Fundamental aspects of transport in
nanostructures and atomic physics
Massimiliano Di Ventra
Department of Physics, University of California, San Diego
Collaborators
Neil Bushong (NC)
Yuriy Pershin (USC)
Chih-Chun (LANL)
Mike Zwolak (OSU)
Roberto D’Agosta (Spain)
Giovanni Vignale (U. Missouri)
Outline
• Introduction to the transport problem
• Many-body effects related to viscosity of the electron liquid
(large for structures with smaller transmissions)
• Properties of steady states and predictions
Theory: Microcanonical picture of transport
Experiments: Atomic gases in optical lattices
Outline
• Introduction to the transport problem
• Many-body effects related to viscosity of the electron liquid
(large for structures with smaller transmissions)
• Properties of steady states and predictions
Theory: Microcanonical picture of transport
Experiments: Atomic gases in optical lattices
Field: nanoscale electronics
Nanotubes/wires
Atomic point contacts
Molecular junctions
Tans et al. (1997)
Organic electronics
Scheer et al. (1998)
from Nitzan et al. (2003)
Fast DNA sequencing
Poly(3-hexylthiophene)
source
polymer
drain
insulator
Doped Si
Lagerqvist et al. (2006)
Z.Q. Li et al. (2006)
What do we want to describe ?
R
R 1  G 
dI
dV
Solved ? Not quite, especially at the atomic level !
Major difference with macro/mesoscopic systems
 ph
e
Pw
I th
e
I th
forces
Large current densities
increased e-e, e-ph scattering
Why is the problem difficult (and interesting) ?
• The system is out of equilibrium
(non-equilibrium statistical mechanics is still an open subject;
do we need to go beyond Hamiltonian dynamics?)
• Interactions among electrons
(Coulomb blockade, correlations, non-Fermi liquid behavior)
• Interactions among electrons and ions
(e.g., el-phonon scattering, current-induced forces)
• Interaction with the environment
(dissipation and dephasing)
• Physical properties are quite sensitive to atomic details
From experiment to model system
Approximation 1: open quantum systems
Closed system
H  H S  H Battery  Hint
S  T rBattery
Open system: dynamical interaction with reservoirs
From experiment to model system
Approximation 1: open quantum systems
H  H S  H Battery  Hint
S  T rBattery
Battery dense spectrum
No initial correlations
Small interaction
t S  S
In general, no closed equation of motion for s
From experiment to model system
Approximation 2: ideal steady state
Assume existence of at least one steady state solution


 




ss
I  Tr  S (t ) I  Tr  I  I  constant
t
Still many-body open quantum system !
From experiment to model system
Approximation 3: “openness” vs boundary conditions
Loss of information
closedsystem(with battery) opensystem closed and infinitedifferentsystem
From experiment to model system
Approximation 4: mean-field approximation
With or w/o interaction
Non-interacting electrons
Non-interacting electrons
If leads are interacting NO closed equation of motion for the current !
From experiment to model system
Approximation 5: independent channels and energy filling
Non-interacting electrons
Non-interacting electrons
With or w/o interaction
The Landauer current
Non-interacting electrons
From scattering theory
I

e
e






dE
f
(
E
)

f
(
E
)
T
(
E
)

dE
f
(
E
)

f
(
E
)
Tr

G

G
L
R
L
R
R
L
 
 
This formula has nothing to do with NEGF !!!

Interacting sample
Non-interacting electrons
Non-interacting electrons
With interactions
From NEGF
I


ie dE







Tr

(
E
)


(
E
)
G

f
(
E
)

(
E
)

f

(
E
)
G
(
E
)

G
(E)
L
R
L
L
R
R
  2
Meir and Wingreen, 1992

Physical origin of many-body corrections:
linear-response theory
ji (r)     ij (r,r') E j (r') dr'
j
xˆ 2
V
xˆ 3
Current conservation


C2

i ij (r,r')  0
C3
Gauge invariance
C1
C4
xˆ 1

xˆ 4

'j ij (r,r')  0

Gmn = -  dr  dr' xˆ mi  ij (r,r') xˆ nj

Cm
Cn
mn
Physical origin of many-body corrections:
linear-response theory
 ij (r, r')  e lim
2
 0
˜ ij (r, r'; )
Im 

