade 

master equation, perturbation theory to leading order: 

• expand to second order in 

or 

• send 

• secular approximation: for degenerate eigenvalues of , choice of basis 

in degenerate subspace is arbitrary → diagonal (Pauli) projection unjustified 

→ keep off-diagonal components of 

connecting degenerate states 

non-secular components lead to fast processes;

Transmitting dot (on resonance), weak electron-vibron coupling ¸ = 1 

stationary state 

spectrum 

current 

vibrational quantum number 

charge 

• gap in spectrum – no very slow modes 

• slowest decaying modes at small V : threefold generate, spin polarizations

Coulomb blockade, weak electron-vibron coupling ¸ = 1 


spectrum 

current 


charge 

• gap closes – very slow modes 

• slowest decaying modes at small V : spin polarizations 

very slow since one electron must tunnel out, another one in

Deep Coulomb blockade, weak electron-vibron coupling ¸ = 1 


spectrum 

current 


charge 

• deep in the Coulomb blockade more modes become slow: 

excited-state-to-excited-state transitions become blocked 

• slow modes are mostly vibrational excitations

Franck-Condon blockade (on resonance), strong electron-vibron coupling ¸ = 4 


spectrum 

current 


charge 

• many slow modes (exponentially small Franck-Condon matrix elements) 

• slowest decaying modes are vibrational excitations

Overview 





• Summary

Realistic modeling for endohedral N@C 60 

nitrogen retains spin S N = 3/2 (Hund‘s 1 st rule), essentially isotropic 

break-junction experiment motivated by F. Elste and C.T., PRB 71, 155403 

(2005) 

J. E. Grose, E. Tam, C.T., M. 

Scheloske, B. Ulgut, J. J. Parks, H. D. 

Abruña, W. Harneit, and D. C. Ralph, 

Nature Materials 7, 884 (2008) 

• production by ion implantation and 

enrichment: Harneit group, FU Berlin 

• device fabrication, measurements: 

Ralph group, Cornell 

• theory: C.T.

Bias Voltage [V] 

Bias Voltage [V] 

Differential conductance: experiment 

B = 0T 

Vg = -890 mV 

0.00500 

0.00500 

0.00375 

0.00375 

0.00250 

0.00250 

0.00125 

CB 

0.00125 

0.00000 

-0.00125 

CB 

0.00000 

-0.00125 

CB 

kinks 

-0.00250 

-0.00375 

-0.00250 

-0.00375 

-905 -900 -895 -890 -885 -880 

Gate Voltage [mV] 

0 1250 2500 3750 5000 6250 7500 

Magnetic Field [mT] 

0 1 2 3 4 5 

Conductance [uS] 

-2.0 0.0 2.0 4.0 

Conductance [uS]

Modeling 

Coulomb repulsion on C 60 

local potential (asym. coupling) 

exchange between electron and N spin 

N = 1, s e = 1/2, S = 2 

E 

E 

N = 1, s e = 1/2, S = 1 

2|J| 

2|J| 

N = 2, s e = 0, S = 3/2 

-2 -1 0 

1 

2 

m 

-2 -1 0 

1 

2 

m

a 

a b c 

N = 1 

E 

D 

C 

B 

A 

Calculations: master equation 

• sequential tunneling (second order in ) 

b c 

• Pauli master equation (stationary state is 

N = 2 

diagonal, no additional assumption) 

• model parameters from experiment 

Fine structure: Very soft oscillations? 

2|J| 

E 

-2 -1 0 

C 

D 

C 

B 

1 

A 

E 

2 

E 

D 

B 

m 

C 

A 

b 

c 

c 

b 

a 

a 

-2 -1 0 

1 

2 

m

experiment 

theory 

0.00500 

0.00375 

0.00250 

0.00500 

0.00375 

0.00250 

0.00125 

0.00000 

-0.00125 

-0.00250 

-0.00375 

additional 

physics 

-905 -900 -895 -890 -885 -880 

0.00125 

0.00000 

-0.00125 

-0.00250 

-0.00375 

0 1250 2500 3750 5000 6250 7500

Sequential and cotunneling through N@C 60 

N. Roch, R. Vincent, F. Elste, W. Harneit, W. Wernsdorfer, C.T., and F. Balestro, 

Phys. Rev. B 83, 081407(R) (2011) 

exp 

th 

exp 

th 

• reproduce experiment of Grose et al.—not trivial in this field 

• observe inelastic cotunneling due to spin flip 

• calculate cotunneling in picture of Grose et al. (N = 1 to 2, J < 0)