“Proper” currentcurrent response
function
˜ ij (r,r'; )  j (r)



j j (r')
i
+
+

For a non-interacting system
 ij (r,r')  e 2 

f ( )
     Wi* (r)Wj (r')


e
e








I
dE
f
(
E
)

f
(
E
)
T
(
E
)

dE
f
(
E
)

f
(
E
)
Tr

G

G
L
R
L
R
R
L




+…
Outline
• Introduction to the transport problem
• Many-body effects related to viscosity of the electron liquid
(large for structures with smaller transmissions)
• Properties of steady states and predictions
Theory: Microcanonical picture of transport
Experiments: Atomic gases in optical lattices
Interactions in the whole system:
the microcanonical picture of transport
 j  
n
t
I exact   jexactds    jexactdv     jKS dv   jKS ds  I KS
S
V
V
M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004)
S
Fast relaxation of momentum
w
momentum relaxation time
1/nc  tc  ħ/E  m w2/2ħ2  1 fs
Bushong, Sai and M. Di Ventra (Nano Letters 2005)
Comparison with Landauer formula
Chen, Zwolak and Di Ventra , in preparation
Entanglement entropy
R
L
Gaussian
 L  TrR Z
S L  Tr[  L log( L )]
Binomial
Exact
C(t) = correlation matrix
PL = projection operator
Chen, Zwolak and Di Ventra , in preparation
Approximate
Klich and Levitov, PRL 2009
Electron flow
Quasi-2D electron liquid,
TDDFT
V= 0.2V
2
 
 KS
1 
e

  i  Axc (r , t )   Vext (r , t )k (r , t )  0
i 
c

 t 2m 

Sai, Bushong, Hatcher, and Di Ventra , PRB 2007
Hydrodynamics of the electron liquid
d (t )
i
 H (t )
dt
Exact!
n = density
v = j/n
Dt n  n  v  0
continuity
mnDt v j  i Pij  n jVext  0
F=ma
Information on all e-e interactions (generally unknown)
A hydrodynamic formulation is more natural in QM than in classical physics
Martin and Schwinger, Phys. Rev. (1959)
Anticipates TDDFT by many years !
Fast relaxation of momentum
w
momentum relaxation time
1/nc  tc  ħ/E  m w2/2ħ2  1 fs
Bushong, Sai and M. Di Ventra (Nano Letters 2005)
Quantum Navier-Stokes equations
Dt n  n  v  0
mnDt v j  i Pij  n jVext  0
Pij  F[Pijk  F[Pijkl  ]]
1)
1
I [ f ] pi p j  Dt Pij  Pij  v  Pik  k v j  Pkj k vi   k Pijk(3)

m
2)
v /( L max(,n c ))  1
  n
3)
R. D’Agosta and M. Di Ventra, JPCM (2006)
Conductance quantization from hydrodynamics
Dt n  n  v  0
mnDt v j   j P ~ij  n jVext  0
1D, stationary, non-viscous fluid
2
P n 

n
2m 2
v
P
  Vext  const
2 n
Bernoulli
2
2
e
I V
h
D’Agosta and M. Di Ventra, JPCM (2006)
Quantized conductance is
the one of a 1D ideal,
non-viscous charged fluid
Electron Dynamics
Electron dynamics in nanostructures similar to a viscous liquid
Predictions
turbulence
electron heating
effect on ion heating
Turbulence in nanoscale systems
QPCs
molecules
Actual atomic
structure

Approx.
potential
adiabatic
D’Agosta and M. Di Ventra, JPCM (2006)
non-adiabatic
Turbulence in nanoscale systems
mI
Re 
e
Adiabatic constrictions
(e.g., QPC),
laminar flow
laminar
turbulent
e  10 air
2
D’Agosta and M. Di Ventra, JPCM (2006)
Nonadiabatic constrictions
(e.g., molecules),
turbulent flow
Time-Dependent Current DFT
2
 
 KS
1 
e

  i  Axc (r , t )   Vext (r , t )k (r , t )  0
i 
c

 t 2m 

 xc,ij (r, t )
e Axc,i (r, t )
1
ALDA
 eExc,i (r, t ) 
 iVxc
(r , t ) 
 r
c
t
n(r, t ) j
j
no memory, Markov approx.
2


 xc,ij (r , t )   (n, r , t )  j vi  i v j   ij k vk    (n, r , t )  v ij
3


bulk viscosity
Vignale and Kohn PRL 1996; Vignale, Ullrich and Conti (1997)
Electron turbulence
TDCDFT
Closed system, quasi-2D electron liquid, TDCDFT
Eddies size
 Rcr 

 Re 
0 ~ l 
3/ 4
nanoscale, for fully developed
turbulence
Bushong, Gamble, and M. Di Ventra (Nano Lett. 2007); D’Agosta and Di Ventra JPCM (2006)
NS
0.02 V (Laminar)
Bushong, Gamble and M. Di Ventra (Nano Lett. 2007)
3 V (Turbulent)
Bushong, Gamble and M. Di Ventra (Nano Lett. 2007)
Possible exp. verification
Laminar
Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)
Possible exp. verification
Turbulent
Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)
Possible exp. verification
Bushong, Pershin and M. Di Ventra (Phys. Rev. Lett. 2007)
Electron Dynamics
Electron dynamics in nanostructures similar to a viscous liquid
Predictions
turbulence
electron heating
effect on ion heating
Electron heating: elementary considerations
Power in the junction:
Heat dissipated in
the electrodes:
2
Vbias
Pe  
,   1
R
I th  nTe2
I th  Pe  Te   eVbias
D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)
Pw
I th
I th
Electron heating from hydrodynamics
Heat equation (no turbulence)
~ij jivi   [kTe ]  0
Pw
I th
I th
Thermal conductivity
Te   eVbias
e.g. Au QPC
e  
viscosity
D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)
 e  65K / V
Electron Dynamics
Electron dynamics in nanostructures similar to a viscous liquid
Predictions
turbulence
electron heating
effect on ion heating
Ionic Heating: elementary considerations
Power in the junction:
Heat dissipated in
the electrodes:
2
Vbias
Pw  
,   1
R
I th  nTw4
I th  Pw  Tw   Vbias
Y-C Chen, M. Zwolak, M. Di Ventra
Nano Letters (2003)
T.N. Todorov Phil. Mag, (1998)
Pw
I th
I th
Effect of heating on shot noise and current
?
Tw   Vbias
G
 Vbias
0
G
F
S
 Vbias
2eI
I th  
I th  
Chen, Di Ventra, PRB (2003); Phys. Rev. Lett. (2005)
Agrait et al., Phys. Rev. Lett. (2002)
Effect of el. and ion heating on inelastic scattering
Exp.
Djukic et al PRB (2005)
H2
e--e-
e--ph
Th.
D’Agosta and M. Di Ventra, J. Phys. Cond. Matt. (2008)
Effect on the ionic temperature: theory
Pw
I th
I th
Tw  [ 4e pV 2   4eV 4 ]1/ 4
e--ph
D’Agosta, Sai and M. Di Ventra, Nano Lett. (2006)
e--e-
Effect on the ionic temperature: exp.
c
Teff  [ 4e pV 2   4eV 4 ]1/ 4
325
a
Teff /K
315
F *  f  ln(toff / f  )  f  ln rF (1)
toff
E
 t D exp( b )
K BTeff
Teff 
Eb
KB
1
f
F*
ln  
rF t D f 
t D  1.5  10
10
(2)
305
C6
C8
C10
295
0.0
s, Eb  0.52eV , f   0.26nN
0.4
0.8
Vbias /V
Huang et al. Nano Lett. (2006); Huang et al. (Nature Natotech., 2007)
1.2
1.6
Outline
• Introduction to the transport problem
• Many-body effects related to viscosity of the electron liquid
(large for structures with smaller transmissions)
• Properties of steady states and predictions
Theory: Microcanonical picture of transport
Experiments: Atomic gases in optical lattices
Interactions in the whole system:
the microcanonical picture of transport
 j  
n
t
I exact   jexactds    jexactdv     jKS dv   jKS ds  I KS
S
V
V
M. Di Ventra, T.N. Todorov, (J. Phys. Cond. Matt. 2004)
S
Formation of steady states
w
~ G0
The formation of a steady
state has nothing to do
with the infinite nature of
momentum relaxation time
1/nc  tc  ħ/E  m w2/2ħ2  1 fs the electrodes
Bushong, Sai and M. Di Ventra (Nano Letters 2005)
Cold atoms are ideal systems for
studying transport phenomena
• You can choose:
1.fermions or bosons
2.harmonic trap, optical lattice, or both (Box-potential is
coming soon!)
3.single or multi components or species
4.dimensions (3D, 2D, 1D, or mixed)
• You can tune:
1.interactions among atoms (via Feshbach resonance)
2.trap depth or lattice constant
3.temperature
4.density / filling factor
Experimental tests: cold atoms
Observation of Fermi surfaces
I.Bloch, Nat. Phys. 1, 23 (2005)
Superfluid-Mott insulator transition
Hambury-Brown-Twiss
experiment
Quantum ratchet
Science 326, 1241 (2009)
Density-induced transport
1. Loading non-interacting
single-species atoms into the
ground state.
2. Remove particles on the right
half using photons.
T=0, N particles:
fermions:
bosons:
Bosonic vs fermionic currents
bosons (quasi-condensate)
fermions (Fermi sea)
More on the current
• Fermionic QSSC: Robust against trap and T
• Bosonic current: Never reaches a finite QSSC
even in the thermodynamic limit at T=0
IN
1/ 2
Interaction-induced transport
1. Loading non-interacting twocomponent fermions into the ground
state.
2. Turning on interactions on half of
the lattice.
Optically controlled collisions:
Optical Feshbach resonance (coupling to auxiliary channels by
photons. Realized in Yb and Sr, proposed for Li.): PRL 105, 050405;
PRA 79, 021601; PRL 107,073202; PRL 108, 010401.
Equations of motion
Using Wick’s decompositions at 2-particle level or
3-particle level show similar results.
QSSC and conducting-nonconducting transition
Similar to the negative differential resistance
in semiconductor devices.
(Phys. Today 23, 35 (1970), IBM J. Res. Develop. 14,
61 (1970).)
Mismatch of energy spectra
1. The switch-on of U changes the
energy dispersion of the left-half
lattice.
2. Moving a high-energy particle to
a low-energy state or vice versa is
forbidden by the underlying
quantum dynamics.
3. The blockade is dynamical.
(unlike Mott insulator in equilibrium
which is due to energy minimization.)
Conclusions
• Microcanonical picture of transport
• Many-body effects related to viscosity of the electron liquid
(large for structures with smaller transmissions)
• Predictions (some verifiable in atomic lattices):
• Turbulence of the electron liquid
• Quasi-steady state formation also in finite systems
• Fermi distributions not necessary for the existence of a QSS
• Conducting-nonconducting transition due to interactions
Future direction:
transport in a liquid environment
Idea: Transverse Transport
Voltage Biased
translocaton
DNA
-A-C- T-G -
ACTG
FIB Milled Channels
(10-50-nm width)
AFM Formed Channel
(1-5-nm width and length)
(b)
(a)
(c)
750 nm
Microchannels
Microchannels
STM Formed Nanoelectrodes
2-nm width
Idea: Transverse Transport
e-