Thanks to 

Agnieszka Donabidowicz-Kolkowska 

TU Dresden 

Tim Ludwig 

TU Dresden 

Thomas Brumme TU Dresden 

Stefan Lange 

TU Dresden → PIK 

Massimiliano Di Ventra UC San Diego 

Binhe Wu 

Donghua U, Shanghai 

Guichao Hu 

Shandong Normal U 

Jens Kortus 

TUBA Freiberg 

Torsten Hahn 

TUBA Freiberg 

Florian Elste 

Columbia U → Deutsche Bank 

Felix von Oppen FU Berlin 

Daniel C. Ralph 

Cornell U 

Wolfgang Wernsdorfer CNRS Grenoble 

Christian Hess 

IFW Dresden 

FOR 1154 

Towards Molecular 

Spintronics

Summary 

source 

drain 


Nakajima-Zwanzig-Markov vs. time-convolutionless 

gate 


TCL Pauli master equation converges order by order 

agrees with Nakajima-Zwanzig-Markov only to 2 nd order 


…for Franck-Condon system: characteristic of dynamics 


break-junction experiments (Ralph, Wernsdorfer) 

can be described quantitatively

Spin relaxation and amplification 

molecules with magnetic 

anisotropy 

Mn 12 

cluster 

Co-phthalocyanine 

easy-axis 

anisotropy 

anisotropy → energy 

barrier for spin reversal 

→ slow spin relaxation 

C.T. and F. Elste, PRB 

73, 235304 (2006)

apid intermediate Toward memory devices slow 

gate 

ac bias voltage 

C.T. and M. Di Ventra, 

in progress

dI/dV 

Anisotropy and magnetic field 

dI/dV for arbitrary orientation of magnetic 

field vs. easy axis—usually unknown 

terms do not commute 

→ no conserved quantities 

except electron number 

→ degeneracies lifted and 

many allowed transitions 

C.T., PRB 76, 014421 (2007)

Anisotropy and magnetic field 

dI/dV for arbitrary orientation of magnetic 

field vs. easy axis—usually unknown 

dI/dV 

terms do not commute 

→ no conserved quantities 

except electron number 

→ degeneracies lifted and 

many allowed transitions 

C.T., PRB 76, 014421 (2007)

STM modeling for large U 

with T. Ludwig (TUD), J. Kortus and T. Hahn 

(TUBA Freiberg) 

strong interactions, 

sharp molecular levels 

• insulating layer, e.g., NaCl 

• ligands as tunneling barriers 

• probably for HOPG substrate (→ IFW) 

usual (Green-function) approaches fail 

Goal: Master equation with dependence 

on tip position 

• model for molecule and substrate based 

on DFT (energies and orbitals) 

• Bardeen approach++ → tunnel matrix 

elements tip–molecule, tip–substrate 

metal, semiconductor… 

related work: 

Sobczyk et al., PRB 85, 205408 (2012); 

Donarini et al., arXiv:1206.2664

Example: Co- and Mn-phthalocyanine 

Co II 

Mn II

CoPC 

MnPC 

• tip height 

(over Mn/Co) 

h = const 

• scale: nm 

h = 1 Å 

h = 1 Å 

• here no substrate 

structure 

• CoPC, MnPC: 

different HOMOs 

• MnPC Jahn- 

Teller distorted 

h = 10 Å 

h = 10 Å

MnPc with model substrate 

constant current, I = const 

proof of principle: 

many-particle-approach 

with Coulomb blockade 

and spatial resolution 

experiments in 

progress 

Hess, Büchner 

(IFW Dresden)

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