E

E||
M. Zwolak and M. Di Ventra,
Nano Lett. 5, 421 (2005)
MD + Quantum Transport
Hamiltonian
Transverse Transport (dynamics)
J. Lagerqvist, M. Zwolak, and M. Di Ventra, Nano Letters 2006
Current Distributions
Accuracy 99.9 %
106 measurements/ s
Genome seq. time < 3 days
No parallelization
3 10 bases (70counts) s  3days
9
106 m easurem ents
N
Error  1  P  

I n  X  A,T ,C ,G
J. Lagerqvist, M. Zwolak, and M. Di Ventra, Nano Letters 2006
X
4
P
n
X
n 1
N
N
N
N
P P P P
n
A
n 1
n
T
n 1
n
C
n 1
n
G
n 1
Idea: Transverse Transport
Voltage Biased
translocaton
DNA
-A-C- T-G -
ACTG
FIB Milled Channels
(10-50-nm width)
http://www.mcb.harvard.edu/branton/
Microchannels
Microchannels
STM Formed Nanoelectrodes
2-nm width
AFM Formed Channel
(1-5-nm width and length)
(b)
(a)
(c)
750 nm
M. Ramsey, UNC
Branton et al., Harvard
S. Lindsay
T. Kawai
S.Y. Chou
U. Penn, IBM, Samsung…..